Asymptotic Analysis of Quadratic Forms
Introduction
These lecture notes cover asymptotic analysis of quadratic forms for random vectors with
heavy-tailed distributions.
Asymptotic Behavior of Quadratic Forms
Main Theorem
Theorem 8.2.1: Let 𝑋 be a 𝑑-dimensional random vector with independent and identically
distributed components having a Student-like distribution. Let 𝐶 be a 𝑑×𝑑 matrix. Define
𝐽1={𝑗:𝐶𝑗,𝑗 >0}. If 𝐽1 is not empty, then:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}∼𝐾𝑠,𝛼𝛼(𝛼−1)/2 2
𝑡𝛼/2∑𝐶𝑗,𝑗
𝛼/2
𝑗∈𝐽1
as 𝑡→∞
Key points:
• Applies to Student-like distributions (heavy-tailed)
• Asymptotic behavior depends on positive diagonal elements of 𝐶
• Power law decay with exponent 𝛼/2
Proof Strategy
1. Change of variables: Transform the problem to work with standard normal distribution
2. Apply Theorem 5.1 (not provided in the excerpt) 3. Analyze the boundary of the
transformed set
Change of Variables
Define:
• 𝑌: Random vector with centered normal distribution (identity covariance)
• 𝛷(𝑦): Normal cumulative distribution function
• 𝑆𝛼(𝑦): Student-like cumulative distribution function
• 𝑆𝛼
←, 𝛷←: Inverse functions of 𝑆𝛼 and 𝛷 respectively
Key transformation:
𝑋=
𝑑𝑆𝛼
←∘𝛷(𝑌)
Define sets: 𝐴𝑡={𝑥∈ℝ𝑑:⟨𝐶𝑥,𝑥⟩>𝑡}=√𝑡𝐴1
𝐵𝑡={𝑦∈ℝ𝑑:⟨𝐶𝑆𝛼
←∘𝛷(𝑦),𝑆𝛼
←∘𝛷(𝑦)⟩>𝑡}=𝛷←∘𝑆𝛼(𝐴𝑡)
Important result:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}=∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥𝑖=∫ 𝑒−|𝑦|2/2
(2𝜋)𝑑/2
𝐵𝑡𝑑𝑦
Asymptotic Analysis
Asymptotic Equivalence
Key result:
(𝛷←∘𝑆𝛼(𝑥))2=2𝛼log𝑥−loglog𝑥−2log(𝐾𝑠,𝛼𝛼𝛼/2)−2log(2√𝜋)+𝑜(1)
as 𝑥→∞.
Boundary Parameterization
Lemma 8.2.2: Provides parameterizations for ∂𝐴𝑡 and ∂𝐵𝑡 near specific points.
For ∂𝐴𝑡 near 𝑝𝜖,𝑗,𝑡 =𝜖√𝑡/𝐶𝑗,𝑗𝑒𝑗:
𝑝(𝑣)=𝜖√𝑡
𝐶𝑗,𝑗[1− 1
2𝑡⟨𝐶𝑣,𝑣⟩+𝑂(⟨𝐶𝑣,𝑣⟩
𝑡2)]𝑒𝑗+𝑣
For ∂𝐵𝑡 near 𝑞𝜖,𝑗,𝑡 =𝛷←∘𝑆𝛼(𝑝𝜖,𝑗,𝑡):
𝑞(𝑣)=𝜖√𝛼log 𝑡
𝐶𝑗,𝑗−loglog√𝑡
2√𝛼log𝑡−log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋)
√𝛼log𝑡+𝑜( 1
√log𝑡)𝑒𝑗
+ ∑ 𝛷←
𝑑
𝑖=1,𝑖≠𝑗 ∘𝑆𝛼(𝑣𝑖)𝑒𝑖
Note: These parameterizations are crucial for applying Theorem 5.1 and deriving the
asymptotic behavior.
These lecture notes cover asymptotic analysis of quadratic forms for random vectors with
heavy-tailed distributions.
Asymptotic Behavior of Quadratic Forms
Main Theorem
Theorem 8.2.1: Let 𝑋 be a 𝑑-dimensional random vector with independent and identically
distributed components having a Student-like distribution. Let 𝐶 be a 𝑑×𝑑 matrix. Define
𝐽1={𝑗:𝐶𝑗,𝑗 >0}. If 𝐽1 is not empty, then:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}∼𝐾𝑠,𝛼𝛼(𝛼−1)/2 2
𝑡𝛼/2∑𝐶𝑗,𝑗
𝛼/2
𝑗∈𝐽1
as 𝑡→∞
Key points:
• Applies to Student-like distributions (heavy-tailed)
• Asymptotic behavior depends on positive diagonal elements of 𝐶
• Power law decay with exponent 𝛼/2
Proof Strategy
1. Change of variables: Transform the problem to work with standard normal distribution
2. Apply Theorem 5.1 (not provided in the excerpt) 3. Analyze the boundary of the
transformed set
Change of Variables
Define:
• 𝑌: Random vector with centered normal distribution (identity covariance)
• 𝛷(𝑦): Normal cumulative distribution function
• 𝑆𝛼(𝑦): Student-like cumulative distribution function
• 𝑆𝛼
←, 𝛷←: Inverse functions of 𝑆𝛼 and 𝛷 respectively
Key transformation:
𝑋=
𝑑𝑆𝛼
←∘𝛷(𝑌)
Define sets: 𝐴𝑡={𝑥∈ℝ𝑑:⟨𝐶𝑥,𝑥⟩>𝑡}=√𝑡𝐴1
𝐵𝑡={𝑦∈ℝ𝑑:⟨𝐶𝑆𝛼
←∘𝛷(𝑦),𝑆𝛼
←∘𝛷(𝑦)⟩>𝑡}=𝛷←∘𝑆𝛼(𝐴𝑡)
Important result:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}=∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥𝑖=∫ 𝑒−|𝑦|2/2
(2𝜋)𝑑/2
𝐵𝑡𝑑𝑦
Asymptotic Analysis
Asymptotic Equivalence
Key result:
(𝛷←∘𝑆𝛼(𝑥))2=2𝛼log𝑥−loglog𝑥−2log(𝐾𝑠,𝛼𝛼𝛼/2)−2log(2√𝜋)+𝑜(1)
as 𝑥→∞.
Boundary Parameterization
Lemma 8.2.2: Provides parameterizations for ∂𝐴𝑡 and ∂𝐵𝑡 near specific points.
For ∂𝐴𝑡 near 𝑝𝜖,𝑗,𝑡 =𝜖√𝑡/𝐶𝑗,𝑗𝑒𝑗:
𝑝(𝑣)=𝜖√𝑡
𝐶𝑗,𝑗[1− 1
2𝑡⟨𝐶𝑣,𝑣⟩+𝑂(⟨𝐶𝑣,𝑣⟩
𝑡2)]𝑒𝑗+𝑣
For ∂𝐵𝑡 near 𝑞𝜖,𝑗,𝑡 =𝛷←∘𝑆𝛼(𝑝𝜖,𝑗,𝑡):
𝑞(𝑣)=𝜖√𝛼log 𝑡
𝐶𝑗,𝑗−loglog√𝑡
2√𝛼log𝑡−log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋)
√𝛼log𝑡+𝑜( 1
√log𝑡)𝑒𝑗
+ ∑ 𝛷←
𝑑
𝑖=1,𝑖≠𝑗 ∘𝑆𝛼(𝑣𝑖)𝑒𝑖
Note: These parameterizations are crucial for applying Theorem 5.1 and deriving the
asymptotic behavior.
These lecture notes cover asymptotic analysis of quadratic forms for random vectors with
heavy-tailed distributions.
Asymptotic Behavior of Quadratic Forms
Main Theorem
Theorem 8.2.1: Let 𝑋 be a 𝑑-dimensional random vector with independent and identically
distributed components having a Student-like distribution. Let 𝐶 be a 𝑑×𝑑 matrix. Define
𝐽1={𝑗:𝐶𝑗,𝑗 >0}. If 𝐽1 is not empty, then:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}∼𝐾𝑠,𝛼𝛼(𝛼−1)/2 2
𝑡𝛼/2∑𝐶𝑗,𝑗
𝛼/2
𝑗∈𝐽1
as 𝑡→∞
Key points:
• Applies to Student-like distributions (heavy-tailed)
• Asymptotic behavior depends on positive diagonal elements of 𝐶
• Power law decay with exponent 𝛼/2
Proof Strategy
1. Change of variables: Transform the problem to work with standard normal distribution
2. Apply Theorem 5.1 (not provided in the excerpt) 3. Analyze the boundary of the
transformed set
Change of Variables
Define:
• 𝑌: Random vector with centered normal distribution (identity covariance)
• 𝛷(𝑦): Normal cumulative distribution function
• 𝑆𝛼(𝑦): Student-like cumulative distribution function
• 𝑆𝛼
←, 𝛷←: Inverse functions of 𝑆𝛼 and 𝛷 respectively
Key transformation:
𝑋=
𝑑𝑆𝛼
←∘𝛷(𝑌)
Define sets: 𝐴𝑡={𝑥∈ℝ𝑑:⟨𝐶𝑥,𝑥⟩>𝑡}=√𝑡𝐴1
𝐵𝑡={𝑦∈ℝ𝑑:⟨𝐶𝑆𝛼
←∘𝛷(𝑦),𝑆𝛼
←∘𝛷(𝑦)⟩>𝑡}=𝛷←∘𝑆𝛼(𝐴𝑡)
Important result:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}=∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥𝑖=∫ 𝑒−|𝑦|2/2
(2𝜋)𝑑/2
𝐵𝑡𝑑𝑦
Asymptotic Analysis
Asymptotic Equivalence
Key result:
(𝛷←∘𝑆𝛼(𝑥))2=2𝛼log𝑥−loglog𝑥−2log(𝐾𝑠,𝛼𝛼𝛼/2)−2log(2√𝜋)+𝑜(1)
as 𝑥→∞.
Boundary Parameterization
Lemma 8.2.2: Provides parameterizations for ∂𝐴𝑡 and ∂𝐵𝑡 near specific points.
For ∂𝐴𝑡 near 𝑝𝜖,𝑗,𝑡 =𝜖√𝑡/𝐶𝑗,𝑗𝑒𝑗:
𝑝(𝑣)=𝜖√𝑡
𝐶𝑗,𝑗[1− 1
2𝑡⟨𝐶𝑣,𝑣⟩+𝑂(⟨𝐶𝑣,𝑣⟩
𝑡2)]𝑒𝑗+𝑣
For ∂𝐵𝑡 near 𝑞𝜖,𝑗,𝑡 =𝛷←∘𝑆𝛼(𝑝𝜖,𝑗,𝑡):
𝑞(𝑣)=𝜖√𝛼log 𝑡
𝐶𝑗,𝑗−loglog√𝑡
2√𝛼log𝑡−log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋)
√𝛼log𝑡+𝑜( 1
√log𝑡)𝑒𝑗
+ ∑ 𝛷←
𝑑
𝑖=1,𝑖≠𝑗 ∘𝑆𝛼(𝑣𝑖)𝑒𝑖
Note: These parameterizations are crucial for applying Theorem 5.1 and deriving the
asymptotic behavior.
These lecture notes cover asymptotic analysis of quadratic forms for random vectors with
heavy-tailed distributions.
Asymptotic Behavior of Quadratic Forms
Main Theorem
Theorem 8.2.1: Let 𝑋 be a 𝑑-dimensional random vector with independent and identically
distributed components having a Student-like distribution. Let 𝐶 be a 𝑑×𝑑 matrix. Define
𝐽1={𝑗:𝐶𝑗,𝑗 >0}. If 𝐽1 is not empty, then:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}∼𝐾𝑠,𝛼𝛼(𝛼−1)/2 2
𝑡𝛼/2∑𝐶𝑗,𝑗
𝛼/2
𝑗∈𝐽1
as 𝑡→∞
Key points:
• Applies to Student-like distributions (heavy-tailed)
• Asymptotic behavior depends on positive diagonal elements of 𝐶
• Power law decay with exponent 𝛼/2
Proof Strategy
1. Change of variables: Transform the problem to work with standard normal distribution
2. Apply Theorem 5.1 (not provided in the excerpt) 3. Analyze the boundary of the
transformed set
Change of Variables
Define:
• 𝑌: Random vector with centered normal distribution (identity covariance)
• 𝛷(𝑦): Normal cumulative distribution function
• 𝑆𝛼(𝑦): Student-like cumulative distribution function
• 𝑆𝛼
←, 𝛷←: Inverse functions of 𝑆𝛼 and 𝛷 respectively
Key transformation:
𝑋=
𝑑𝑆𝛼
←∘𝛷(𝑌)
Define sets: 𝐴𝑡={𝑥∈ℝ𝑑:⟨𝐶𝑥,𝑥⟩>𝑡}=√𝑡𝐴1
𝐵𝑡={𝑦∈ℝ𝑑:⟨𝐶𝑆𝛼
←∘𝛷(𝑦),𝑆𝛼
←∘𝛷(𝑦)⟩>𝑡}=𝛷←∘𝑆𝛼(𝐴𝑡)
Important result:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}=∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥𝑖=∫ 𝑒−|𝑦|2/2
(2𝜋)𝑑/2
𝐵𝑡𝑑𝑦
Asymptotic Analysis
Asymptotic Equivalence
Key result:
(𝛷←∘𝑆𝛼(𝑥))2=2𝛼log𝑥−loglog𝑥−2log(𝐾𝑠,𝛼𝛼𝛼/2)−2log(2√𝜋)+𝑜(1)
as 𝑥→∞.
Boundary Parameterization
Lemma 8.2.2: Provides parameterizations for ∂𝐴𝑡 and ∂𝐵𝑡 near specific points.
For ∂𝐴𝑡 near 𝑝𝜖,𝑗,𝑡 =𝜖√𝑡/𝐶𝑗,𝑗𝑒𝑗:
𝑝(𝑣)=𝜖√𝑡
𝐶𝑗,𝑗[1− 1
2𝑡⟨𝐶𝑣,𝑣⟩+𝑂(⟨𝐶𝑣,𝑣⟩
𝑡2)]𝑒𝑗+𝑣
For ∂𝐵𝑡 near 𝑞𝜖,𝑗,𝑡 =𝛷←∘𝑆𝛼(𝑝𝜖,𝑗,𝑡):
𝑞(𝑣)=𝜖√𝛼log 𝑡
𝐶𝑗,𝑗−loglog√𝑡
2√𝛼log𝑡−log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋)
√𝛼log𝑡+𝑜( 1
√log𝑡)𝑒𝑗
+ ∑ 𝛷←
𝑑
𝑖=1,𝑖≠𝑗 ∘𝑆𝛼(𝑣𝑖)𝑒𝑖
Note: These parameterizations are crucial for applying Theorem 5.1 and deriving the
asymptotic behavior.
These lecture notes cover asymptotic analysis of quadratic forms for random vectors with
heavy-tailed distributions.
Asymptotic Behavior of Quadratic Forms
Main Theorem
Theorem 8.2.1: Let 𝑋 be a 𝑑-dimensional random vector with independent and identically
distributed components having a Student-like distribution. Let 𝐶 be a 𝑑×𝑑 matrix. Define
𝐽1={𝑗:𝐶𝑗,𝑗 >0}. If 𝐽1 is not empty, then:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}∼𝐾𝑠,𝛼𝛼(𝛼−1)/2 2
𝑡𝛼/2∑𝐶𝑗,𝑗
𝛼/2
𝑗∈𝐽1
as 𝑡→∞
Key points:
• Applies to Student-like distributions (heavy-tailed)
• Asymptotic behavior depends on positive diagonal elements of 𝐶
• Power law decay with exponent 𝛼/2
Proof Strategy
1. Change of variables: Transform the problem to work with standard normal distribution
2. Apply Theorem 5.1 (not provided in the excerpt) 3. Analyze the boundary of the
transformed set
Change of Variables
Define:
• 𝑌: Random vector with centered normal distribution (identity covariance)
• 𝛷(𝑦): Normal cumulative distribution function
• 𝑆𝛼(𝑦): Student-like cumulative distribution function
• 𝑆𝛼
←, 𝛷←: Inverse functions of 𝑆𝛼 and 𝛷 respectively
Key transformation:
𝑋=
𝑑𝑆𝛼
←∘𝛷(𝑌)
Define sets: 𝐴𝑡={𝑥∈ℝ𝑑:⟨𝐶𝑥,𝑥⟩>𝑡}=√𝑡𝐴1
𝐵𝑡={𝑦∈ℝ𝑑:⟨𝐶𝑆𝛼
←∘𝛷(𝑦),𝑆𝛼
←∘𝛷(𝑦)⟩>𝑡}=𝛷←∘𝑆𝛼(𝐴𝑡)
Important result:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}=∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥𝑖=∫ 𝑒−|𝑦|2/2
(2𝜋)𝑑/2
𝐵𝑡𝑑𝑦
Asymptotic Analysis
Asymptotic Equivalence
Key result:
(𝛷←∘𝑆𝛼(𝑥))2=2𝛼log𝑥−loglog𝑥−2log(𝐾𝑠,𝛼𝛼𝛼/2)−2log(2√𝜋)+𝑜(1)
as 𝑥→∞.
Boundary Parameterization
Lemma 8.2.2: Provides parameterizations for ∂𝐴𝑡 and ∂𝐵𝑡 near specific points.
For ∂𝐴𝑡 near 𝑝𝜖,𝑗,𝑡 =𝜖√𝑡/𝐶𝑗,𝑗𝑒𝑗:
𝑝(𝑣)=𝜖√𝑡
𝐶𝑗,𝑗[1− 1
2𝑡⟨𝐶𝑣,𝑣⟩+𝑂(⟨𝐶𝑣,𝑣⟩
𝑡2)]𝑒𝑗+𝑣
For ∂𝐵𝑡 near 𝑞𝜖,𝑗,𝑡 =𝛷←∘𝑆𝛼(𝑝𝜖,𝑗,𝑡):
𝑞(𝑣)=𝜖√𝛼log 𝑡
𝐶𝑗,𝑗−loglog√𝑡
2√𝛼log𝑡−log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋)
√𝛼log𝑡+𝑜( 1
√log𝑡)𝑒𝑗
+ ∑ 𝛷←
𝑑
𝑖=1,𝑖≠𝑗 ∘𝑆𝛼(𝑣𝑖)𝑒𝑖
Note: These parameterizations are crucial for applying Theorem 5.1 and deriving the
asymptotic behavior.
These lecture notes cover asymptotic analysis of quadratic forms for random vectors with
heavy-tailed distributions.
