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Model Definition and Autoregressive Models of Order 2
Introduction to Autoregressive Models
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
1. 𝑎 > 0
2. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
3. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
4. 𝑎 > 0
5. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
6. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
7. 𝑎 > 0
8. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
9. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
10. 𝑎 > 0
11. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
12. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
13. 𝑎 > 0
14. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
15. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
16. 𝑎 > 0
17. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
18. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
19. 𝑎 > 0
20. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
21. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
22. 𝑎 > 0
23. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
24. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
25. 𝑎 > 0
26. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
27. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
28. 𝑎 > 0
29. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
30. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
31. 𝑎 > 0
32. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
33. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
34. 𝑎 > 0
35. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
36. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
37. 𝑎 > 0
38. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
39. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
40. 𝑎 > 0
41. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
42. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
43. 𝑎 > 0
44. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
45. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
46. 𝑎 > 0
47. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
48. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
49. 𝑎 > 0
50. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
51. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
52. 𝑎 > 0
53. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
54. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
55. 𝑎 > 0
56. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
57. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
58. 𝑎 > 0
59. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
60. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
61. 𝑎 > 0
62. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
63. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
64. 𝑎 > 0
65. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
66. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
67. 𝑎 > 0
68. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
69. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
70. 𝑎 > 0
71. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
72. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
73. 𝑎 > 0
74. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
75. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
76. 𝑎 > 0
77. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
78. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
79. 𝑎 > 0
80. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
81. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
82. 𝑎 > 0
83. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
84. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
85. 𝑎 > 0
86. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
87. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
88. 𝑎 > 0
89. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
90. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
91. 𝑎 > 0
92. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
93. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
94. 𝑎 > 0
95. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
96. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
97. 𝑎 > 0
98. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
99. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
100. 𝑎 > 0
101. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
102. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
103. 𝑎 > 0
104. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
105. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
106. 𝑎 > 0
107. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
108. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
109. 𝑎 > 0
110. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
111. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
112. 𝑎 > 0
113. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
114. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
115. 𝑎 > 0
116. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
117. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
118. 𝑎 > 0
119. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
120. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
121. 𝑎 > 0
122. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
123. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
124. 𝑎 > 0
125. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
126. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
127. 𝑎 > 0
128. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
129. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
130. 𝑎 > 0
131. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
132. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
133. 𝑎 > 0
134. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
135. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
136. 𝑎 > 0
137. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
138. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
139. 𝑎 > 0
140. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
141. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
142. 𝑎 > 0
143. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
144. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
145. 𝑎 > 0
146. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
147. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
148. 𝑎 > 0
149. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
150. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
151. 𝑎 > 0
152. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
153. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
154. 𝑎 > 0
155. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
156. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
157. 𝑎 > 0
158. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
159. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
160. 𝑎 > 0
161. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
162. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
163. 𝑎 > 0
164. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
165. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
166. 𝑎 > 0
167. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
168. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
169. 𝑎 > 0
170. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
171. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
172. 𝑎 > 0
173. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
174. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
175. 𝑎 > 0
176. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
177. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
178. 𝑎 > 0
179. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
180. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
181. 𝑎 > 0
182. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
183. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
184. 𝑎 > 0
185. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
186. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
187. 𝑎 > 0
188. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
189. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
190. 𝑎 > 0
191. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
192. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
193. 𝑎 > 0
194. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
195. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
196. 𝑎 > 0
197. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
198. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
199. 𝑎 > 0
200. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
201. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
202. 𝑎 > 0
203. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
204. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
205. 𝑎 > 0
206. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
207. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
208. 𝑎 > 0
209. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
210. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
211. 𝑎 > 0
212. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
213. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
214. 𝑎 > 0
215. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
216. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
217. 𝑎 > 0
218. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
219. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
220. 𝑎 > 0
221. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
222. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
223. 𝑎 > 0
224. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
225. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
226. 𝑎 > 0
227. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
228. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
229. 𝑎 > 0
230. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
231. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
232. 𝑎 > 0
233. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
234. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
235. 𝑎 > 0
236. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
237. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
238. 𝑎 > 0
239. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
240. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
241. 𝑎 > 0
242. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
243. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
244. 𝑎 > 0
245. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
246. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
247. 𝑎 > 0
248. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
249. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
250. 𝑎 > 0
251. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
252. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
253. 𝑎 > 0
254. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
255. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
256. 𝑎 > 0
257. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
258. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
259. 𝑎 > 0
260. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
261. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
262. 𝑎 > 0
263. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
264. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
265. 𝑎 > 0
266. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
267. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
268. 𝑎 > 0
269. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
270. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
271. 𝑎 > 0
272. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
273. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
274. 𝑎 > 0
275. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
276. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
277. 𝑎 > 0
278. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
279. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
280. 𝑎 > 0
281. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
282. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
283. 𝑎 > 0
284. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
285. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
286. 𝑎 > 0
287. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
288. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
289. 𝑎 > 0
290. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
291. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
292. 𝑎 > 0
293. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
294. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
295. 𝑎 > 0
296. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
297. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
298. 𝑎 > 0
299. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
300. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
301. 𝑎 > 0
302. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
303. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
304. 𝑎 > 0
305. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
306. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
307. 𝑎 > 0
308. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
309. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
310. 𝑎 > 0
311. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
312. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
313. 𝑎 > 0
314. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
315. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
316. 𝑎 > 0
317. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
318. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
319. 𝑎 > 0
320. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
321. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
322. 𝑎 > 0
323. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
324. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
325. 𝑎 > 0
326. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
327. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
328. 𝑎 > 0
329. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
330. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
331. 𝑎 > 0
332. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
333. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
334. 𝑎 > 0
335. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
336. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
337. 𝑎 > 0
338. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
339. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
340. 𝑎 > 0
341. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
342. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
343. 𝑎 > 0
344. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
345. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
346. 𝑎 > 0
347. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
348. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
349. 𝑎 > 0
350. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
351. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
352. 𝑎 > 0
353. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
354. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
355. 𝑎 > 0
356. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
357. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
358. 𝑎 > 0
359. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
360. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
361. 𝑎 > 0
362. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
363. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
• Focus on autoregressive models of order 2
• Notation change: (𝑎, 𝑏) instead of (𝜃1, 𝜃2)
Model Definition
The model is defined as:
𝑋1= 𝜖1
𝑋2= 𝑎𝑋1+ 𝜖2
𝑋𝑘= 𝑎𝑋𝑘−1 + 𝑏𝑋𝑘−2 + 𝜖𝑘,  3 ≤ 𝑘 ≤ 𝑛
Regions 𝑅𝑘
𝑅1 and 𝑅2
• 𝑅1= ∅
• 𝑅2= {(𝑎, 𝑏): 𝑎 > 0}
𝑅3
𝑅3 is defined as:
𝑅3= {(𝑎, 𝑏): 𝑎 > 0 and 𝑏 > −1 − 𝑎2; or 𝑎 < 0 and 𝑏 < −1 − 𝑎2}
Union of 𝑅2 and 𝑅3
𝑅2∪ 𝑅3= ((0, ∞)× ℝ) ∪ {(𝑎, 𝑏): 𝑎 < 0, 𝑏 < −1 − 𝑎2}
𝑅4 and Beyond
• Expressions become more complex for 𝑅4 and higher
• Closed-form expressions become cumbersome or nonexistent
Theoretical Understanding of ⋃𝑅𝑘𝑘≥1
Theorem 11.3.2: The closure of ⋃𝑅𝑘𝑘≥1 contains all points (𝑎, 𝑏) satisfying one of:
364. 𝑎 > 0
365. 𝑎 ≤ 0 and 𝑏 < −𝑎2− 1
366. 𝑎 ≤ 0 and 𝑏 < min(−𝑎2/4, −𝑎 − 1)
Tail Behavior
• In the region described by Theorem 11.3.2: 𝑃{𝑛𝛾𝑛(1)> 𝑡} ∼ 𝑡−𝛼/2 for large 𝑛
• In the non-shaded domain: Believed to be 𝑡−𝛼log𝑡 (proven only for 𝑏 > 𝑎2/4)
Proof of Theorem 11.3.2
Lemma 11.3.3
Key points:
• If 𝑎 > 0, the largest diagonal coefficient of 𝐶 is positive
• If 𝑎 ≤ 0, the largest diagonal coefficient of 𝐶 vanishes
• ⋃𝑅𝑘𝑘≥1 contains 𝑎 > 0 but not 𝑎 ≤ 0 and 𝑏 > −𝑎2/4
Lemma 11.3.4
If 𝑎 ≤ 0 and 𝑏 > −1, the largest diagonal coefficient of 𝐶 vanishes.
Analysis for 𝑎 ≤ 0, 𝑏 ≤ −1, and 𝑏 < −𝑎2/4
• Parameterization: 𝑎 = −2𝑟cos𝜙, 𝑏 = −𝑟2, 𝑟 > 0, 0 ≤ 𝜙 ≤ 𝜋/2
• Lemma 11.3.5: Provides a closed formula for diagonal coefficients of 𝐶
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