Asymptotic Behavior of Quadratic Forms
Main Theorem
Theorem 8.2.1: Let 𝑋 be a 𝑑-dimensional random vector with independent and identically
distributed components having a Student-like distribution. Let 𝐶 be a 𝑑×𝑑 matrix. Define
𝐽1={𝑗:𝐶𝑗,𝑗 >0}. If 𝐽1 is not empty, then:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}∼𝐾𝑠,𝛼𝛼(𝛼−1)/2 2
𝑡𝛼/2∑𝐶𝑗,𝑗
𝛼/2
𝑗∈𝐽1
as 𝑡→∞
Key points:
• Applies to Student-like distributions (heavy-tailed)
• Asymptotic behavior depends on positive diagonal elements of 𝐶
• Power law decay with exponent 𝛼/2
Proof Strategy
1. Change of variables: Transform the problem to work with standard normal distribution
2. Apply Theorem 5.1 (not provided in the excerpt) 3. Analyze the boundary of the
transformed set
Change of Variables
Define:
• 𝑌: Random vector with centered normal distribution (identity covariance)
• 𝛷(𝑦): Normal cumulative distribution function
• 𝑆𝛼(𝑦): Student-like cumulative distribution function
• 𝑆𝛼
←, 𝛷←: Inverse functions of 𝑆𝛼 and 𝛷 respectively
Key transformation:
𝑋=
𝑑𝑆𝛼
←∘𝛷(𝑌)
Define sets: 𝐴𝑡={𝑥∈ℝ𝑑:⟨𝐶𝑥,𝑥⟩>𝑡}=√𝑡𝐴1
𝐵𝑡={𝑦∈ℝ𝑑:⟨𝐶𝑆𝛼
←∘𝛷(𝑦),𝑆𝛼
←∘𝛷(𝑦)⟩>𝑡}=𝛷←∘𝑆𝛼(𝐴𝑡)
Important result:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}=∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥𝑖=∫ 𝑒−|𝑦|2/2
(2𝜋)𝑑/2
𝐵𝑡𝑑𝑦
Asymptotic Analysis
Asymptotic Equivalence
Key result:
(𝛷←∘𝑆𝛼(𝑥))2=2𝛼log𝑥−loglog𝑥−2log(𝐾𝑠,𝛼𝛼𝛼/2)−2log(2√𝜋)+𝑜(1)
as 𝑥→∞.
Boundary Parameterization
Lemma 8.2.2: Provides parameterizations for ∂𝐴𝑡 and ∂𝐵𝑡 near specific points.
For ∂𝐴𝑡 near 𝑝𝜖,𝑗,𝑡 =𝜖√𝑡/𝐶𝑗,𝑗𝑒𝑗:
𝑝(𝑣)=𝜖√𝑡
𝐶𝑗,𝑗[1− 1
2𝑡⟨𝐶𝑣,𝑣⟩+𝑂(⟨𝐶𝑣,𝑣⟩
𝑡2)]𝑒𝑗+𝑣
For ∂𝐵𝑡 near 𝑞𝜖,𝑗,𝑡 =𝛷←∘𝑆𝛼(𝑝𝜖,𝑗,𝑡):
𝑞(𝑣)=𝜖√𝛼log 𝑡
𝐶𝑗,𝑗−loglog√𝑡
2√𝛼log𝑡−log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋)
√𝛼log𝑡+𝑜( 1
√log𝑡)𝑒𝑗
+ ∑ 𝛷←
𝑑
𝑖=1,𝑖≠𝑗 ∘𝑆𝛼(𝑣𝑖)𝑒𝑖
Note: These parameterizations are crucial for applying Theorem 5.1 and deriving the
asymptotic behavior.
These lecture notes cover asymptotic analysis of quadratic forms for random vectors with
heavy-tailed distributions.
Asymptotic Behavior of Quadratic Forms
Main Theorem
Theorem 8.2.1: Let 𝑋 be a 𝑑-dimensional random vector with independent and identically
distributed components having a Student-like distribution. Let 𝐶 be a 𝑑×𝑑 matrix. Define
𝐽1={𝑗:𝐶𝑗,𝑗 >0}. If 𝐽1 is not empty, then:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}∼𝐾𝑠,𝛼𝛼(𝛼−1)/2 2
𝑡𝛼/2∑𝐶𝑗,𝑗
𝛼/2
𝑗∈𝐽1
as 𝑡→∞
Key points:
• Applies to Student-like distributions (heavy-tailed)
• Asymptotic behavior depends on positive diagonal elements of 𝐶
• Power law decay with exponent 𝛼/2
Proof Strategy
1. Change of variables: Transform the problem to work with standard normal distribution
2. Apply Theorem 5.1 (not provided in the excerpt) 3. Analyze the boundary of the
transformed set
Change of Variables
Define:
• 𝑌: Random vector with centered normal distribution (identity covariance)
• 𝛷(𝑦): Normal cumulative distribution function
• 𝑆𝛼(𝑦): Student-like cumulative distribution function
• 𝑆𝛼
←, 𝛷←: Inverse functions of 𝑆𝛼 and 𝛷 respectively
Key transformation:
𝑋=
𝑑𝑆𝛼
←∘𝛷(𝑌)
Define sets: 𝐴𝑡={𝑥∈ℝ𝑑:⟨𝐶𝑥,𝑥⟩>𝑡}=√𝑡𝐴1
𝐵𝑡={𝑦∈ℝ𝑑:⟨𝐶𝑆𝛼
←∘𝛷(𝑦),𝑆𝛼
←∘𝛷(𝑦)⟩>𝑡}=𝛷←∘𝑆𝛼(𝐴𝑡)
Important result:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}=∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥𝑖=∫ 𝑒−|𝑦|2/2
(2𝜋)𝑑/2
𝐵𝑡𝑑𝑦
Asymptotic Analysis
Asymptotic Equivalence
Key result:
(𝛷←∘𝑆𝛼(𝑥))2=2𝛼log𝑥−loglog𝑥−2log(𝐾𝑠,𝛼𝛼𝛼/2)−2log(2√𝜋)+𝑜(1)
as 𝑥→∞.
Boundary Parameterization
Lemma 8.2.2: Provides parameterizations for ∂𝐴𝑡 and ∂𝐵𝑡 near specific points.
For ∂𝐴𝑡 near 𝑝𝜖,𝑗,𝑡 =𝜖√𝑡/𝐶𝑗,𝑗𝑒𝑗:
𝑝(𝑣)=𝜖√𝑡
𝐶𝑗,𝑗[1− 1
2𝑡⟨𝐶𝑣,𝑣⟩+𝑂(⟨𝐶𝑣,𝑣⟩
𝑡2)]𝑒𝑗+𝑣
For ∂𝐵𝑡 near 𝑞𝜖,𝑗,𝑡 =𝛷←∘𝑆𝛼(𝑝𝜖,𝑗,𝑡):
𝑞(𝑣)=𝜖√𝛼log 𝑡
𝐶𝑗,𝑗−loglog√𝑡
2√𝛼log𝑡−log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋)
√𝛼log𝑡+𝑜( 1
√log𝑡)𝑒𝑗
+ ∑ 𝛷←
𝑑
𝑖=1,𝑖≠𝑗 ∘𝑆𝛼(𝑣𝑖)𝑒𝑖
Note: These parameterizations are crucial for applying Theorem 5.1 and deriving the
asymptotic behavior.
These lecture notes cover asymptotic analysis of quadratic forms for random vectors with
heavy-tailed distributions.
Asymptotic Behavior of Quadratic Forms
Main Theorem
Theorem 8.2.1: Let 𝑋 be a 𝑑-dimensional random vector with independent and identically
distributed components having a Student-like distribution. Let 𝐶 be a 𝑑×𝑑 matrix. Define
𝐽1={𝑗:𝐶𝑗,𝑗 >0}. If 𝐽1 is not empty, then:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}∼𝐾𝑠,𝛼𝛼(𝛼−1)/2 2
𝑡𝛼/2∑𝐶𝑗,𝑗
𝛼/2
𝑗∈𝐽1
as 𝑡→∞
Key points:
• Applies to Student-like distributions (heavy-tailed)
• Asymptotic behavior depends on positive diagonal elements of 𝐶
• Power law decay with exponent 𝛼/2
Proof Strategy
1. Change of variables: Transform the problem to work with standard normal distribution
2. Apply Theorem 5.1 (not provided in the excerpt) 3. Analyze the boundary of the
transformed set
Change of Variables
Define:
• 𝑌: Random vector with centered normal distribution (identity covariance)
• 𝛷(𝑦): Normal cumulative distribution function
• 𝑆𝛼(𝑦): Student-like cumulative distribution function
• 𝑆𝛼
←, 𝛷←: Inverse functions of 𝑆𝛼 and 𝛷 respectively
Key transformation:
𝑋=
𝑑𝑆𝛼
←∘𝛷(𝑌)
Define sets:
𝐴𝑡={𝑥∈ℝ𝑑:⟨𝐶𝑥,𝑥⟩>𝑡}=√𝑡𝐴1
𝐵𝑡={𝑦∈ℝ𝑑:⟨𝐶𝑆𝛼
←∘𝛷(𝑦),𝑆𝛼
←∘𝛷(𝑦)⟩>𝑡}=𝛷←∘𝑆𝛼(𝐴𝑡)
Important result:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}=∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥𝑖=∫ 𝑒−|𝑦|2/2
(2𝜋)𝑑/2
𝐵𝑡𝑑𝑦
Asymptotic Analysis
Asymptotic Equivalence
Key result:
(𝛷←∘𝑆𝛼(𝑥))2=2𝛼log𝑥−loglog𝑥−2log(𝐾𝑠,𝛼𝛼𝛼/2)−2log(2√𝜋)+𝑜(1)
as 𝑥→∞.
Boundary Parameterization
Lemma 8.2.2: Provides parameterizations for ∂𝐴𝑡 and ∂𝐵𝑡 near specific points.
For ∂𝐴𝑡 near 𝑝𝜖,𝑗,𝑡 =𝜖√𝑡/𝐶𝑗,𝑗𝑒𝑗:
𝑝(𝑣)=𝜖√𝑡
𝐶𝑗,𝑗[1− 1
2𝑡⟨𝐶𝑣,𝑣⟩+𝑂(⟨𝐶𝑣,𝑣⟩
𝑡2)]𝑒𝑗+𝑣
For ∂𝐵𝑡 near 𝑞𝜖,𝑗,𝑡 =𝛷←∘𝑆𝛼(𝑝𝜖,𝑗,𝑡):
𝑞(𝑣)=𝜖√𝛼log 𝑡
𝐶𝑗,𝑗−loglog√𝑡
2√𝛼log𝑡−log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋)
√𝛼log𝑡+𝑜( 1
√log𝑡)𝑒𝑗
+ ∑ 𝛷←
𝑑
𝑖=1,𝑖≠𝑗 ∘𝑆𝛼(𝑣𝑖)𝑒𝑖
Note: These parameterizations are crucial for applying Theorem 5.1 and deriving the
asymptotic behavior.
These lecture notes cover asymptotic analysis of quadratic forms for random vectors with
heavy-tailed distributions.
Asymptotic Behavior of Quadratic Forms
Main Theorem
Theorem 8.2.1: Let 𝑋 be a 𝑑-dimensional random vector with independent and identically
distributed components having a Student-like distribution. Let 𝐶 be a 𝑑×𝑑 matrix. Define
𝐽1={𝑗:𝐶𝑗,𝑗 >0}. If 𝐽1 is not empty, then:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}∼𝐾𝑠,𝛼𝛼(𝛼−1)/2 2
𝑡𝛼/2∑𝐶𝑗,𝑗
𝛼/2
𝑗∈𝐽1
as 𝑡→∞
Key points:
• Applies to Student-like distributions (heavy-tailed)
• Asymptotic behavior depends on positive diagonal elements of 𝐶
• Power law decay with exponent 𝛼/2
Proof Strategy
1. Change of variables: Transform the problem to work with standard normal distribution
2. Apply Theorem 5.1 (not provided in the excerpt) 3. Analyze the boundary of the
transformed set
Change of Variables
Define:
• 𝑌: Random vector with centered normal distribution (identity covariance)
• 𝛷(𝑦): Normal cumulative distribution function
• 𝑆𝛼(𝑦): Student-like cumulative distribution function
• 𝑆𝛼
←, 𝛷←: Inverse functions of 𝑆𝛼 and 𝛷 respectively
Key transformation:
𝑋=
𝑑𝑆𝛼
←∘𝛷(𝑌)
Define sets: 𝐴𝑡={𝑥∈ℝ𝑑:⟨𝐶𝑥,𝑥⟩>𝑡}=√𝑡𝐴1
𝐵𝑡={𝑦∈ℝ𝑑:⟨𝐶𝑆𝛼
←∘𝛷(𝑦),𝑆𝛼
←∘𝛷(𝑦)⟩>𝑡}=𝛷←∘𝑆𝛼(𝐴𝑡)
Important result:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}=∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥𝑖=∫ 𝑒−|𝑦|2/2
(2𝜋)𝑑/2
𝐵𝑡𝑑𝑦
Asymptotic Analysis
Asymptotic Equivalence
Key result:
(𝛷←∘𝑆𝛼(𝑥))2=2𝛼log𝑥−loglog𝑥−2log(𝐾𝑠,𝛼𝛼𝛼/2)−2log(2√𝜋)+𝑜(1)
as 𝑥→∞.
Boundary Parameterization
Lemma 8.2.2: Provides parameterizations for ∂𝐴𝑡 and ∂𝐵𝑡 near specific points.
For ∂𝐴𝑡 near 𝑝𝜖,𝑗,𝑡 =𝜖√𝑡/𝐶𝑗,𝑗𝑒𝑗:
𝑝(𝑣)=𝜖√𝑡
𝐶𝑗,𝑗[1− 1
2𝑡⟨𝐶𝑣,𝑣⟩+𝑂(⟨𝐶𝑣,𝑣⟩
𝑡2)]𝑒𝑗+𝑣
For ∂𝐵𝑡 near 𝑞𝜖,𝑗,𝑡 =𝛷←∘𝑆𝛼(𝑝𝜖,𝑗,𝑡):
𝑞(𝑣)=𝜖√𝛼log 𝑡
𝐶𝑗,𝑗−loglog√𝑡
2√𝛼log𝑡−log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋)
√𝛼log𝑡+𝑜( 1
√log𝑡)𝑒𝑗
+ ∑ 𝛷←
𝑑
𝑖=1,𝑖≠𝑗 ∘𝑆𝛼(𝑣𝑖)𝑒𝑖
Note: These parameterizations are crucial for applying Theorem 5.1 and deriving the
asymptotic behavior.
These lecture notes cover asymptotic analysis of quadratic forms for random vectors with
heavy-tailed distributions.
Asymptotic Behavior of Quadratic Forms
Main Theorem
Theorem 8.2.1: Let 𝑋 be a 𝑑-dimensional random vector with independent and identically
distributed components having a Student-like distribution. Let 𝐶 be a 𝑑×𝑑 matrix. Define
𝐽1={𝑗:𝐶𝑗,𝑗 >0}. If 𝐽1 is not empty, then:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}∼𝐾𝑠,𝛼𝛼(𝛼−1)/2 2
𝑡𝛼/2∑𝐶𝑗,𝑗
𝛼/2
𝑗∈𝐽1
as 𝑡→∞
Key points:
• Applies to Student-like distributions (heavy-tailed)
• Asymptotic behavior depends on positive diagonal elements of 𝐶
• Power law decay with exponent 𝛼/2
Proof Strategy
1. Change of variables: Transform the problem to work with standard normal distribution
2. Apply Theorem 5.1 (not provided in the excerpt) 3. Analyze the boundary of the
transformed set
Change of Variables
Define:
• 𝑌: Random vector with centered normal distribution (identity covariance)
• 𝛷(𝑦): Normal cumulative distribution function
• 𝑆𝛼(𝑦): Student-like cumulative distribution function
• 𝑆𝛼
←, 𝛷←: Inverse functions of 𝑆𝛼 and 𝛷 respectively
Key transformation:
𝑋=
𝑑𝑆𝛼
←∘𝛷(𝑌)
Define sets: 𝐴𝑡={𝑥∈ℝ𝑑:⟨𝐶𝑥,𝑥⟩>𝑡}=√𝑡𝐴1
𝐵𝑡={𝑦∈ℝ𝑑:⟨𝐶𝑆𝛼
←∘𝛷(𝑦),𝑆𝛼
←∘𝛷(𝑦)⟩>𝑡}=𝛷←∘𝑆𝛼(𝐴𝑡)
Important result:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}=∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥𝑖=∫ 𝑒−|𝑦|2/2
(2𝜋)𝑑/2
𝐵𝑡𝑑𝑦
Asymptotic Analysis
Asymptotic Equivalence
Key result:
(𝛷←∘𝑆𝛼(𝑥))2=2𝛼log𝑥−loglog𝑥−2log(𝐾𝑠,𝛼𝛼𝛼/2)−2log(2√𝜋)+𝑜(1)
as 𝑥→∞.
Boundary Parameterization
Lemma 8.2.2: Provides parameterizations for ∂𝐴𝑡 and ∂𝐵𝑡 near specific points.
For ∂𝐴𝑡 near 𝑝𝜖,𝑗,𝑡 =𝜖√𝑡/𝐶𝑗,𝑗𝑒𝑗:
𝑝(𝑣)=𝜖√𝑡
𝐶𝑗,𝑗[1− 1
2𝑡⟨𝐶𝑣,𝑣⟩+𝑂(⟨𝐶𝑣,𝑣⟩
𝑡2)]𝑒𝑗+𝑣
For ∂𝐵𝑡 near 𝑞𝜖,𝑗,𝑡 =𝛷←∘𝑆𝛼(𝑝𝜖,𝑗,𝑡):
𝑞(𝑣)=𝜖√𝛼log 𝑡
𝐶𝑗,𝑗−loglog√𝑡
2√𝛼log𝑡−log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋)
√𝛼log𝑡+𝑜( 1
√log𝑡)𝑒𝑗
+ ∑ 𝛷←
𝑑
𝑖=1,𝑖≠𝑗 ∘𝑆𝛼(𝑣𝑖)𝑒𝑖
Note: These parameterizations are crucial for applying Theorem 5.1 and deriving the
asymptotic behavior.
These lecture notes cover asymptotic analysis of quadratic forms for random vectors with
heavy-tailed distributions.
Asymptotic Behavior of Quadratic Forms
Main Theorem
Theorem 8.2.1: Let 𝑋 be a 𝑑-dimensional random vector with independent and identically
distributed components having a Student-like distribution. Let 𝐶 be a 𝑑×𝑑 matrix. Define
𝐽1={𝑗:𝐶𝑗,𝑗 >0}. If 𝐽1 is not empty, then:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}∼𝐾𝑠,𝛼𝛼(𝛼−1)/2 2
𝑡𝛼/2∑𝐶𝑗,𝑗
𝛼/2
𝑗∈𝐽1
as 𝑡→∞
Key points:
• Applies to Student-like distributions (heavy-tailed)
• Asymptotic behavior depends on positive diagonal elements of 𝐶
• Power law decay with exponent 𝛼/2
Proof Strategy
1. Change of variables: Transform the problem to work with standard normal distribution
2. Apply Theorem 5.1 (not provided in the excerpt) 3. Analyze the boundary of the
transformed set
Change of Variables
Define:
• 𝑌: Random vector with centered normal distribution (identity covariance)
• 𝛷(𝑦): Normal cumulative distribution function
• 𝑆𝛼(𝑦): Student-like cumulative distribution function
• 𝑆𝛼
←, 𝛷←: Inverse functions of 𝑆𝛼 and 𝛷 respectively
Key transformation:
𝑋=
𝑑𝑆𝛼
←∘𝛷(𝑌)
Define sets: 𝐴𝑡={𝑥∈ℝ𝑑:⟨𝐶𝑥,𝑥⟩>𝑡}=√𝑡𝐴1
𝐵𝑡={𝑦∈ℝ𝑑:⟨𝐶𝑆𝛼
←∘𝛷(𝑦),𝑆𝛼
←∘𝛷(𝑦)⟩>𝑡}=𝛷←∘𝑆𝛼(𝐴𝑡)
Important result:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}=∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥𝑖=∫ 𝑒−|𝑦|2/2
(2𝜋)𝑑/2
𝐵𝑡𝑑𝑦
Asymptotic Analysis
Asymptotic Equivalence
Key result:
(𝛷←∘𝑆𝛼(𝑥))2=2𝛼log𝑥−loglog𝑥−2log(𝐾𝑠,𝛼𝛼𝛼/2)−2log(2√𝜋)+𝑜(1)
as 𝑥→∞.
Boundary Parameterization
Lemma 8.2.2: Provides parameterizations for ∂𝐴𝑡 and ∂𝐵𝑡 near specific points.
For ∂𝐴𝑡 near 𝑝𝜖,𝑗,𝑡 =𝜖√𝑡/𝐶𝑗,𝑗𝑒𝑗:
𝑝(𝑣)=𝜖√𝑡
𝐶𝑗,𝑗[1− 1
2𝑡⟨𝐶𝑣,𝑣⟩+𝑂(⟨𝐶𝑣,𝑣⟩
𝑡2)]𝑒𝑗+𝑣
For ∂𝐵𝑡 near 𝑞𝜖,𝑗,𝑡 =𝛷←∘𝑆𝛼(𝑝𝜖,𝑗,𝑡):
𝑞(𝑣)=𝜖√𝛼log 𝑡
𝐶𝑗,𝑗−loglog√𝑡
2√𝛼log𝑡−log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋)
√𝛼log𝑡+𝑜( 1
√log𝑡)𝑒𝑗
+ ∑ 𝛷←
𝑑
𝑖=1,𝑖≠𝑗 ∘𝑆𝛼(𝑣𝑖)𝑒𝑖
Note: These parameterizations are crucial for applying Theorem 5.1 and deriving the
asymptotic behavior.
These lecture notes cover asymptotic analysis of quadratic forms for random vectors with
heavy-tailed distributions.
Asymptotic Behavior of Quadratic Forms
Main Theorem
Theorem 8.2.1: Let 𝑋 be a 𝑑-dimensional random vector with independent and identically
distributed components having a Student-like distribution. Let 𝐶 be a 𝑑×𝑑 matrix. Define
𝐽1={𝑗:𝐶𝑗,𝑗 >0}. If 𝐽1 is not empty, then:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}∼𝐾𝑠,𝛼𝛼(𝛼−1)/2 2
𝑡𝛼/2∑𝐶𝑗,𝑗
𝛼/2
𝑗∈𝐽1
as 𝑡→∞
Key points:
• Applies to Student-like distributions (heavy-tailed)
• Asymptotic behavior depends on positive diagonal elements of 𝐶
• Power law decay with exponent 𝛼/2
Proof Strategy
1. Change of variables: Transform the problem to work with standard normal distribution
2. Apply Theorem 5.1 (not provided in the excerpt) 3. Analyze the boundary of the
transformed set
Change of Variables
Define:
• 𝑌: Random vector with centered normal distribution (identity covariance)
• 𝛷(𝑦): Normal cumulative distribution function
• 𝑆𝛼(𝑦): Student-like cumulative distribution function
• 𝑆𝛼
←, 𝛷←: Inverse functions of 𝑆𝛼 and 𝛷 respectively
Key transformation:
𝑋=
𝑑𝑆𝛼
←∘𝛷(𝑌)
Define sets: 𝐴𝑡={𝑥∈ℝ𝑑:⟨𝐶𝑥,𝑥⟩>𝑡}=√𝑡𝐴1
𝐵𝑡={𝑦∈ℝ𝑑:⟨𝐶𝑆𝛼
←∘𝛷(𝑦),𝑆𝛼
←∘𝛷(𝑦)⟩>𝑡}=𝛷←∘𝑆𝛼(𝐴𝑡)
Important result:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}=∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥𝑖=∫ 𝑒−|𝑦|2/2
(2𝜋)𝑑/2
𝐵𝑡𝑑𝑦
Asymptotic Analysis
Asymptotic Equivalence
Key result:
(𝛷←∘𝑆𝛼(𝑥))2=2𝛼log𝑥−loglog𝑥−2log(𝐾𝑠,𝛼𝛼𝛼/2)−2log(2√𝜋)+𝑜(1)
as 𝑥→∞.
Boundary Parameterization
Lemma 8.2.2: Provides parameterizations for ∂𝐴𝑡 and ∂𝐵𝑡 near specific points.
For ∂𝐴𝑡 near 𝑝𝜖,𝑗,𝑡 =𝜖√𝑡/𝐶𝑗,𝑗𝑒𝑗:
𝑝(𝑣)=𝜖√𝑡
𝐶𝑗,𝑗[1− 1
2𝑡⟨𝐶𝑣,𝑣⟩+𝑂(⟨𝐶𝑣,𝑣⟩
𝑡2)]𝑒𝑗+𝑣
For ∂𝐵𝑡 near 𝑞𝜖,𝑗,𝑡 =𝛷←∘𝑆𝛼(𝑝𝜖,𝑗,𝑡):
𝑞(𝑣)=𝜖√𝛼log 𝑡
𝐶𝑗,𝑗−loglog√𝑡
2√𝛼log𝑡−log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋)
√𝛼log𝑡+𝑜( 1
√log𝑡)𝑒𝑗
+ ∑ 𝛷←
𝑑
𝑖=1,𝑖≠𝑗 ∘𝑆𝛼(𝑣𝑖)𝑒𝑖
Note: These parameterizations are crucial for applying Theorem 5.1 and deriving the
asymptotic behavior.
These lecture notes cover asymptotic analysis of quadratic forms for random vectors with
heavy-tailed distributions.
Asymptotic Behavior of Quadratic Forms
Main Theorem
Theorem 8.2.1: Let 𝑋 be a 𝑑-dimensional random vector with independent and identically
distributed components having a Student-like distribution. Let 𝐶 be a 𝑑×𝑑 matrix. Define
𝐽1={𝑗:𝐶𝑗,𝑗 >0}. If 𝐽1 is not empty, then:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}∼𝐾𝑠,𝛼𝛼(𝛼−1)/2 2
𝑡𝛼/2∑𝐶𝑗,𝑗
𝛼/2
𝑗∈𝐽1
as 𝑡→∞
Key points:
• Applies to Student-like distributions (heavy-tailed)
• Asymptotic behavior depends on positive diagonal elements of 𝐶
• Power law decay with exponent 𝛼/2
Proof Strategy
1. Change of variables: Transform the problem to work with standard normal distribution
2. Apply Theorem 5.1 (not provided in the excerpt) 3. Analyze the boundary of the
transformed set
Change of Variables
Define:
• 𝑌: Random vector with centered normal distribution (identity covariance)
• 𝛷(𝑦): Normal cumulative distribution function
• 𝑆𝛼(𝑦): Student-like cumulative distribution function
• 𝑆𝛼
←, 𝛷←: Inverse functions of 𝑆𝛼 and 𝛷 respectively
Key transformation:
𝑋=
𝑑𝑆𝛼
←∘𝛷(𝑌)
Define sets: 𝐴𝑡={𝑥∈ℝ𝑑:⟨𝐶𝑥,𝑥⟩>𝑡}=√𝑡𝐴1
𝐵𝑡={𝑦∈ℝ𝑑:⟨𝐶𝑆𝛼
←∘𝛷(𝑦),𝑆𝛼
←∘𝛷(𝑦)⟩>𝑡}=𝛷←∘𝑆𝛼(𝐴𝑡)
Important result:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}=∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥𝑖=∫ 𝑒−|𝑦|2/2
(2𝜋)𝑑/2
𝐵𝑡𝑑𝑦
Asymptotic Analysis
Asymptotic Equivalence
Key result:
(𝛷←∘𝑆𝛼(𝑥))2=2𝛼log𝑥−loglog𝑥−2log(𝐾𝑠,𝛼𝛼𝛼/2)−2log(2√𝜋)+𝑜(1)
as 𝑥→∞.
Boundary Parameterization
Lemma 8.2.2: Provides parameterizations for ∂𝐴𝑡 and ∂𝐵𝑡 near specific points.
For ∂𝐴𝑡 near 𝑝𝜖,𝑗,𝑡 =𝜖√𝑡/𝐶𝑗,𝑗𝑒𝑗:
𝑝(𝑣)=𝜖√𝑡
𝐶𝑗,𝑗[1− 1
2𝑡⟨𝐶𝑣,𝑣⟩+𝑂(⟨𝐶𝑣,𝑣⟩
𝑡2)]𝑒𝑗+𝑣
For ∂𝐵𝑡 near 𝑞𝜖,𝑗,𝑡 =𝛷←∘𝑆𝛼(𝑝𝜖,𝑗,𝑡):
𝑞(𝑣)=𝜖√𝛼log 𝑡
𝐶𝑗,𝑗−loglog√𝑡
2√𝛼log𝑡−log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋)
√𝛼log𝑡+𝑜( 1
√log𝑡)𝑒𝑗
+ ∑ 𝛷←
𝑑
𝑖=1,𝑖≠𝑗 ∘𝑆𝛼(𝑣𝑖)𝑒𝑖
Note: These parameterizations are crucial for applying Theorem 5.1 and deriving the
asymptotic behavior.
These lecture notes cover asymptotic analysis of quadratic forms for random vectors with
heavy-tailed distributions.
Asymptotic Behavior of Quadratic Forms
Main Theorem
Theorem 8.2.1: Let 𝑋 be a 𝑑-dimensional random vector with independent and identically
distributed components having a Student-like distribution. Let 𝐶 be a 𝑑×𝑑 matrix. Define
𝐽1={𝑗:𝐶𝑗,𝑗 >0}. If 𝐽1 is not empty, then:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}∼𝐾𝑠,𝛼𝛼(𝛼−1)/2 2
𝑡𝛼/2∑𝐶𝑗,𝑗
𝛼/2
𝑗∈𝐽1
as 𝑡→∞
Key points:
• Applies to Student-like distributions (heavy-tailed)
• Asymptotic behavior depends on positive diagonal elements of 𝐶
• Power law decay with exponent 𝛼/2
Proof Strategy
1. Change of variables: Transform the problem to work with standard normal distribution
2. Apply Theorem 5.1 (not provided in the excerpt) 3. Analyze the boundary of the
transformed set
Change of Variables
Define:
• 𝑌: Random vector with centered normal distribution (identity covariance)
• 𝛷(𝑦): Normal cumulative distribution function
• 𝑆𝛼(𝑦): Student-like cumulative distribution function
• 𝑆𝛼
←, 𝛷←: Inverse functions of 𝑆𝛼 and 𝛷 respectively
Key transformation:
𝑋=
𝑑𝑆𝛼
←∘𝛷(𝑌)
Define sets: 𝐴𝑡={𝑥∈ℝ𝑑:⟨𝐶𝑥,𝑥⟩>𝑡}=√𝑡𝐴1
𝐵𝑡={𝑦∈ℝ𝑑:⟨𝐶𝑆𝛼
←∘𝛷(𝑦),𝑆𝛼
←∘𝛷(𝑦)⟩>𝑡}=𝛷←∘𝑆𝛼(𝐴𝑡)
Important result:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}=∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥𝑖=∫ 𝑒−|𝑦|2/2
(2𝜋)𝑑/2
𝐵𝑡𝑑𝑦
Asymptotic Analysis
Asymptotic Equivalence
Key result:
(𝛷←∘𝑆𝛼(𝑥))2=2𝛼log𝑥−loglog𝑥−2log(𝐾𝑠,𝛼𝛼𝛼/2)−2log(2√𝜋)+𝑜(1)
as 𝑥→∞.
Boundary Parameterization
Lemma 8.2.2: Provides parameterizations for ∂𝐴𝑡 and ∂𝐵𝑡 near specific points.
For ∂𝐴𝑡 near 𝑝𝜖,𝑗,𝑡 =𝜖√𝑡/𝐶𝑗,𝑗𝑒𝑗:
𝑝(𝑣)=𝜖√𝑡
𝐶𝑗,𝑗[1− 1
2𝑡⟨𝐶𝑣,𝑣⟩+𝑂(⟨𝐶𝑣,𝑣⟩
𝑡2)]𝑒𝑗+𝑣
For ∂𝐵𝑡 near 𝑞𝜖,𝑗,𝑡 =𝛷←∘𝑆𝛼(𝑝𝜖,𝑗,𝑡):
𝑞(𝑣)=𝜖√𝛼log 𝑡
𝐶𝑗,𝑗−loglog√𝑡
2√𝛼log𝑡−log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋)
√𝛼log𝑡+𝑜( 1
√log𝑡)𝑒𝑗
+ ∑ 𝛷←
𝑑
𝑖=1,𝑖≠𝑗 ∘𝑆𝛼(𝑣𝑖)𝑒𝑖
Note: These parameterizations are crucial for applying Theorem 5.1 and deriving the
asymptotic behavior.
These lecture notes cover asymptotic analysis of quadratic forms for random vectors with
heavy-tailed distributions.
Asymptotic Behavior of Quadratic Forms
Main Theorem
Theorem 8.2.1: Let 𝑋 be a 𝑑-dimensional random vector with independent and identically
distributed components having a Student-like distribution. Let 𝐶 be a 𝑑×𝑑 matrix. Define
𝐽1={𝑗:𝐶𝑗,𝑗 >0}. If 𝐽1 is not empty, then:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}∼𝐾𝑠,𝛼𝛼(𝛼−1)/2 2
𝑡𝛼/2∑𝐶𝑗,𝑗
𝛼/2
𝑗∈𝐽1
as 𝑡→∞
Key points:
• Applies to Student-like distributions (heavy-tailed)
• Asymptotic behavior depends on positive diagonal elements of 𝐶
• Power law decay with exponent 𝛼/2
Proof Strategy
1. Change of variables: Transform the problem to work with standard normal distribution
2. Apply Theorem 5.1 (not provided in the excerpt) 3. Analyze the boundary of the
transformed set
Change of Variables
Define:
• 𝑌: Random vector with centered normal distribution (identity covariance)
• 𝛷(𝑦): Normal cumulative distribution function
• 𝑆𝛼(𝑦): Student-like cumulative distribution function
• 𝑆𝛼
←, 𝛷←: Inverse functions of 𝑆𝛼 and 𝛷 respectively
Key transformation:
𝑋=
𝑑𝑆𝛼
←∘𝛷(𝑌)
Define sets:
𝐴𝑡={𝑥∈ℝ𝑑:⟨𝐶𝑥,𝑥⟩>𝑡}=√𝑡𝐴1
𝐵𝑡={𝑦∈ℝ𝑑:⟨𝐶𝑆𝛼
←∘𝛷(𝑦),𝑆𝛼
←∘𝛷(𝑦)⟩>𝑡}=𝛷←∘𝑆𝛼(𝐴𝑡)
Important result:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}=∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥𝑖=∫ 𝑒−|𝑦|2/2
(2𝜋)𝑑/2
𝐵𝑡𝑑𝑦
Asymptotic Analysis
Asymptotic Equivalence
Key result:
(𝛷←∘𝑆𝛼(𝑥))2=2𝛼log𝑥−loglog𝑥−2log(𝐾𝑠,𝛼𝛼𝛼/2)−2log(2√𝜋)+𝑜(1)
as 𝑥→∞.
Boundary Parameterization
Lemma 8.2.2: Provides parameterizations for ∂𝐴𝑡 and ∂𝐵𝑡 near specific points.
For ∂𝐴𝑡 near 𝑝𝜖,𝑗,𝑡 =𝜖√𝑡/𝐶𝑗,𝑗𝑒𝑗:
𝑝(𝑣)=𝜖√𝑡
𝐶𝑗,𝑗[1− 1
2𝑡⟨𝐶𝑣,𝑣⟩+𝑂(⟨𝐶𝑣,𝑣⟩
𝑡2)]𝑒𝑗+𝑣
For ∂𝐵𝑡 near 𝑞𝜖,𝑗,𝑡 =𝛷←∘𝑆𝛼(𝑝𝜖,𝑗,𝑡):
𝑞(𝑣)=𝜖√𝛼log 𝑡
𝐶𝑗,𝑗−loglog√𝑡
2√𝛼log𝑡−log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋)
√𝛼log𝑡+𝑜( 1
√log𝑡)𝑒𝑗
+ ∑ 𝛷←
𝑑
𝑖=1,𝑖≠𝑗 ∘𝑆𝛼(𝑣𝑖)𝑒𝑖
Note: These parameterizations are crucial for applying Theorem 5.1 and deriving the
asymptotic behavior.
These lecture notes cover asymptotic analysis of quadratic forms for random vectors with
heavy-tailed distributions.
Asymptotic Behavior of Quadratic Forms
Main Theorem
Theorem 8.2.1: Let 𝑋 be a 𝑑-dimensional random vector with independent and identically
distributed components having a Student-like distribution. Let 𝐶 be a 𝑑×𝑑 matrix. Define
𝐽1={𝑗:𝐶𝑗,𝑗 >0}. If 𝐽1 is not empty, then:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}∼𝐾𝑠,𝛼𝛼(𝛼−1)/2 2
𝑡𝛼/2∑𝐶𝑗,𝑗
𝛼/2
𝑗∈𝐽1
as 𝑡→∞
Key points:
• Applies to Student-like distributions (heavy-tailed)
• Asymptotic behavior depends on positive diagonal elements of 𝐶
• Power law decay with exponent 𝛼/2
Proof Strategy
1. Change of variables: Transform the problem to work with standard normal distribution
2. Apply Theorem 5.1 (not provided in the excerpt) 3. Analyze the boundary of the
transformed set
Change of Variables
Define:
• 𝑌: Random vector with centered normal distribution (identity covariance)
• 𝛷(𝑦): Normal cumulative distribution function
• 𝑆𝛼(𝑦): Student-like cumulative distribution function
• 𝑆𝛼
←, 𝛷←: Inverse functions of 𝑆𝛼 and 𝛷 respectively
Key transformation:
𝑋=
𝑑𝑆𝛼
←∘𝛷(𝑌)
Define sets: 𝐴𝑡={𝑥∈ℝ𝑑:⟨𝐶𝑥,𝑥⟩>𝑡}=√𝑡𝐴1
𝐵𝑡={𝑦∈ℝ𝑑:⟨𝐶𝑆𝛼
←∘𝛷(𝑦),𝑆𝛼
←∘𝛷(𝑦)⟩>𝑡}=𝛷←∘𝑆𝛼(𝐴𝑡)
Important result:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}=∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥𝑖=∫ 𝑒−|𝑦|2/2
(2𝜋)𝑑/2
𝐵𝑡𝑑𝑦
Asymptotic Analysis
Asymptotic Equivalence
Key result:
(𝛷←∘𝑆𝛼(𝑥))2=2𝛼log𝑥−loglog𝑥−2log(𝐾𝑠,𝛼𝛼𝛼/2)−2log(2√𝜋)+𝑜(1)
as 𝑥→∞.
Boundary Parameterization
Lemma 8.2.2: Provides parameterizations for ∂𝐴𝑡 and ∂𝐵𝑡 near specific points.
For ∂𝐴𝑡 near 𝑝𝜖,𝑗,𝑡 =𝜖√𝑡/𝐶𝑗,𝑗𝑒𝑗:
𝑝(𝑣)=𝜖√𝑡
𝐶𝑗,𝑗[1− 1
2𝑡⟨𝐶𝑣,𝑣⟩+𝑂(⟨𝐶𝑣,𝑣⟩
𝑡2)]𝑒𝑗+𝑣
For ∂𝐵𝑡 near 𝑞𝜖,𝑗,𝑡 =𝛷←∘𝑆𝛼(𝑝𝜖,𝑗,𝑡):
𝑞(𝑣)=𝜖√𝛼log 𝑡
𝐶𝑗,𝑗−loglog√𝑡
2√𝛼log𝑡−log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋)
√𝛼log𝑡+𝑜( 1
√log𝑡)𝑒𝑗
+ ∑ 𝛷←
𝑑
𝑖=1,𝑖≠𝑗 ∘𝑆𝛼(𝑣𝑖)𝑒𝑖
Note: These parameterizations are crucial for applying Theorem 5.1 and deriving the
asymptotic behavior.
These lecture notes cover asymptotic analysis of quadratic forms for random vectors with
heavy-tailed distributions.
Asymptotic Behavior of Quadratic Forms
Main Theorem
Theorem 8.2.1: Let 𝑋 be a 𝑑-dimensional random vector with independent and identically
distributed components having a Student-like distribution. Let 𝐶 be a 𝑑×𝑑 matrix. Define
𝐽1={𝑗:𝐶𝑗,𝑗 >0}. If 𝐽1 is not empty, then:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}∼𝐾𝑠,𝛼𝛼(𝛼−1)/2 2
𝑡𝛼/2∑𝐶𝑗,𝑗
𝛼/2
𝑗∈𝐽1
as 𝑡→∞
Key points:
• Applies to Student-like distributions (heavy-tailed)
• Asymptotic behavior depends on positive diagonal elements of 𝐶
• Power law decay with exponent 𝛼/2
Proof Strategy
1. Change of variables: Transform the problem to work with standard normal distribution
2. Apply Theorem 5.1 (not provided in the excerpt) 3. Analyze the boundary of the
transformed set
Change of Variables
Define:
• 𝑌: Random vector with centered normal distribution (identity covariance)
• 𝛷(𝑦): Normal cumulative distribution function
• 𝑆𝛼(𝑦): Student-like cumulative distribution function
• 𝑆𝛼
←, 𝛷←: Inverse functions of 𝑆𝛼 and 𝛷 respectively
Key transformation:
𝑋=
𝑑𝑆𝛼
←∘𝛷(𝑌)
Define sets: 𝐴𝑡={𝑥∈ℝ𝑑:⟨𝐶𝑥,𝑥⟩>𝑡}=√𝑡𝐴1
𝐵𝑡={𝑦∈ℝ𝑑:⟨𝐶𝑆𝛼
←∘𝛷(𝑦),𝑆𝛼
←∘𝛷(𝑦)⟩>𝑡}=𝛷←∘𝑆𝛼(𝐴𝑡)
Important result:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}=∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥𝑖=∫ 𝑒−|𝑦|2/2
(2𝜋)𝑑/2
𝐵𝑡𝑑𝑦
Asymptotic Analysis
Asymptotic Equivalence
Key result:
(𝛷←∘𝑆𝛼(𝑥))2=2𝛼log𝑥−loglog𝑥−2log(𝐾𝑠,𝛼𝛼𝛼/2)−2log(2√𝜋)+𝑜(1)
as 𝑥→∞.
Boundary Parameterization
Lemma 8.2.2: Provides parameterizations for ∂𝐴𝑡 and ∂𝐵𝑡 near specific points.
For ∂𝐴𝑡 near 𝑝𝜖,𝑗,𝑡 =𝜖√𝑡/𝐶𝑗,𝑗𝑒𝑗:
𝑝(𝑣)=𝜖√𝑡
𝐶𝑗,𝑗[1− 1
2𝑡⟨𝐶𝑣,𝑣⟩+𝑂(⟨𝐶𝑣,𝑣⟩
𝑡2)]𝑒𝑗+𝑣
For ∂𝐵𝑡 near 𝑞𝜖,𝑗,𝑡 =𝛷←∘𝑆𝛼(𝑝𝜖,𝑗,𝑡):
𝑞(𝑣)=𝜖√𝛼log 𝑡
𝐶𝑗,𝑗−loglog√𝑡
2√𝛼log𝑡−log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋)
√𝛼log𝑡+𝑜( 1
√log𝑡)𝑒𝑗
+ ∑ 𝛷←
𝑑
𝑖=1,𝑖≠𝑗 ∘𝑆𝛼(𝑣𝑖)𝑒𝑖
Note: These parameterizations are crucial for applying Theorem 5.1 and deriving the
asymptotic behavior.
These lecture notes cover asymptotic analysis of quadratic forms for random vectors with
heavy-tailed distributions.
Asymptotic Behavior of Quadratic Forms
Main Theorem
Theorem 8.2.1: Let 𝑋 be a 𝑑-dimensional random vector with independent and identically
distributed components having a Student-like distribution. Let 𝐶 be a 𝑑×𝑑 matrix. Define
𝐽1={𝑗:𝐶𝑗,𝑗 >0}. If 𝐽1 is not empty, then:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}∼𝐾𝑠,𝛼𝛼(𝛼−1)/2 2
𝑡𝛼/2∑𝐶𝑗,𝑗
𝛼/2
𝑗∈𝐽1
as 𝑡→∞
Key points:
• Applies to Student-like distributions (heavy-tailed)
• Asymptotic behavior depends on positive diagonal elements of 𝐶
• Power law decay with exponent 𝛼/2
Proof Strategy
1. Change of variables: Transform the problem to work with standard normal distribution
2. Apply Theorem 5.1 (not provided in the excerpt) 3. Analyze the boundary of the
transformed set
Change of Variables
Define:
• 𝑌: Random vector with centered normal distribution (identity covariance)
• 𝛷(𝑦): Normal cumulative distribution function
• 𝑆𝛼(𝑦): Student-like cumulative distribution function
• 𝑆𝛼
←, 𝛷←: Inverse functions of 𝑆𝛼 and 𝛷 respectively
Key transformation:
𝑋=
𝑑𝑆𝛼
←∘𝛷(𝑌)
Define sets: 𝐴𝑡={𝑥∈ℝ𝑑:⟨𝐶𝑥,𝑥⟩>𝑡}=√𝑡𝐴1
𝐵𝑡={𝑦∈ℝ𝑑:⟨𝐶𝑆𝛼
←∘𝛷(𝑦),𝑆𝛼
←∘𝛷(𝑦)⟩>𝑡}=𝛷←∘𝑆𝛼(𝐴𝑡)
Important result:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}=∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥𝑖=∫ 𝑒−|𝑦|2/2
(2𝜋)𝑑/2
𝐵𝑡𝑑𝑦
Asymptotic Analysis
Asymptotic Equivalence
Key result:
(𝛷←∘𝑆𝛼(𝑥))2=2𝛼log𝑥−loglog𝑥−2log(𝐾𝑠,𝛼𝛼𝛼/2)−2log(2√𝜋)+𝑜(1)
as 𝑥→∞.
Boundary Parameterization
Lemma 8.2.2: Provides parameterizations for ∂𝐴𝑡 and ∂𝐵𝑡 near specific points.
For ∂𝐴𝑡 near 𝑝𝜖,𝑗,𝑡 =𝜖√𝑡/𝐶𝑗,𝑗𝑒𝑗:
𝑝(𝑣)=𝜖√𝑡
𝐶𝑗,𝑗[1− 1
2𝑡⟨𝐶𝑣,𝑣⟩+𝑂(⟨𝐶𝑣,𝑣⟩
𝑡2)]𝑒𝑗+𝑣
For ∂𝐵𝑡 near 𝑞𝜖,𝑗,𝑡 =𝛷←∘𝑆𝛼(𝑝𝜖,𝑗,𝑡):
𝑞(𝑣)=𝜖√𝛼log 𝑡
𝐶𝑗,𝑗−loglog√𝑡
2√𝛼log𝑡−log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋)
√𝛼log𝑡+𝑜( 1
√log𝑡)𝑒𝑗
+ ∑ 𝛷←
𝑑
𝑖=1,𝑖≠𝑗 ∘𝑆𝛼(𝑣𝑖)𝑒𝑖
Note: These parameterizations are crucial for applying Theorem 5.1 and deriving the
asymptotic behavior.
These lecture notes cover asymptotic analysis of quadratic forms for random vectors with
heavy-tailed distributions.
Asymptotic Behavior of Quadratic Forms
Main Theorem
Theorem 8.2.1: Let 𝑋 be a 𝑑-dimensional random vector with independent and identically
distributed components having a Student-like distribution. Let 𝐶 be a 𝑑×𝑑 matrix. Define
𝐽1={𝑗:𝐶𝑗,𝑗 >0}. If 𝐽1 is not empty, then:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}∼𝐾𝑠,𝛼𝛼(𝛼−1)/2 2
𝑡𝛼/2∑𝐶𝑗,𝑗
𝛼/2
𝑗∈𝐽1
as 𝑡→∞
Key points:
• Applies to Student-like distributions (heavy-tailed)
• Asymptotic behavior depends on positive diagonal elements of 𝐶
• Power law decay with exponent 𝛼/2
Proof Strategy
1. Change of variables: Transform the problem to work with standard normal distribution
2. Apply Theorem 5.1 (not provided in the excerpt) 3. Analyze the boundary of the
transformed set
Change of Variables
Define:
• 𝑌: Random vector with centered normal distribution (identity covariance)
• 𝛷(𝑦): Normal cumulative distribution function
• 𝑆𝛼(𝑦): Student-like cumulative distribution function
• 𝑆𝛼
←, 𝛷←: Inverse functions of 𝑆𝛼 and 𝛷 respectively
Key transformation:
𝑋=
𝑑𝑆𝛼
←∘𝛷(𝑌)
Define sets: 𝐴𝑡={𝑥∈ℝ𝑑:⟨𝐶𝑥,𝑥⟩>𝑡}=√𝑡𝐴1
𝐵𝑡={𝑦∈ℝ𝑑:⟨𝐶𝑆𝛼
←∘𝛷(𝑦),𝑆𝛼
←∘𝛷(𝑦)⟩>𝑡}=𝛷←∘𝑆𝛼(𝐴𝑡)
Important result:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}=∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥𝑖=∫ 𝑒−|𝑦|2/2
(2𝜋)𝑑/2
𝐵𝑡𝑑𝑦
Asymptotic Analysis
Asymptotic Equivalence
Key result:
(𝛷←∘𝑆𝛼(𝑥))2=2𝛼log𝑥−loglog𝑥−2log(𝐾𝑠,𝛼𝛼𝛼/2)−2log(2√𝜋)+𝑜(1)
as 𝑥→∞.
Boundary Parameterization
Lemma 8.2.2: Provides parameterizations for ∂𝐴𝑡 and ∂𝐵𝑡 near specific points.
For ∂𝐴𝑡 near 𝑝𝜖,𝑗,𝑡 =𝜖√𝑡/𝐶𝑗,𝑗𝑒𝑗:
𝑝(𝑣)=𝜖√𝑡
𝐶𝑗,𝑗[1− 1
2𝑡⟨𝐶𝑣,𝑣⟩+𝑂(⟨𝐶𝑣,𝑣⟩
𝑡2)]𝑒𝑗+𝑣
For ∂𝐵𝑡 near 𝑞𝜖,𝑗,𝑡 =𝛷←∘𝑆𝛼(𝑝𝜖,𝑗,𝑡):
𝑞(𝑣)=𝜖√𝛼log 𝑡
𝐶𝑗,𝑗−loglog√𝑡
2√𝛼log𝑡−log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋)
√𝛼log𝑡+𝑜( 1
√log𝑡)𝑒𝑗
+ ∑ 𝛷←
𝑑
𝑖=1,𝑖≠𝑗 ∘𝑆𝛼(𝑣𝑖)𝑒𝑖
Note: These parameterizations are crucial for applying Theorem 5.1 and deriving the
asymptotic behavior.
These lecture notes cover asymptotic analysis of quadratic forms for random vectors with
heavy-tailed distributions.
Asymptotic Behavior of Quadratic Forms
Main Theorem
Theorem 8.2.1: Let 𝑋 be a 𝑑-dimensional random vector with independent and identically
distributed components having a Student-like distribution. Let 𝐶 be a 𝑑×𝑑 matrix. Define
𝐽1={𝑗:𝐶𝑗,𝑗 >0}. If 𝐽1 is not empty, then:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}∼𝐾𝑠,𝛼𝛼(𝛼−1)/2 2
𝑡𝛼/2∑𝐶𝑗,𝑗
𝛼/2
𝑗∈𝐽1
as 𝑡→∞
Key points:
• Applies to Student-like distributions (heavy-tailed)
• Asymptotic behavior depends on positive diagonal elements of 𝐶
• Power law decay with exponent 𝛼/2
Proof Strategy
1. Change of variables: Transform the problem to work with standard normal distribution
2. Apply Theorem 5.1 (not provided in the excerpt) 3. Analyze the boundary of the
transformed set
Change of Variables
Define:
• 𝑌: Random vector with centered normal distribution (identity covariance)
• 𝛷(𝑦): Normal cumulative distribution function
• 𝑆𝛼(𝑦): Student-like cumulative distribution function
• 𝑆𝛼
←, 𝛷←: Inverse functions of 𝑆𝛼 and 𝛷 respectively
Key transformation:
𝑋=
𝑑𝑆𝛼
←∘𝛷(𝑌)
Define sets: 𝐴𝑡={𝑥∈ℝ𝑑:⟨𝐶𝑥,𝑥⟩>𝑡}=√𝑡𝐴1
𝐵𝑡={𝑦∈ℝ𝑑:⟨𝐶𝑆𝛼
←∘𝛷(𝑦),𝑆𝛼
←∘𝛷(𝑦)⟩>𝑡}=𝛷←∘𝑆𝛼(𝐴𝑡)
Important result:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}=∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥𝑖=∫ 𝑒−|𝑦|2/2
(2𝜋)𝑑/2
𝐵𝑡𝑑𝑦
Asymptotic Analysis
Asymptotic Equivalence
Key result:
(𝛷←∘𝑆𝛼(𝑥))2=2𝛼log𝑥−loglog𝑥−2log(𝐾𝑠,𝛼𝛼𝛼/2)−2log(2√𝜋)+𝑜(1)
as 𝑥→∞.
Boundary Parameterization
Lemma 8.2.2: Provides parameterizations for ∂𝐴𝑡 and ∂𝐵𝑡 near specific points.
For ∂𝐴𝑡 near 𝑝𝜖,𝑗,𝑡 =𝜖√𝑡/𝐶𝑗,𝑗𝑒𝑗:
𝑝(𝑣)=𝜖√𝑡
𝐶𝑗,𝑗[1− 1
2𝑡⟨𝐶𝑣,𝑣⟩+𝑂(⟨𝐶𝑣,𝑣⟩
𝑡2)]𝑒𝑗+𝑣
For ∂𝐵𝑡 near 𝑞𝜖,𝑗,𝑡 =𝛷←∘𝑆𝛼(𝑝𝜖,𝑗,𝑡):
𝑞(𝑣)=𝜖√𝛼log 𝑡
𝐶𝑗,𝑗−loglog√𝑡
2√𝛼log𝑡−log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋)
√𝛼log𝑡+𝑜( 1
√log𝑡)𝑒𝑗
+ ∑ 𝛷←
𝑑
𝑖=1,𝑖≠𝑗 ∘𝑆𝛼(𝑣𝑖)𝑒𝑖
Note: These parameterizations are crucial for applying Theorem 5.1 and deriving the
asymptotic behavior.
These lecture notes cover asymptotic analysis of quadratic forms for random vectors with
heavy-tailed distributions.
Asymptotic Behavior of Quadratic Forms
Main Theorem
Theorem 8.2.1: Let 𝑋 be a 𝑑-dimensional random vector with independent and identically
distributed components having a Student-like distribution. Let 𝐶 be a 𝑑×𝑑 matrix. Define
𝐽1={𝑗:𝐶𝑗,𝑗 >0}. If 𝐽1 is not empty, then:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}∼𝐾𝑠,𝛼𝛼(𝛼−1)/2 2
𝑡𝛼/2∑𝐶𝑗,𝑗
𝛼/2
𝑗∈𝐽1
as 𝑡→∞
Key points:
• Applies to Student-like distributions (heavy-tailed)
• Asymptotic behavior depends on positive diagonal elements of 𝐶
• Power law decay with exponent 𝛼/2
Proof Strategy
1. Change of variables: Transform the problem to work with standard normal distribution
2. Apply Theorem 5.1 (not provided in the excerpt) 3. Analyze the boundary of the
transformed set
Change of Variables
Define:
• 𝑌: Random vector with centered normal distribution (identity covariance)
• 𝛷(𝑦): Normal cumulative distribution function
• 𝑆𝛼(𝑦): Student-like cumulative distribution function
• 𝑆𝛼
←, 𝛷←: Inverse functions of 𝑆𝛼 and 𝛷 respectively
Key transformation:
𝑋=
𝑑𝑆𝛼
←∘𝛷(𝑌)
Define sets: 𝐴𝑡={𝑥∈ℝ𝑑:⟨𝐶𝑥,𝑥⟩>𝑡}=√𝑡𝐴1
𝐵𝑡={𝑦∈ℝ𝑑:⟨𝐶𝑆𝛼
←∘𝛷(𝑦),𝑆𝛼
←∘𝛷(𝑦)⟩>𝑡}=𝛷←∘𝑆𝛼(𝐴𝑡)
Important result:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}=∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥𝑖=∫ 𝑒−|𝑦|2/2
(2𝜋)𝑑/2
𝐵𝑡𝑑𝑦
Asymptotic Analysis
Asymptotic Equivalence
Key result:
(𝛷←∘𝑆𝛼(𝑥))2=2𝛼log𝑥−loglog𝑥−2log(𝐾𝑠,𝛼𝛼𝛼/2)−2log(2√𝜋)+𝑜(1)
as 𝑥→∞.
Boundary Parameterization
Lemma 8.2.2: Provides parameterizations for ∂𝐴𝑡 and ∂𝐵𝑡 near specific points.
For ∂𝐴𝑡 near 𝑝𝜖,𝑗,𝑡 =𝜖√𝑡/𝐶𝑗,𝑗𝑒𝑗:
𝑝(𝑣)=𝜖√𝑡
𝐶𝑗,𝑗[1− 1
2𝑡⟨𝐶𝑣,𝑣⟩+𝑂(⟨𝐶𝑣,𝑣⟩
𝑡2)]𝑒𝑗+𝑣
For ∂𝐵𝑡 near 𝑞𝜖,𝑗,𝑡 =𝛷←∘𝑆𝛼(𝑝𝜖,𝑗,𝑡):
𝑞(𝑣)=𝜖√𝛼log 𝑡
𝐶𝑗,𝑗−loglog√𝑡
2√𝛼log𝑡−log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋)
√𝛼log𝑡+𝑜( 1
√log𝑡)𝑒𝑗
+ ∑ 𝛷←
𝑑
𝑖=1,𝑖≠𝑗 ∘𝑆𝛼(𝑣𝑖)𝑒𝑖
Note: These parameterizations are crucial for applying Theorem 5.1 and deriving the
asymptotic behavior.
These lecture notes cover asymptotic analysis of quadratic forms for random vectors with
heavy-tailed distributions.
Asymptotic Behavior of Quadratic Forms
Main Theorem
Theorem 8.2.1: Let 𝑋 be a 𝑑-dimensional random vector with independent and identically
distributed components having a Student-like distribution. Let 𝐶 be a 𝑑×𝑑 matrix. Define
𝐽1={𝑗:𝐶𝑗,𝑗 >0}. If 𝐽1 is not empty, then:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}∼𝐾𝑠,𝛼𝛼(𝛼−1)/2 2
𝑡𝛼/2∑𝐶𝑗,𝑗
𝛼/2
𝑗∈𝐽1
as 𝑡→∞
Key points:
• Applies to Student-like distributions (heavy-tailed)
• Asymptotic behavior depends on positive diagonal elements of 𝐶
• Power law decay with exponent 𝛼/2
Proof Strategy
1. Change of variables: Transform the problem to work with standard normal distribution
2. Apply Theorem 5.1 (not provided in the excerpt) 3. Analyze the boundary of the
transformed set
Change of Variables
Define:
• 𝑌: Random vector with centered normal distribution (identity covariance)
• 𝛷(𝑦): Normal cumulative distribution function
• 𝑆𝛼(𝑦): Student-like cumulative distribution function
• 𝑆𝛼
←, 𝛷←: Inverse functions of 𝑆𝛼 and 𝛷 respectively
Key transformation:
𝑋=
𝑑𝑆𝛼
←∘𝛷(𝑌)
Define sets:
𝐴𝑡={𝑥∈ℝ𝑑:⟨𝐶𝑥,𝑥⟩>𝑡}=√𝑡𝐴1
𝐵𝑡={𝑦∈ℝ𝑑:⟨𝐶𝑆𝛼
←∘𝛷(𝑦),𝑆𝛼
←∘𝛷(𝑦)⟩>𝑡}=𝛷←∘𝑆𝛼(𝐴𝑡)
Important result:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}=∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥𝑖=∫ 𝑒−|𝑦|2/2
(2𝜋)𝑑/2
𝐵𝑡𝑑𝑦
Asymptotic Analysis
Asymptotic Equivalence
Key result:
(𝛷←∘𝑆𝛼(𝑥))2=2𝛼log𝑥−loglog𝑥−2log(𝐾𝑠,𝛼𝛼𝛼/2)−2log(2√𝜋)+𝑜(1)
as 𝑥→∞.
Boundary Parameterization
Lemma 8.2.2: Provides parameterizations for ∂𝐴𝑡 and ∂𝐵𝑡 near specific points.
For ∂𝐴𝑡 near 𝑝𝜖,𝑗,𝑡 =𝜖√𝑡/𝐶𝑗,𝑗𝑒𝑗:
𝑝(𝑣)=𝜖√𝑡
𝐶𝑗,𝑗[1− 1
2𝑡⟨𝐶𝑣,𝑣⟩+𝑂(⟨𝐶𝑣,𝑣⟩
𝑡2)]𝑒𝑗+𝑣
For ∂𝐵𝑡 near 𝑞𝜖,𝑗,𝑡 =𝛷←∘𝑆𝛼(𝑝𝜖,𝑗,𝑡):
𝑞(𝑣)=𝜖√𝛼log 𝑡
𝐶𝑗,𝑗−loglog√𝑡
2√𝛼log𝑡−log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋)
√𝛼log𝑡+𝑜( 1
√log𝑡)𝑒𝑗
+ ∑ 𝛷←
𝑑
𝑖=1,𝑖≠𝑗 ∘𝑆𝛼(𝑣𝑖)𝑒𝑖
Note: These parameterizations are crucial for applying Theorem 5.1 and deriving the
asymptotic behavior.
These lecture notes cover asymptotic analysis of quadratic forms for random vectors with
heavy-tailed distributions.
Asymptotic Behavior of Quadratic Forms
Main Theorem
Theorem 8.2.1: Let 𝑋 be a 𝑑-dimensional random vector with independent and identically
distributed components having a Student-like distribution. Let 𝐶 be a 𝑑×𝑑 matrix. Define
𝐽1={𝑗:𝐶𝑗,𝑗 >0}. If 𝐽1 is not empty, then:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}∼𝐾𝑠,𝛼𝛼(𝛼−1)/2 2
𝑡𝛼/2∑𝐶𝑗,𝑗
𝛼/2
𝑗∈𝐽1
as 𝑡→∞
Key points:
• Applies to Student-like distributions (heavy-tailed)
• Asymptotic behavior depends on positive diagonal elements of 𝐶
• Power law decay with exponent 𝛼/2
Proof Strategy
1. Change of variables: Transform the problem to work with standard normal distribution
2. Apply Theorem 5.1 (not provided in the excerpt) 3. Analyze the boundary of the
transformed set
Change of Variables
Define:
• 𝑌: Random vector with centered normal distribution (identity covariance)
• 𝛷(𝑦): Normal cumulative distribution function
• 𝑆𝛼(𝑦): Student-like cumulative distribution function
• 𝑆𝛼
←, 𝛷←: Inverse functions of 𝑆𝛼 and 𝛷 respectively
Key transformation:
𝑋=
𝑑𝑆𝛼
←∘𝛷(𝑌)
Define sets: 𝐴𝑡={𝑥∈ℝ𝑑:⟨𝐶𝑥,𝑥⟩>𝑡}=√𝑡𝐴1
𝐵𝑡={𝑦∈ℝ𝑑:⟨𝐶𝑆𝛼
←∘𝛷(𝑦),𝑆𝛼
←∘𝛷(𝑦)⟩>𝑡}=𝛷←∘𝑆𝛼(𝐴𝑡)
Important result:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}=∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥𝑖=∫ 𝑒−|𝑦|2/2
(2𝜋)𝑑/2
𝐵𝑡𝑑𝑦
Asymptotic Analysis
Asymptotic Equivalence
Key result:
(𝛷←∘𝑆𝛼(𝑥))2=2𝛼log𝑥−loglog𝑥−2log(𝐾𝑠,𝛼𝛼𝛼/2)−2log(2√𝜋)+𝑜(1)
as 𝑥→∞.
Boundary Parameterization
Lemma 8.2.2: Provides parameterizations for ∂𝐴𝑡 and ∂𝐵𝑡 near specific points.
For ∂𝐴𝑡 near 𝑝𝜖,𝑗,𝑡 =𝜖√𝑡/𝐶𝑗,𝑗𝑒𝑗:
𝑝(𝑣)=𝜖√𝑡
𝐶𝑗,𝑗[1− 1
2𝑡⟨𝐶𝑣,𝑣⟩+𝑂(⟨𝐶𝑣,𝑣⟩
𝑡2)]𝑒𝑗+𝑣
For ∂𝐵𝑡 near 𝑞𝜖,𝑗,𝑡 =𝛷←∘𝑆𝛼(𝑝𝜖,𝑗,𝑡):
𝑞(𝑣)=𝜖√𝛼log 𝑡
𝐶𝑗,𝑗−loglog√𝑡
2√𝛼log𝑡−log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋)
√𝛼log𝑡+𝑜( 1
√log𝑡)𝑒𝑗
+ ∑ 𝛷←
𝑑
𝑖=1,𝑖≠𝑗 ∘𝑆𝛼(𝑣𝑖)𝑒𝑖
Note: These parameterizations are crucial for applying Theorem 5.1 and deriving the
asymptotic behavior.
These lecture notes cover asymptotic analysis of quadratic forms for random vectors with
heavy-tailed distributions.
Asymptotic Behavior of Quadratic Forms
Main Theorem
Theorem 8.2.1: Let 𝑋 be a 𝑑-dimensional random vector with independent and identically
distributed components having a Student-like distribution. Let 𝐶 be a 𝑑×𝑑 matrix. Define
𝐽1={𝑗:𝐶𝑗,𝑗 >0}. If 𝐽1 is not empty, then:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}∼𝐾𝑠,𝛼𝛼(𝛼−1)/2 2
𝑡𝛼/2∑𝐶𝑗,𝑗
𝛼/2
𝑗∈𝐽1
as 𝑡→∞
Key points:
• Applies to Student-like distributions (heavy-tailed)
• Asymptotic behavior depends on positive diagonal elements of 𝐶
• Power law decay with exponent 𝛼/2
Proof Strategy
1. Change of variables: Transform the problem to work with standard normal distribution
2. Apply Theorem 5.1 (not provided in the excerpt) 3. Analyze the boundary of the
transformed set
Change of Variables
Define:
• 𝑌: Random vector with centered normal distribution (identity covariance)
• 𝛷(𝑦): Normal cumulative distribution function
• 𝑆𝛼(𝑦): Student-like cumulative distribution function
• 𝑆𝛼
←, 𝛷←: Inverse functions of 𝑆𝛼 and 𝛷 respectively
Key transformation:
𝑋=
𝑑𝑆𝛼
←∘𝛷(𝑌)
Define sets: 𝐴𝑡={𝑥∈ℝ𝑑:⟨𝐶𝑥,𝑥⟩>𝑡}=√𝑡𝐴1
𝐵𝑡={𝑦∈ℝ𝑑:⟨𝐶𝑆𝛼
←∘𝛷(𝑦),𝑆𝛼
←∘𝛷(𝑦)⟩>𝑡}=𝛷←∘𝑆𝛼(𝐴𝑡)
Important result:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}=∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥𝑖=∫ 𝑒−|𝑦|2/2
(2𝜋)𝑑/2
𝐵𝑡𝑑𝑦
Asymptotic Analysis
Asymptotic Equivalence
Key result:
(𝛷←∘𝑆𝛼(𝑥))2=2𝛼log𝑥−loglog𝑥−2log(𝐾𝑠,𝛼𝛼𝛼/2)−2log(2√𝜋)+𝑜(1)
as 𝑥→∞.
Boundary Parameterization
Lemma 8.2.2: Provides parameterizations for ∂𝐴𝑡 and ∂𝐵𝑡 near specific points.
For ∂𝐴𝑡 near 𝑝𝜖,𝑗,𝑡 =𝜖√𝑡/𝐶𝑗,𝑗𝑒𝑗:
𝑝(𝑣)=𝜖√𝑡
𝐶𝑗,𝑗[1− 1
2𝑡⟨𝐶𝑣,𝑣⟩+𝑂(⟨𝐶𝑣,𝑣⟩
𝑡2)]𝑒𝑗+𝑣
For ∂𝐵𝑡 near 𝑞𝜖,𝑗,𝑡 =𝛷←∘𝑆𝛼(𝑝𝜖,𝑗,𝑡):
𝑞(𝑣)=𝜖√𝛼log 𝑡
𝐶𝑗,𝑗−loglog√𝑡
2√𝛼log𝑡−log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋)
√𝛼log𝑡+𝑜( 1
√log𝑡)𝑒𝑗
+ ∑ 𝛷←
𝑑
𝑖=1,𝑖≠𝑗 ∘𝑆𝛼(𝑣𝑖)𝑒𝑖
Note: These parameterizations are crucial for applying Theorem 5.1 and deriving the
asymptotic behavior.
These lecture notes cover asymptotic analysis of quadratic forms for random vectors with
heavy-tailed distributions.
Asymptotic Behavior of Quadratic Forms
Main Theorem
Theorem 8.2.1: Let 𝑋 be a 𝑑-dimensional random vector with independent and identically
distributed components having a Student-like distribution. Let 𝐶 be a 𝑑×𝑑 matrix. Define
𝐽1={𝑗:𝐶𝑗,𝑗 >0}. If 𝐽1 is not empty, then:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}∼𝐾𝑠,𝛼𝛼(𝛼−1)/2 2
𝑡𝛼/2∑𝐶𝑗,𝑗
𝛼/2
𝑗∈𝐽1
as 𝑡→∞
Key points:
• Applies to Student-like distributions (heavy-tailed)
• Asymptotic behavior depends on positive diagonal elements of 𝐶
• Power law decay with exponent 𝛼/2
Proof Strategy
1. Change of variables: Transform the problem to work with standard normal distribution
2. Apply Theorem 5.1 (not provided in the excerpt) 3. Analyze the boundary of the
transformed set
Change of Variables
Define:
• 𝑌: Random vector with centered normal distribution (identity covariance)
• 𝛷(𝑦): Normal cumulative distribution function
• 𝑆𝛼(𝑦): Student-like cumulative distribution function
• 𝑆𝛼
←, 𝛷←: Inverse functions of 𝑆𝛼 and 𝛷 respectively
Key transformation:
𝑋=
𝑑𝑆𝛼
←∘𝛷(𝑌)
Define sets: 𝐴𝑡={𝑥∈ℝ𝑑:⟨𝐶𝑥,𝑥⟩>𝑡}=√𝑡𝐴1
𝐵𝑡={𝑦∈ℝ𝑑:⟨𝐶𝑆𝛼
←∘𝛷(𝑦),𝑆𝛼
←∘𝛷(𝑦)⟩>𝑡}=𝛷←∘𝑆𝛼(𝐴𝑡)
Important result:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}=∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥𝑖=∫ 𝑒−|𝑦|2/2
(2𝜋)𝑑/2
𝐵𝑡𝑑𝑦
Asymptotic Analysis
Asymptotic Equivalence
Key result:
(𝛷←∘𝑆𝛼(𝑥))2=2𝛼log𝑥−loglog𝑥−2log(𝐾𝑠,𝛼𝛼𝛼/2)−2log(2√𝜋)+𝑜(1)
as 𝑥→∞.
Boundary Parameterization
Lemma 8.2.2: Provides parameterizations for ∂𝐴𝑡 and ∂𝐵𝑡 near specific points.
For ∂𝐴𝑡 near 𝑝𝜖,𝑗,𝑡 =𝜖√𝑡/𝐶𝑗,𝑗𝑒𝑗:
𝑝(𝑣)=𝜖√𝑡
𝐶𝑗,𝑗[1− 1
2𝑡⟨𝐶𝑣,𝑣⟩+𝑂(⟨𝐶𝑣,𝑣⟩
𝑡2)]𝑒𝑗+𝑣
For ∂𝐵𝑡 near 𝑞𝜖,𝑗,𝑡 =𝛷←∘𝑆𝛼(𝑝𝜖,𝑗,𝑡):
𝑞(𝑣)=𝜖√𝛼log 𝑡
𝐶𝑗,𝑗−loglog√𝑡
2√𝛼log𝑡−log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋)
√𝛼log𝑡+𝑜( 1
√log𝑡)𝑒𝑗
+ ∑ 𝛷←
𝑑
𝑖=1,𝑖≠𝑗 ∘𝑆𝛼(𝑣𝑖)𝑒𝑖
Note: These parameterizations are crucial for applying Theorem 5.1 and deriving the
asymptotic behavior.
These lecture notes cover asymptotic analysis of quadratic forms for random vectors with
heavy-tailed distributions.
Asymptotic Behavior of Quadratic Forms
Main Theorem
Theorem 8.2.1: Let 𝑋 be a 𝑑-dimensional random vector with independent and identically
distributed components having a Student-like distribution. Let 𝐶 be a 𝑑×𝑑 matrix. Define
𝐽1={𝑗:𝐶𝑗,𝑗 >0}. If 𝐽1 is not empty, then:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}∼𝐾𝑠,𝛼𝛼(𝛼−1)/2 2
𝑡𝛼/2∑𝐶𝑗,𝑗
𝛼/2
𝑗∈𝐽1
as 𝑡→∞
Key points:
• Applies to Student-like distributions (heavy-tailed)
• Asymptotic behavior depends on positive diagonal elements of 𝐶
• Power law decay with exponent 𝛼/2
Proof Strategy
1. Change of variables: Transform the problem to work with standard normal distribution
2. Apply Theorem 5.1 (not provided in the excerpt) 3. Analyze the boundary of the
transformed set
Change of Variables
Define:
• 𝑌: Random vector with centered normal distribution (identity covariance)
• 𝛷(𝑦): Normal cumulative distribution function
• 𝑆𝛼(𝑦): Student-like cumulative distribution function
• 𝑆𝛼
←, 𝛷←: Inverse functions of 𝑆𝛼 and 𝛷 respectively
Key transformation:
𝑋=
𝑑𝑆𝛼
←∘𝛷(𝑌)
Define sets: 𝐴𝑡={𝑥∈ℝ𝑑:⟨𝐶𝑥,𝑥⟩>𝑡}=√𝑡𝐴1
𝐵𝑡={𝑦∈ℝ𝑑:⟨𝐶𝑆𝛼
←∘𝛷(𝑦),𝑆𝛼
←∘𝛷(𝑦)⟩>𝑡}=𝛷←∘𝑆𝛼(𝐴𝑡)
Important result:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}=∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥𝑖=∫ 𝑒−|𝑦|2/2
(2𝜋)𝑑/2
𝐵𝑡𝑑𝑦
Asymptotic Analysis
Asymptotic Equivalence
Key result:
(𝛷←∘𝑆𝛼(𝑥))2=2𝛼log𝑥−loglog𝑥−2log(𝐾𝑠,𝛼𝛼𝛼/2)−2log(2√𝜋)+𝑜(1)
as 𝑥→∞.
Boundary Parameterization
Lemma 8.2.2: Provides parameterizations for ∂𝐴𝑡 and ∂𝐵𝑡 near specific points.
For ∂𝐴𝑡 near 𝑝𝜖,𝑗,𝑡 =𝜖√𝑡/𝐶𝑗,𝑗𝑒𝑗:
𝑝(𝑣)=𝜖√𝑡
𝐶𝑗,𝑗[1− 1
2𝑡⟨𝐶𝑣,𝑣⟩+𝑂(⟨𝐶𝑣,𝑣⟩
𝑡2)]𝑒𝑗+𝑣
For ∂𝐵𝑡 near 𝑞𝜖,𝑗,𝑡 =𝛷←∘𝑆𝛼(𝑝𝜖,𝑗,𝑡):
𝑞(𝑣)=𝜖√𝛼log 𝑡
𝐶𝑗,𝑗−loglog√𝑡
2√𝛼log𝑡−log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋)
√𝛼log𝑡+𝑜( 1
√log𝑡)𝑒𝑗
+ ∑ 𝛷←
𝑑
𝑖=1,𝑖≠𝑗 ∘𝑆𝛼(𝑣𝑖)𝑒𝑖
Note: These parameterizations are crucial for applying Theorem 5.1 and deriving the
asymptotic behavior.
These lecture notes cover asymptotic analysis of quadratic forms for random vectors with
heavy-tailed distributions.
Asymptotic Behavior of Quadratic Forms
Main Theorem
Theorem 8.2.1: Let 𝑋 be a 𝑑-dimensional random vector with independent and identically
distributed components having a Student-like distribution. Let 𝐶 be a 𝑑×𝑑 matrix. Define
𝐽1={𝑗:𝐶𝑗,𝑗 >0}. If 𝐽1 is not empty, then:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}∼𝐾𝑠,𝛼𝛼(𝛼−1)/2 2
𝑡𝛼/2∑𝐶𝑗,𝑗
𝛼/2
𝑗∈𝐽1
as 𝑡→∞
Key points:
• Applies to Student-like distributions (heavy-tailed)
• Asymptotic behavior depends on positive diagonal elements of 𝐶
• Power law decay with exponent 𝛼/2
Proof Strategy
1. Change of variables: Transform the problem to work with standard normal distribution
2. Apply Theorem 5.1 (not provided in the excerpt) 3. Analyze the boundary of the
transformed set
Change of Variables
Define:
• 𝑌: Random vector with centered normal distribution (identity covariance)
• 𝛷(𝑦): Normal cumulative distribution function
• 𝑆𝛼(𝑦): Student-like cumulative distribution function
• 𝑆𝛼
←, 𝛷←: Inverse functions of 𝑆𝛼 and 𝛷 respectively
Key transformation:
𝑋=
𝑑𝑆𝛼
←∘𝛷(𝑌)
Define sets: 𝐴𝑡={𝑥∈ℝ𝑑:⟨𝐶𝑥,𝑥⟩>𝑡}=√𝑡𝐴1
𝐵𝑡={𝑦∈ℝ𝑑:⟨𝐶𝑆𝛼
←∘𝛷(𝑦),𝑆𝛼
←∘𝛷(𝑦)⟩>𝑡}=𝛷←∘𝑆𝛼(𝐴𝑡)
Important result:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}=∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥𝑖=∫ 𝑒−|𝑦|2/2
(2𝜋)𝑑/2
𝐵𝑡𝑑𝑦
Asymptotic Analysis
Asymptotic Equivalence
Key result:
(𝛷←∘𝑆𝛼(𝑥))2=2𝛼log𝑥−loglog𝑥−2log(𝐾𝑠,𝛼𝛼𝛼/2)−2log(2√𝜋)+𝑜(1)
as 𝑥→∞.
Boundary Parameterization
Lemma 8.2.2: Provides parameterizations for ∂𝐴𝑡 and ∂𝐵𝑡 near specific points.
For ∂𝐴𝑡 near 𝑝𝜖,𝑗,𝑡 =𝜖√𝑡/𝐶𝑗,𝑗𝑒𝑗:
𝑝(𝑣)=𝜖√𝑡
𝐶𝑗,𝑗[1− 1
2𝑡⟨𝐶𝑣,𝑣⟩+𝑂(⟨𝐶𝑣,𝑣⟩
𝑡2)]𝑒𝑗+𝑣
For ∂𝐵𝑡 near 𝑞𝜖,𝑗,𝑡 =𝛷←∘𝑆𝛼(𝑝𝜖,𝑗,𝑡):
𝑞(𝑣)=𝜖√𝛼log 𝑡
𝐶𝑗,𝑗−loglog√𝑡
2√𝛼log𝑡−log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋)
√𝛼log𝑡+𝑜( 1
√log𝑡)𝑒𝑗
+ ∑ 𝛷←
𝑑
𝑖=1,𝑖≠𝑗 ∘𝑆𝛼(𝑣𝑖)𝑒𝑖
Note: These parameterizations are crucial for applying Theorem 5.1 and deriving the
asymptotic behavior.
These lecture notes cover asymptotic analysis of quadratic forms for random vectors with
heavy-tailed distributions.
Asymptotic Behavior of Quadratic Forms
Main Theorem
Theorem 8.2.1: Let 𝑋 be a 𝑑-dimensional random vector with independent and identically
distributed components having a Student-like distribution. Let 𝐶 be a 𝑑×𝑑 matrix. Define
𝐽1={𝑗:𝐶𝑗,𝑗 >0}. If 𝐽1 is not empty, then:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}∼𝐾𝑠,𝛼𝛼(𝛼−1)/2 2
𝑡𝛼/2∑𝐶𝑗,𝑗
𝛼/2
𝑗∈𝐽1
as 𝑡→∞
Key points:
• Applies to Student-like distributions (heavy-tailed)
• Asymptotic behavior depends on positive diagonal elements of 𝐶
• Power law decay with exponent 𝛼/2
Proof Strategy
1. Change of variables: Transform the problem to work with standard normal distribution
2. Apply Theorem 5.1 (not provided in the excerpt) 3. Analyze the boundary of the
transformed set
Change of Variables
Define:
• 𝑌: Random vector with centered normal distribution (identity covariance)
• 𝛷(𝑦): Normal cumulative distribution function
• 𝑆𝛼(𝑦): Student-like cumulative distribution function
• 𝑆𝛼
←, 𝛷←: Inverse functions of 𝑆𝛼 and 𝛷 respectively
Key transformation:
𝑋=
𝑑𝑆𝛼
←∘𝛷(𝑌)
Define sets: 𝐴𝑡={𝑥∈ℝ𝑑:⟨𝐶𝑥,𝑥⟩>𝑡}=√𝑡𝐴1
𝐵𝑡={𝑦∈ℝ𝑑:⟨𝐶𝑆𝛼
←∘𝛷(𝑦),𝑆𝛼
←∘𝛷(𝑦)⟩>𝑡}=𝛷←∘𝑆𝛼(𝐴𝑡)
Important result:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}=∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥𝑖=∫ 𝑒−|𝑦|2/2
(2𝜋)𝑑/2
𝐵𝑡𝑑𝑦
Asymptotic Analysis
Asymptotic Equivalence
Key result:
(𝛷←∘𝑆𝛼(𝑥))2=2𝛼log𝑥−loglog𝑥−2log(𝐾𝑠,𝛼𝛼𝛼/2)−2log(2√𝜋)+𝑜(1)
as 𝑥→∞.
Boundary Parameterization
Lemma 8.2.2: Provides parameterizations for ∂𝐴𝑡 and ∂𝐵𝑡 near specific points.
For ∂𝐴𝑡 near 𝑝𝜖,𝑗,𝑡 =𝜖√𝑡/𝐶𝑗,𝑗𝑒𝑗:
𝑝(𝑣)=𝜖√𝑡
𝐶𝑗,𝑗[1− 1
2𝑡⟨𝐶𝑣,𝑣⟩+𝑂(⟨𝐶𝑣,𝑣⟩
𝑡2)]𝑒𝑗+𝑣
For ∂𝐵𝑡 near 𝑞𝜖,𝑗,𝑡 =𝛷←∘𝑆𝛼(𝑝𝜖,𝑗,𝑡):
𝑞(𝑣)=𝜖√𝛼log 𝑡
𝐶𝑗,𝑗−loglog√𝑡
2√𝛼log𝑡−log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋)
√𝛼log𝑡+𝑜( 1
√log𝑡)𝑒𝑗
+ ∑ 𝛷←
𝑑
𝑖=1,𝑖≠𝑗 ∘𝑆𝛼(𝑣𝑖)𝑒𝑖
Note: These parameterizations are crucial for applying Theorem 5.1 and deriving the
asymptotic behavior.
These lecture notes cover asymptotic analysis of quadratic forms for random vectors with
heavy-tailed distributions.
Asymptotic Behavior of Quadratic Forms
Main Theorem
Theorem 8.2.1: Let 𝑋 be a 𝑑-dimensional random vector with independent and identically
distributed components having a Student-like distribution. Let 𝐶 be a 𝑑×𝑑 matrix. Define
𝐽1={𝑗:𝐶𝑗,𝑗 >0}. If 𝐽1 is not empty, then:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}∼𝐾𝑠,𝛼𝛼(𝛼−1)/2 2
𝑡𝛼/2∑𝐶𝑗,𝑗
𝛼/2
𝑗∈𝐽1
as 𝑡→∞
Key points:
• Applies to Student-like distributions (heavy-tailed)
• Asymptotic behavior depends on positive diagonal elements of 𝐶
• Power law decay with exponent 𝛼/2
Proof Strategy
1. Change of variables: Transform the problem to work with standard normal distribution
2. Apply Theorem 5.1 (not provided in the excerpt) 3. Analyze the boundary of the
transformed set
Change of Variables
Define:
• 𝑌: Random vector with centered normal distribution (identity covariance)
• 𝛷(𝑦): Normal cumulative distribution function
• 𝑆𝛼(𝑦): Student-like cumulative distribution function
• 𝑆𝛼
←, 𝛷←: Inverse functions of 𝑆𝛼 and 𝛷 respectively
Key transformation:
𝑋=
𝑑𝑆𝛼
←∘𝛷(𝑌)
Define sets:
𝐴𝑡={𝑥∈ℝ𝑑:⟨𝐶𝑥,𝑥⟩>𝑡}=√𝑡𝐴1
𝐵𝑡={𝑦∈ℝ𝑑:⟨𝐶𝑆𝛼
←∘𝛷(𝑦),𝑆𝛼
←∘𝛷(𝑦)⟩>𝑡}=𝛷←∘𝑆𝛼(𝐴𝑡)
Important result:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}=∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥𝑖=∫ 𝑒−|𝑦|2/2
(2𝜋)𝑑/2
𝐵𝑡𝑑𝑦
Asymptotic Analysis
Asymptotic Equivalence
Key result:
(𝛷←∘𝑆𝛼(𝑥))2=2𝛼log𝑥−loglog𝑥−2log(𝐾𝑠,𝛼𝛼𝛼/2)−2log(2√𝜋)+𝑜(1)
as 𝑥→∞.
Boundary Parameterization
Lemma 8.2.2: Provides parameterizations for ∂𝐴𝑡 and ∂𝐵𝑡 near specific points.
For ∂𝐴𝑡 near 𝑝𝜖,𝑗,𝑡 =𝜖√𝑡/𝐶𝑗,𝑗𝑒𝑗:
𝑝(𝑣)=𝜖√𝑡
𝐶𝑗,𝑗[1− 1
2𝑡⟨𝐶𝑣,𝑣⟩+𝑂(⟨𝐶𝑣,𝑣⟩
𝑡2)]𝑒𝑗+𝑣
For ∂𝐵𝑡 near 𝑞𝜖,𝑗,𝑡 =𝛷←∘𝑆𝛼(𝑝𝜖,𝑗,𝑡):
𝑞(𝑣)=𝜖√𝛼log 𝑡
𝐶𝑗,𝑗−loglog√𝑡
2√𝛼log𝑡−log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋)
√𝛼log𝑡+𝑜( 1
√log𝑡)𝑒𝑗
+ ∑ 𝛷←
𝑑
𝑖=1,𝑖≠𝑗 ∘𝑆𝛼(𝑣𝑖)𝑒𝑖
Note: These parameterizations are crucial for applying Theorem 5.1 and deriving the
asymptotic behavior.
These lecture notes cover asymptotic analysis of quadratic forms for random vectors with
heavy-tailed distributions.
Asymptotic Behavior of Quadratic Forms
Main Theorem
Theorem 8.2.1: Let 𝑋 be a 𝑑-dimensional random vector with independent and identically
distributed components having a Student-like distribution. Let 𝐶 be a 𝑑×𝑑 matrix. Define
𝐽1={𝑗:𝐶𝑗,𝑗 >0}. If 𝐽1 is not empty, then:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}∼𝐾𝑠,𝛼𝛼(𝛼−1)/2 2
𝑡𝛼/2∑𝐶𝑗,𝑗
𝛼/2
𝑗∈𝐽1
as 𝑡→∞
Key points:
• Applies to Student-like distributions (heavy-tailed)
• Asymptotic behavior depends on positive diagonal elements of 𝐶
• Power law decay with exponent 𝛼/2
Proof Strategy
1. Change of variables: Transform the problem to work with standard normal distribution
2. Apply Theorem 5.1 (not provided in the excerpt) 3. Analyze the boundary of the
transformed set
Change of Variables
Define:
• 𝑌: Random vector with centered normal distribution (identity covariance)
• 𝛷(𝑦): Normal cumulative distribution function
• 𝑆𝛼(𝑦): Student-like cumulative distribution function
• 𝑆𝛼
←, 𝛷←: Inverse functions of 𝑆𝛼 and 𝛷 respectively
Key transformation:
𝑋=
𝑑𝑆𝛼
←∘𝛷(𝑌)
Define sets: 𝐴𝑡={𝑥∈ℝ𝑑:⟨𝐶𝑥,𝑥⟩>𝑡}=√𝑡𝐴1
𝐵𝑡={𝑦∈ℝ𝑑:⟨𝐶𝑆𝛼
←∘𝛷(𝑦),𝑆𝛼
←∘𝛷(𝑦)⟩>𝑡}=𝛷←∘𝑆𝛼(𝐴𝑡)
Important result:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}=∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥𝑖=∫ 𝑒−|𝑦|2/2
(2𝜋)𝑑/2
𝐵𝑡𝑑𝑦
Asymptotic Analysis
Asymptotic Equivalence
Key result:
(𝛷←∘𝑆𝛼(𝑥))2=2𝛼log𝑥−loglog𝑥−2log(𝐾𝑠,𝛼𝛼𝛼/2)−2log(2√𝜋)+𝑜(1)
as 𝑥→∞.
Boundary Parameterization
Lemma 8.2.2: Provides parameterizations for ∂𝐴𝑡 and ∂𝐵𝑡 near specific points.
For ∂𝐴𝑡 near 𝑝𝜖,𝑗,𝑡 =𝜖√𝑡/𝐶𝑗,𝑗𝑒𝑗:
𝑝(𝑣)=𝜖√𝑡
𝐶𝑗,𝑗[1− 1
2𝑡⟨𝐶𝑣,𝑣⟩+𝑂(⟨𝐶𝑣,𝑣⟩
𝑡2)]𝑒𝑗+𝑣
For ∂𝐵𝑡 near 𝑞𝜖,𝑗,𝑡 =𝛷←∘𝑆𝛼(𝑝𝜖,𝑗,𝑡):
𝑞(𝑣)=𝜖√𝛼log 𝑡
𝐶𝑗,𝑗−loglog√𝑡
2√𝛼log𝑡−log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋)
√𝛼log𝑡+𝑜( 1
√log𝑡)𝑒𝑗
+ ∑ 𝛷←
𝑑
𝑖=1,𝑖≠𝑗 ∘𝑆𝛼(𝑣𝑖)𝑒𝑖
Note: These parameterizations are crucial for applying Theorem 5.1 and deriving the
asymptotic behavior.
These lecture notes cover asymptotic analysis of quadratic forms for random vectors with
heavy-tailed distributions.
Asymptotic Behavior of Quadratic Forms
Main Theorem
Theorem 8.2.1: Let 𝑋 be a 𝑑-dimensional random vector with independent and identically
distributed components having a Student-like distribution. Let 𝐶 be a 𝑑×𝑑 matrix. Define
𝐽1={𝑗:𝐶𝑗,𝑗 >0}. If 𝐽1 is not empty, then:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}∼𝐾𝑠,𝛼𝛼(𝛼−1)/2 2
𝑡𝛼/2∑𝐶𝑗,𝑗
𝛼/2
𝑗∈𝐽1
as 𝑡→∞
Key points:
• Applies to Student-like distributions (heavy-tailed)
• Asymptotic behavior depends on positive diagonal elements of 𝐶
• Power law decay with exponent 𝛼/2
Proof Strategy
1. Change of variables: Transform the problem to work with standard normal distribution
2. Apply Theorem 5.1 (not provided in the excerpt) 3. Analyze the boundary of the
transformed set
Change of Variables
Define:
• 𝑌: Random vector with centered normal distribution (identity covariance)
• 𝛷(𝑦): Normal cumulative distribution function
• 𝑆𝛼(𝑦): Student-like cumulative distribution function
• 𝑆𝛼
←, 𝛷←: Inverse functions of 𝑆𝛼 and 𝛷 respectively
Key transformation:
𝑋=
𝑑𝑆𝛼
←∘𝛷(𝑌)
Define sets: 𝐴𝑡={𝑥∈ℝ𝑑:⟨𝐶𝑥,𝑥⟩>𝑡}=√𝑡𝐴1
𝐵𝑡={𝑦∈ℝ𝑑:⟨𝐶𝑆𝛼
←∘𝛷(𝑦),𝑆𝛼
←∘𝛷(𝑦)⟩>𝑡}=𝛷←∘𝑆𝛼(𝐴𝑡)
Important result:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}=∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥𝑖=∫ 𝑒−|𝑦|2/2
(2𝜋)𝑑/2
𝐵𝑡𝑑𝑦
Asymptotic Analysis
Asymptotic Equivalence
Key result:
(𝛷←∘𝑆𝛼(𝑥))2=2𝛼log𝑥−loglog𝑥−2log(𝐾𝑠,𝛼𝛼𝛼/2)−2log(2√𝜋)+𝑜(1)
as 𝑥→∞.
Boundary Parameterization
Lemma 8.2.2: Provides parameterizations for ∂𝐴𝑡 and ∂𝐵𝑡 near specific points.
For ∂𝐴𝑡 near 𝑝𝜖,𝑗,𝑡 =𝜖√𝑡/𝐶𝑗,𝑗𝑒𝑗:
𝑝(𝑣)=𝜖√𝑡
𝐶𝑗,𝑗[1− 1
2𝑡⟨𝐶𝑣,𝑣⟩+𝑂(⟨𝐶𝑣,𝑣⟩
𝑡2)]𝑒𝑗+𝑣
For ∂𝐵𝑡 near 𝑞𝜖,𝑗,𝑡 =𝛷←∘𝑆𝛼(𝑝𝜖,𝑗,𝑡):
𝑞(𝑣)=𝜖√𝛼log 𝑡
𝐶𝑗,𝑗−loglog√𝑡
2√𝛼log𝑡−log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋)
√𝛼log𝑡+𝑜( 1
√log𝑡)𝑒𝑗
+ ∑ 𝛷←
𝑑
𝑖=1,𝑖≠𝑗 ∘𝑆𝛼(𝑣𝑖)𝑒𝑖
Note: These parameterizations are crucial for applying Theorem 5.1 and deriving the
asymptotic behavior.
These lecture notes cover asymptotic analysis of quadratic forms for random vectors with
heavy-tailed distributions.
Asymptotic Behavior of Quadratic Forms
Main Theorem
Theorem 8.2.1: Let 𝑋 be a 𝑑-dimensional random vector with independent and identically
distributed components having a Student-like distribution. Let 𝐶 be a 𝑑×𝑑 matrix. Define
𝐽1={𝑗:𝐶𝑗,𝑗 >0}. If 𝐽1 is not empty, then:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}∼𝐾𝑠,𝛼𝛼(𝛼−1)/2 2
𝑡𝛼/2∑𝐶𝑗,𝑗
𝛼/2
𝑗∈𝐽1
as 𝑡→∞
Key points:
• Applies to Student-like distributions (heavy-tailed)
• Asymptotic behavior depends on positive diagonal elements of 𝐶
• Power law decay with exponent 𝛼/2
Proof Strategy
1. Change of variables: Transform the problem to work with standard normal distribution
2. Apply Theorem 5.1 (not provided in the excerpt) 3. Analyze the boundary of the
transformed set
Change of Variables
Define:
• 𝑌: Random vector with centered normal distribution (identity covariance)
• 𝛷(𝑦): Normal cumulative distribution function
• 𝑆𝛼(𝑦): Student-like cumulative distribution function
• 𝑆𝛼
←, 𝛷←: Inverse functions of 𝑆𝛼 and 𝛷 respectively
Key transformation:
𝑋=
𝑑𝑆𝛼
←∘𝛷(𝑌)
Define sets: 𝐴𝑡={𝑥∈ℝ𝑑:⟨𝐶𝑥,𝑥⟩>𝑡}=√𝑡𝐴1
𝐵𝑡={𝑦∈ℝ𝑑:⟨𝐶𝑆𝛼
←∘𝛷(𝑦),𝑆𝛼
←∘𝛷(𝑦)⟩>𝑡}=𝛷←∘𝑆𝛼(𝐴𝑡)
Important result:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}=∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥𝑖=∫ 𝑒−|𝑦|2/2
(2𝜋)𝑑/2
𝐵𝑡𝑑𝑦
Asymptotic Analysis
Asymptotic Equivalence
Key result:
(𝛷←∘𝑆𝛼(𝑥))2=2𝛼log𝑥−loglog𝑥−2log(𝐾𝑠,𝛼𝛼𝛼/2)−2log(2√𝜋)+𝑜(1)
as 𝑥→∞.
Boundary Parameterization
Lemma 8.2.2: Provides parameterizations for ∂𝐴𝑡 and ∂𝐵𝑡 near specific points.
For ∂𝐴𝑡 near 𝑝𝜖,𝑗,𝑡 =𝜖√𝑡/𝐶𝑗,𝑗𝑒𝑗:
𝑝(𝑣)=𝜖√𝑡
𝐶𝑗,𝑗[1− 1
2𝑡⟨𝐶𝑣,𝑣⟩+𝑂(⟨𝐶𝑣,𝑣⟩
𝑡2)]𝑒𝑗+𝑣
For ∂𝐵𝑡 near 𝑞𝜖,𝑗,𝑡 =𝛷←∘𝑆𝛼(𝑝𝜖,𝑗,𝑡):
𝑞(𝑣)=𝜖√𝛼log 𝑡
𝐶𝑗,𝑗−loglog√𝑡
2√𝛼log𝑡−log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋)
√𝛼log𝑡+𝑜( 1
√log𝑡)𝑒𝑗
+ ∑ 𝛷←
𝑑
𝑖=1,𝑖≠𝑗 ∘𝑆𝛼(𝑣𝑖)𝑒𝑖
Note: These parameterizations are crucial for applying Theorem 5.1 and deriving the
asymptotic behavior.
These lecture notes cover asymptotic analysis of quadratic forms for random vectors with
heavy-tailed distributions.
Asymptotic Behavior of Quadratic Forms
Main Theorem
Theorem 8.2.1: Let 𝑋 be a 𝑑-dimensional random vector with independent and identically
distributed components having a Student-like distribution. Let 𝐶 be a 𝑑×𝑑 matrix. Define
𝐽1={𝑗:𝐶𝑗,𝑗 >0}. If 𝐽1 is not empty, then:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}∼𝐾𝑠,𝛼𝛼(𝛼−1)/2 2
𝑡𝛼/2∑𝐶𝑗,𝑗
𝛼/2
𝑗∈𝐽1
as 𝑡→∞
Key points:
• Applies to Student-like distributions (heavy-tailed)
• Asymptotic behavior depends on positive diagonal elements of 𝐶
• Power law decay with exponent 𝛼/2
Proof Strategy
1. Change of variables: Transform the problem to work with standard normal distribution
2. Apply Theorem 5.1 (not provided in the excerpt) 3. Analyze the boundary of the
transformed set
Change of Variables
Define:
• 𝑌: Random vector with centered normal distribution (identity covariance)
• 𝛷(𝑦): Normal cumulative distribution function
• 𝑆𝛼(𝑦): Student-like cumulative distribution function
• 𝑆𝛼
←, 𝛷←: Inverse functions of 𝑆𝛼 and 𝛷 respectively
Key transformation:
𝑋=
𝑑𝑆𝛼
←∘𝛷(𝑌)
Define sets: 𝐴𝑡={𝑥∈ℝ𝑑:⟨𝐶𝑥,𝑥⟩>𝑡}=√𝑡𝐴1
𝐵𝑡={𝑦∈ℝ𝑑:⟨𝐶𝑆𝛼
←∘𝛷(𝑦),𝑆𝛼
←∘𝛷(𝑦)⟩>𝑡}=𝛷←∘𝑆𝛼(𝐴𝑡)
Important result:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}=∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥𝑖=∫ 𝑒−|𝑦|2/2
(2𝜋)𝑑/2
𝐵𝑡𝑑𝑦
Asymptotic Analysis
Asymptotic Equivalence
Key result:
(𝛷←∘𝑆𝛼(𝑥))2=2𝛼log𝑥−loglog𝑥−2log(𝐾𝑠,𝛼𝛼𝛼/2)−2log(2√𝜋)+𝑜(1)
as 𝑥→∞.
Boundary Parameterization
Lemma 8.2.2: Provides parameterizations for ∂𝐴𝑡 and ∂𝐵𝑡 near specific points.
For ∂𝐴𝑡 near 𝑝𝜖,𝑗,𝑡 =𝜖√𝑡/𝐶𝑗,𝑗𝑒𝑗:
𝑝(𝑣)=𝜖√𝑡
𝐶𝑗,𝑗[1− 1
2𝑡⟨𝐶𝑣,𝑣⟩+𝑂(⟨𝐶𝑣,𝑣⟩
𝑡2)]𝑒𝑗+𝑣
For ∂𝐵𝑡 near 𝑞𝜖,𝑗,𝑡 =𝛷←∘𝑆𝛼(𝑝𝜖,𝑗,𝑡):
𝑞(𝑣)=𝜖√𝛼log 𝑡
𝐶𝑗,𝑗−loglog√𝑡
2√𝛼log𝑡−log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋)
√𝛼log𝑡+𝑜( 1
√log𝑡)𝑒𝑗
+ ∑ 𝛷←
𝑑
𝑖=1,𝑖≠𝑗 ∘𝑆𝛼(𝑣𝑖)𝑒𝑖
Note: These parameterizations are crucial for applying Theorem 5.1 and deriving the
asymptotic behavior.
These lecture notes cover asymptotic analysis of quadratic forms for random vectors with
heavy-tailed distributions.
Asymptotic Behavior of Quadratic Forms
Main Theorem
Theorem 8.2.1: Let 𝑋 be a 𝑑-dimensional random vector with independent and identically
distributed components having a Student-like distribution. Let 𝐶 be a 𝑑×𝑑 matrix. Define
𝐽1={𝑗:𝐶𝑗,𝑗 >0}. If 𝐽1 is not empty, then:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}∼𝐾𝑠,𝛼𝛼(𝛼−1)/2 2
𝑡𝛼/2∑𝐶𝑗,𝑗
𝛼/2
𝑗∈𝐽1
as 𝑡→∞
Key points:
• Applies to Student-like distributions (heavy-tailed)
• Asymptotic behavior depends on positive diagonal elements of 𝐶
• Power law decay with exponent 𝛼/2
Proof Strategy
1. Change of variables: Transform the problem to work with standard normal distribution
2. Apply Theorem 5.1 (not provided in the excerpt) 3. Analyze the boundary of the
transformed set
Change of Variables
Define:
• 𝑌: Random vector with centered normal distribution (identity covariance)
• 𝛷(𝑦): Normal cumulative distribution function
• 𝑆𝛼(𝑦): Student-like cumulative distribution function
• 𝑆𝛼
←, 𝛷←: Inverse functions of 𝑆𝛼 and 𝛷 respectively
Key transformation:
𝑋=
𝑑𝑆𝛼
←∘𝛷(𝑌)
Define sets: 𝐴𝑡={𝑥∈ℝ𝑑:⟨𝐶𝑥,𝑥⟩>𝑡}=√𝑡𝐴1
𝐵𝑡={𝑦∈ℝ𝑑:⟨𝐶𝑆𝛼
←∘𝛷(𝑦),𝑆𝛼
←∘𝛷(𝑦)⟩>𝑡}=𝛷←∘𝑆𝛼(𝐴𝑡)
Important result:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}=∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥𝑖=∫ 𝑒−|𝑦|2/2
(2𝜋)𝑑/2
𝐵𝑡𝑑𝑦
Asymptotic Analysis
Asymptotic Equivalence
Key result:
(𝛷←∘𝑆𝛼(𝑥))2=2𝛼log𝑥−loglog𝑥−2log(𝐾𝑠,𝛼𝛼𝛼/2)−2log(2√𝜋)+𝑜(1)
as 𝑥→∞.
Boundary Parameterization
Lemma 8.2.2: Provides parameterizations for ∂𝐴𝑡 and ∂𝐵𝑡 near specific points.
For ∂𝐴𝑡 near 𝑝𝜖,𝑗,𝑡 =𝜖√𝑡/𝐶𝑗,𝑗𝑒𝑗:
𝑝(𝑣)=𝜖√𝑡
𝐶𝑗,𝑗[1− 1
2𝑡⟨𝐶𝑣,𝑣⟩+𝑂(⟨𝐶𝑣,𝑣⟩
𝑡2)]𝑒𝑗+𝑣
For ∂𝐵𝑡 near 𝑞𝜖,𝑗,𝑡 =𝛷←∘𝑆𝛼(𝑝𝜖,𝑗,𝑡):
𝑞(𝑣)=𝜖√𝛼log 𝑡
𝐶𝑗,𝑗−loglog√𝑡
2√𝛼log𝑡−log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋)
√𝛼log𝑡+𝑜( 1
√log𝑡)𝑒𝑗
+ ∑ 𝛷←
𝑑
𝑖=1,𝑖≠𝑗 ∘𝑆𝛼(𝑣𝑖)𝑒𝑖
Note: These parameterizations are crucial for applying Theorem 5.1 and deriving the
asymptotic behavior.
These lecture notes cover asymptotic analysis of quadratic forms for random vectors with
heavy-tailed distributions.
Asymptotic Behavior of Quadratic Forms
Main Theorem
Theorem 8.2.1: Let 𝑋 be a 𝑑-dimensional random vector with independent and identically
distributed components having a Student-like distribution. Let 𝐶 be a 𝑑×𝑑 matrix. Define
𝐽1={𝑗:𝐶𝑗,𝑗 >0}. If 𝐽1 is not empty, then:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}∼𝐾𝑠,𝛼𝛼(𝛼−1)/2 2
𝑡𝛼/2∑𝐶𝑗,𝑗
𝛼/2
𝑗∈𝐽1
as 𝑡→∞
Key points:
• Applies to Student-like distributions (heavy-tailed)
• Asymptotic behavior depends on positive diagonal elements of 𝐶
• Power law decay with exponent 𝛼/2
Proof Strategy
1. Change of variables: Transform the problem to work with standard normal distribution
2. Apply Theorem 5.1 (not provided in the excerpt) 3. Analyze the boundary of the
transformed set
Change of Variables
Define:
• 𝑌: Random vector with centered normal distribution (identity covariance)
• 𝛷(𝑦): Normal cumulative distribution function
• 𝑆𝛼(𝑦): Student-like cumulative distribution function
• 𝑆𝛼
←, 𝛷←: Inverse functions of 𝑆𝛼 and 𝛷 respectively
Key transformation:
𝑋=
𝑑𝑆𝛼
←∘𝛷(𝑌)
Define sets: 𝐴𝑡={𝑥∈ℝ𝑑:⟨𝐶𝑥,𝑥⟩>𝑡}=√𝑡𝐴1
𝐵𝑡={𝑦∈ℝ𝑑:⟨𝐶𝑆𝛼
←∘𝛷(𝑦),𝑆𝛼
←∘𝛷(𝑦)⟩>𝑡}=𝛷←∘𝑆𝛼(𝐴𝑡)
Important result:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}=∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥𝑖=∫ 𝑒−|𝑦|2/2
(2𝜋)𝑑/2
𝐵𝑡𝑑𝑦
Asymptotic Analysis
Asymptotic Equivalence
Key result:
(𝛷←∘𝑆𝛼(𝑥))2=2𝛼log𝑥−loglog𝑥−2log(𝐾𝑠,𝛼𝛼𝛼/2)−2log(2√𝜋)+𝑜(1)
as 𝑥→∞.
Boundary Parameterization
Lemma 8.2.2: Provides parameterizations for ∂𝐴𝑡 and ∂𝐵𝑡 near specific points.
For ∂𝐴𝑡 near 𝑝𝜖,𝑗,𝑡 =𝜖√𝑡/𝐶𝑗,𝑗𝑒𝑗:
𝑝(𝑣)=𝜖√𝑡
𝐶𝑗,𝑗[1− 1
2𝑡⟨𝐶𝑣,𝑣⟩+𝑂(⟨𝐶𝑣,𝑣⟩
𝑡2)]𝑒𝑗+𝑣
For ∂𝐵𝑡 near 𝑞𝜖,𝑗,𝑡 =𝛷←∘𝑆𝛼(𝑝𝜖,𝑗,𝑡):
𝑞(𝑣)=𝜖√𝛼log 𝑡
𝐶𝑗,𝑗−loglog√𝑡
2√𝛼log𝑡−log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋)
√𝛼log𝑡+𝑜( 1
√log𝑡)𝑒𝑗
+ ∑ 𝛷←
𝑑
𝑖=1,𝑖≠𝑗 ∘𝑆𝛼(𝑣𝑖)𝑒𝑖
Note: These parameterizations are crucial for applying Theorem 5.1 and deriving the
asymptotic behavior.
These lecture notes cover asymptotic analysis of quadratic forms for random vectors with
heavy-tailed distributions.
Asymptotic Behavior of Quadratic Forms
Main Theorem
Theorem 8.2.1: Let 𝑋 be a 𝑑-dimensional random vector with independent and identically
distributed components having a Student-like distribution. Let 𝐶 be a 𝑑×𝑑 matrix. Define
𝐽1={𝑗:𝐶𝑗,𝑗 >0}. If 𝐽1 is not empty, then:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}∼𝐾𝑠,𝛼𝛼(𝛼−1)/2 2
𝑡𝛼/2∑𝐶𝑗,𝑗
𝛼/2
𝑗∈𝐽1
as 𝑡→∞
Key points:
• Applies to Student-like distributions (heavy-tailed)
• Asymptotic behavior depends on positive diagonal elements of 𝐶
• Power law decay with exponent 𝛼/2
Proof Strategy
1. Change of variables: Transform the problem to work with standard normal distribution
2. Apply Theorem 5.1 (not provided in the excerpt) 3. Analyze the boundary of the
transformed set
Change of Variables
Define:
• 𝑌: Random vector with centered normal distribution (identity covariance)
• 𝛷(𝑦): Normal cumulative distribution function
• 𝑆𝛼(𝑦): Student-like cumulative distribution function
• 𝑆𝛼
←, 𝛷←: Inverse functions of 𝑆𝛼 and 𝛷 respectively
Key transformation:
𝑋=
𝑑𝑆𝛼
←∘𝛷(𝑌)
Define sets:
𝐴𝑡={𝑥∈ℝ𝑑:⟨𝐶𝑥,𝑥⟩>𝑡}=√𝑡𝐴1
𝐵𝑡={𝑦∈ℝ𝑑:⟨𝐶𝑆𝛼
←∘𝛷(𝑦),𝑆𝛼
←∘𝛷(𝑦)⟩>𝑡}=𝛷←∘𝑆𝛼(𝐴𝑡)
Important result:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}=∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥𝑖=∫ 𝑒−|𝑦|2/2
(2𝜋)𝑑/2
𝐵𝑡𝑑𝑦
Asymptotic Analysis
Asymptotic Equivalence
Key result:
(𝛷←∘𝑆𝛼(𝑥))2=2𝛼log𝑥−loglog𝑥−2log(𝐾𝑠,𝛼𝛼𝛼/2)−2log(2√𝜋)+𝑜(1)
as 𝑥→∞.
Boundary Parameterization
Lemma 8.2.2: Provides parameterizations for ∂𝐴𝑡 and ∂𝐵𝑡 near specific points.
For ∂𝐴𝑡 near 𝑝𝜖,𝑗,𝑡 =𝜖√𝑡/𝐶𝑗,𝑗𝑒𝑗:
𝑝(𝑣)=𝜖√𝑡
𝐶𝑗,𝑗[1− 1
2𝑡⟨𝐶𝑣,𝑣⟩+𝑂(⟨𝐶𝑣,𝑣⟩
𝑡2)]𝑒𝑗+𝑣
For ∂𝐵𝑡 near 𝑞𝜖,𝑗,𝑡 =𝛷←∘𝑆𝛼(𝑝𝜖,𝑗,𝑡):
𝑞(𝑣)=𝜖√𝛼log 𝑡
𝐶𝑗,𝑗−loglog√𝑡
2√𝛼log𝑡−log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋)
√𝛼log𝑡+𝑜( 1
√log𝑡)𝑒𝑗
+ ∑ 𝛷←
𝑑
𝑖=1,𝑖≠𝑗 ∘𝑆𝛼(𝑣𝑖)𝑒𝑖
Note: These parameterizations are crucial for applying Theorem 5.1 and deriving the
asymptotic behavior.
These lecture notes cover asymptotic analysis of quadratic forms for random vectors with
heavy-tailed distributions.
Asymptotic Behavior of Quadratic Forms
Main Theorem
Theorem 8.2.1: Let 𝑋 be a 𝑑-dimensional random vector with independent and identically
distributed components having a Student-like distribution. Let 𝐶 be a 𝑑×𝑑 matrix. Define
𝐽1={𝑗:𝐶𝑗,𝑗 >0}. If 𝐽1 is not empty, then:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}∼𝐾𝑠,𝛼𝛼(𝛼−1)/2 2
𝑡𝛼/2∑𝐶𝑗,𝑗
𝛼/2
𝑗∈𝐽1
as 𝑡→∞
Key points:
• Applies to Student-like distributions (heavy-tailed)
• Asymptotic behavior depends on positive diagonal elements of 𝐶
• Power law decay with exponent 𝛼/2
Proof Strategy
1. Change of variables: Transform the problem to work with standard normal distribution
2. Apply Theorem 5.1 (not provided in the excerpt) 3. Analyze the boundary of the
transformed set
Change of Variables
Define:
• 𝑌: Random vector with centered normal distribution (identity covariance)
• 𝛷(𝑦): Normal cumulative distribution function
• 𝑆𝛼(𝑦): Student-like cumulative distribution function
• 𝑆𝛼
←, 𝛷←: Inverse functions of 𝑆𝛼 and 𝛷 respectively
Key transformation:
𝑋=
𝑑𝑆𝛼
←∘𝛷(𝑌)
Define sets: 𝐴𝑡={𝑥∈ℝ𝑑:⟨𝐶𝑥,𝑥⟩>𝑡}=√𝑡𝐴1
𝐵𝑡={𝑦∈ℝ𝑑:⟨𝐶𝑆𝛼
←∘𝛷(𝑦),𝑆𝛼
←∘𝛷(𝑦)⟩>𝑡}=𝛷←∘𝑆𝛼(𝐴𝑡)
Important result:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}=∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥𝑖=∫ 𝑒−|𝑦|2/2
(2𝜋)𝑑/2
𝐵𝑡𝑑𝑦
Asymptotic Analysis
Asymptotic Equivalence
Key result:
(𝛷←∘𝑆𝛼(𝑥))2=2𝛼log𝑥−loglog𝑥−2log(𝐾𝑠,𝛼𝛼𝛼/2)−2log(2√𝜋)+𝑜(1)
as 𝑥→∞.
Boundary Parameterization
Lemma 8.2.2: Provides parameterizations for ∂𝐴𝑡 and ∂𝐵𝑡 near specific points.
For ∂𝐴𝑡 near 𝑝𝜖,𝑗,𝑡 =𝜖√𝑡/𝐶𝑗,𝑗𝑒𝑗:
𝑝(𝑣)=𝜖√𝑡
𝐶𝑗,𝑗[1− 1
2𝑡⟨𝐶𝑣,𝑣⟩+𝑂(⟨𝐶𝑣,𝑣⟩
𝑡2)]𝑒𝑗+𝑣
For ∂𝐵𝑡 near 𝑞𝜖,𝑗,𝑡 =𝛷←∘𝑆𝛼(𝑝𝜖,𝑗,𝑡):
𝑞(𝑣)=𝜖√𝛼log 𝑡
𝐶𝑗,𝑗−loglog√𝑡
2√𝛼log𝑡−log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋)
√𝛼log𝑡+𝑜( 1
√log𝑡)𝑒𝑗
+ ∑ 𝛷←
𝑑
𝑖=1,𝑖≠𝑗 ∘𝑆𝛼(𝑣𝑖)𝑒𝑖
Note: These parameterizations are crucial for applying Theorem 5.1 and deriving the
asymptotic behavior.
These lecture notes cover asymptotic analysis of quadratic forms for random vectors with
heavy-tailed distributions.
Asymptotic Behavior of Quadratic Forms
Main Theorem
Theorem 8.2.1: Let 𝑋 be a 𝑑-dimensional random vector with independent and identically
distributed components having a Student-like distribution. Let 𝐶 be a 𝑑×𝑑 matrix. Define
𝐽1={𝑗:𝐶𝑗,𝑗 >0}. If 𝐽1 is not empty, then:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}∼𝐾𝑠,𝛼𝛼(𝛼−1)/2 2
𝑡𝛼/2∑𝐶𝑗,𝑗
𝛼/2
𝑗∈𝐽1
as 𝑡→∞
Key points:
• Applies to Student-like distributions (heavy-tailed)
• Asymptotic behavior depends on positive diagonal elements of 𝐶
• Power law decay with exponent 𝛼/2
Proof Strategy
1. Change of variables: Transform the problem to work with standard normal distribution
2. Apply Theorem 5.1 (not provided in the excerpt) 3. Analyze the boundary of the
transformed set
Change of Variables
Define:
• 𝑌: Random vector with centered normal distribution (identity covariance)
• 𝛷(𝑦): Normal cumulative distribution function
• 𝑆𝛼(𝑦): Student-like cumulative distribution function
• 𝑆𝛼
←, 𝛷←: Inverse functions of 𝑆𝛼 and 𝛷 respectively
Key transformation:
𝑋=
𝑑𝑆𝛼
←∘𝛷(𝑌)
Define sets: 𝐴𝑡={𝑥∈ℝ𝑑:⟨𝐶𝑥,𝑥⟩>𝑡}=√𝑡𝐴1
𝐵𝑡={𝑦∈ℝ𝑑:⟨𝐶𝑆𝛼
←∘𝛷(𝑦),𝑆𝛼
←∘𝛷(𝑦)⟩>𝑡}=𝛷←∘𝑆𝛼(𝐴𝑡)
Important result:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}=∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥𝑖=∫ 𝑒−|𝑦|2/2
(2𝜋)𝑑/2
𝐵𝑡𝑑𝑦
Asymptotic Analysis
Asymptotic Equivalence
Key result:
(𝛷←∘𝑆𝛼(𝑥))2=2𝛼log𝑥−loglog𝑥−2log(𝐾𝑠,𝛼𝛼𝛼/2)−2log(2√𝜋)+𝑜(1)
as 𝑥→∞.
Boundary Parameterization
Lemma 8.2.2: Provides parameterizations for ∂𝐴𝑡 and ∂𝐵𝑡 near specific points.
For ∂𝐴𝑡 near 𝑝𝜖,𝑗,𝑡 =𝜖√𝑡/𝐶𝑗,𝑗𝑒𝑗:
𝑝(𝑣)=𝜖√𝑡
𝐶𝑗,𝑗[1− 1
2𝑡⟨𝐶𝑣,𝑣⟩+𝑂(⟨𝐶𝑣,𝑣⟩
𝑡2)]𝑒𝑗+𝑣
For ∂𝐵𝑡 near 𝑞𝜖,𝑗,𝑡 =𝛷←∘𝑆𝛼(𝑝𝜖,𝑗,𝑡):
𝑞(𝑣)=𝜖√𝛼log 𝑡
𝐶𝑗,𝑗−loglog√𝑡
2√𝛼log𝑡−log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋)
√𝛼log𝑡+𝑜( 1
√log𝑡)𝑒𝑗
+ ∑ 𝛷←
𝑑
𝑖=1,𝑖≠𝑗 ∘𝑆𝛼(𝑣𝑖)𝑒𝑖
Note: These parameterizations are crucial for applying Theorem 5.1 and deriving the
asymptotic behavior.
These lecture notes cover asymptotic analysis of quadratic forms for random vectors with
heavy-tailed distributions.
Asymptotic Behavior of Quadratic Forms
Main Theorem
Theorem 8.2.1: Let 𝑋 be a 𝑑-dimensional random vector with independent and identically
distributed components having a Student-like distribution. Let 𝐶 be a 𝑑×𝑑 matrix. Define
𝐽1={𝑗:𝐶𝑗,𝑗 >0}. If 𝐽1 is not empty, then:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}∼𝐾𝑠,𝛼𝛼(𝛼−1)/2 2
𝑡𝛼/2∑𝐶𝑗,𝑗
𝛼/2
𝑗∈𝐽1
as 𝑡→∞
Key points:
• Applies to Student-like distributions (heavy-tailed)
• Asymptotic behavior depends on positive diagonal elements of 𝐶
• Power law decay with exponent 𝛼/2
Proof Strategy
1. Change of variables: Transform the problem to work with standard normal distribution
2. Apply Theorem 5.1 (not provided in the excerpt) 3. Analyze the boundary of the
transformed set
Change of Variables
Define:
• 𝑌: Random vector with centered normal distribution (identity covariance)
• 𝛷(𝑦): Normal cumulative distribution function
• 𝑆𝛼(𝑦): Student-like cumulative distribution function
• 𝑆𝛼
←, 𝛷←: Inverse functions of 𝑆𝛼 and 𝛷 respectively
Key transformation:
𝑋=
𝑑𝑆𝛼
←∘𝛷(𝑌)
Define sets: 𝐴𝑡={𝑥∈ℝ𝑑:⟨𝐶𝑥,𝑥⟩>𝑡}=√𝑡𝐴1
𝐵𝑡={𝑦∈ℝ𝑑:⟨𝐶𝑆𝛼
←∘𝛷(𝑦),𝑆𝛼
←∘𝛷(𝑦)⟩>𝑡}=𝛷←∘𝑆𝛼(𝐴𝑡)
Important result:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}=∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥𝑖=∫ 𝑒−|𝑦|2/2
(2𝜋)𝑑/2
𝐵𝑡𝑑𝑦
Asymptotic Analysis
Asymptotic Equivalence
Key result:
(𝛷←∘𝑆𝛼(𝑥))2=2𝛼log𝑥−loglog𝑥−2log(𝐾𝑠,𝛼𝛼𝛼/2)−2log(2√𝜋)+𝑜(1)
as 𝑥→∞.
Boundary Parameterization
Lemma 8.2.2: Provides parameterizations for ∂𝐴𝑡 and ∂𝐵𝑡 near specific points.
For ∂𝐴𝑡 near 𝑝𝜖,𝑗,𝑡 =𝜖√𝑡/𝐶𝑗,𝑗𝑒𝑗:
𝑝(𝑣)=𝜖√𝑡
𝐶𝑗,𝑗[1− 1
2𝑡⟨𝐶𝑣,𝑣⟩+𝑂(⟨𝐶𝑣,𝑣⟩
𝑡2)]𝑒𝑗+𝑣
For ∂𝐵𝑡 near 𝑞𝜖,𝑗,𝑡 =𝛷←∘𝑆𝛼(𝑝𝜖,𝑗,𝑡):
𝑞(𝑣)=𝜖√𝛼log 𝑡
𝐶𝑗,𝑗−loglog√𝑡
2√𝛼log𝑡−log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋)
√𝛼log𝑡+𝑜( 1
√log𝑡)𝑒𝑗
+ ∑ 𝛷←
𝑑
𝑖=1,𝑖≠𝑗 ∘𝑆𝛼(𝑣𝑖)𝑒𝑖
Note: These parameterizations are crucial for applying Theorem 5.1 and deriving the
asymptotic behavior.
These lecture notes cover asymptotic analysis of quadratic forms for random vectors with
heavy-tailed distributions.
Asymptotic Behavior of Quadratic Forms
Main Theorem
Theorem 8.2.1: Let 𝑋 be a 𝑑-dimensional random vector with independent and identically
distributed components having a Student-like distribution. Let 𝐶 be a 𝑑×𝑑 matrix. Define
𝐽1={𝑗:𝐶𝑗,𝑗 >0}. If 𝐽1 is not empty, then:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}∼𝐾𝑠,𝛼𝛼(𝛼−1)/2 2
𝑡𝛼/2∑𝐶𝑗,𝑗
𝛼/2
𝑗∈𝐽1
as 𝑡→∞
Key points:
• Applies to Student-like distributions (heavy-tailed)
• Asymptotic behavior depends on positive diagonal elements of 𝐶
• Power law decay with exponent 𝛼/2
Proof Strategy
1. Change of variables: Transform the problem to work with standard normal distribution
2. Apply Theorem 5.1 (not provided in the excerpt) 3. Analyze the boundary of the
transformed set
Change of Variables
Define:
• 𝑌: Random vector with centered normal distribution (identity covariance)
• 𝛷(𝑦): Normal cumulative distribution function
• 𝑆𝛼(𝑦): Student-like cumulative distribution function
• 𝑆𝛼
←, 𝛷←: Inverse functions of 𝑆𝛼 and 𝛷 respectively
Key transformation:
𝑋=
𝑑𝑆𝛼
←∘𝛷(𝑌)
Define sets: 𝐴𝑡={𝑥∈ℝ𝑑:⟨𝐶𝑥,𝑥⟩>𝑡}=√𝑡𝐴1
𝐵𝑡={𝑦∈ℝ𝑑:⟨𝐶𝑆𝛼
←∘𝛷(𝑦),𝑆𝛼
←∘𝛷(𝑦)⟩>𝑡}=𝛷←∘𝑆𝛼(𝐴𝑡)
Important result:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}=∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥𝑖=∫ 𝑒−|𝑦|2/2
(2𝜋)𝑑/2
𝐵𝑡𝑑𝑦
Asymptotic Analysis
Asymptotic Equivalence
Key result:
(𝛷←∘𝑆𝛼(𝑥))2=2𝛼log𝑥−loglog𝑥−2log(𝐾𝑠,𝛼𝛼𝛼/2)−2log(2√𝜋)+𝑜(1)
as 𝑥→∞.
Boundary Parameterization
Lemma 8.2.2: Provides parameterizations for ∂𝐴𝑡 and ∂𝐵𝑡 near specific points.
For ∂𝐴𝑡 near 𝑝𝜖,𝑗,𝑡 =𝜖√𝑡/𝐶𝑗,𝑗𝑒𝑗:
𝑝(𝑣)=𝜖√𝑡
𝐶𝑗,𝑗[1− 1
2𝑡⟨𝐶𝑣,𝑣⟩+𝑂(⟨𝐶𝑣,𝑣⟩
𝑡2)]𝑒𝑗+𝑣
For ∂𝐵𝑡 near 𝑞𝜖,𝑗,𝑡 =𝛷←∘𝑆𝛼(𝑝𝜖,𝑗,𝑡):
𝑞(𝑣)=𝜖√𝛼log 𝑡
𝐶𝑗,𝑗−loglog√𝑡
2√𝛼log𝑡−log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋)
√𝛼log𝑡+𝑜( 1
√log𝑡)𝑒𝑗
+ ∑ 𝛷←
𝑑
𝑖=1,𝑖≠𝑗 ∘𝑆𝛼(𝑣𝑖)𝑒𝑖
Note: These parameterizations are crucial for applying Theorem 5.1 and deriving the
asymptotic behavior.
These lecture notes cover asymptotic analysis of quadratic forms for random vectors with
heavy-tailed distributions.
Asymptotic Behavior of Quadratic Forms
Main Theorem
Theorem 8.2.1: Let 𝑋 be a 𝑑-dimensional random vector with independent and identically
distributed components having a Student-like distribution. Let 𝐶 be a 𝑑×𝑑 matrix. Define
𝐽1={𝑗:𝐶𝑗,𝑗 >0}. If 𝐽1 is not empty, then:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}∼𝐾𝑠,𝛼𝛼(𝛼−1)/2 2
𝑡𝛼/2∑𝐶𝑗,𝑗
𝛼/2
𝑗∈𝐽1
as 𝑡→∞
Key points:
• Applies to Student-like distributions (heavy-tailed)
• Asymptotic behavior depends on positive diagonal elements of 𝐶
• Power law decay with exponent 𝛼/2
Proof Strategy
1. Change of variables: Transform the problem to work with standard normal distribution
2. Apply Theorem 5.1 (not provided in the excerpt) 3. Analyze the boundary of the
transformed set
Change of Variables
Define:
• 𝑌: Random vector with centered normal distribution (identity covariance)
• 𝛷(𝑦): Normal cumulative distribution function
• 𝑆𝛼(𝑦): Student-like cumulative distribution function
• 𝑆𝛼
←, 𝛷←: Inverse functions of 𝑆𝛼 and 𝛷 respectively
Key transformation:
𝑋=
𝑑𝑆𝛼
←∘𝛷(𝑌)
Define sets: 𝐴𝑡={𝑥∈ℝ𝑑:⟨𝐶𝑥,𝑥⟩>𝑡}=√𝑡𝐴1
𝐵𝑡={𝑦∈ℝ𝑑:⟨𝐶𝑆𝛼
←∘𝛷(𝑦),𝑆𝛼
←∘𝛷(𝑦)⟩>𝑡}=𝛷←∘𝑆𝛼(𝐴𝑡)
Important result:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}=∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥𝑖=∫ 𝑒−|𝑦|2/2
(2𝜋)𝑑/2
𝐵𝑡𝑑𝑦
Asymptotic Analysis
Asymptotic Equivalence
Key result:
(𝛷←∘𝑆𝛼(𝑥))2=2𝛼log𝑥−loglog𝑥−2log(𝐾𝑠,𝛼𝛼𝛼/2)−2log(2√𝜋)+𝑜(1)
as 𝑥→∞.
Boundary Parameterization
Lemma 8.2.2: Provides parameterizations for ∂𝐴𝑡 and ∂𝐵𝑡 near specific points.
For ∂𝐴𝑡 near 𝑝𝜖,𝑗,𝑡 =𝜖√𝑡/𝐶𝑗,𝑗𝑒𝑗:
𝑝(𝑣)=𝜖√𝑡
𝐶𝑗,𝑗[1− 1
2𝑡⟨𝐶𝑣,𝑣⟩+𝑂(⟨𝐶𝑣,𝑣⟩
𝑡2)]𝑒𝑗+𝑣
For ∂𝐵𝑡 near 𝑞𝜖,𝑗,𝑡 =𝛷←∘𝑆𝛼(𝑝𝜖,𝑗,𝑡):
𝑞(𝑣)=𝜖√𝛼log 𝑡
𝐶𝑗,𝑗−loglog√𝑡
2√𝛼log𝑡−log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋)
√𝛼log𝑡+𝑜( 1
√log𝑡)𝑒𝑗
+ ∑ 𝛷←
𝑑
𝑖=1,𝑖≠𝑗 ∘𝑆𝛼(𝑣𝑖)𝑒𝑖
Note: These parameterizations are crucial for applying Theorem 5.1 and deriving the
asymptotic behavior.
These lecture notes cover asymptotic analysis of quadratic forms for random vectors with
heavy-tailed distributions.
Asymptotic Behavior of Quadratic Forms
Main Theorem
Theorem 8.2.1: Let 𝑋 be a 𝑑-dimensional random vector with independent and identically
distributed components having a Student-like distribution. Let 𝐶 be a 𝑑×𝑑 matrix. Define
𝐽1={𝑗:𝐶𝑗,𝑗 >0}. If 𝐽1 is not empty, then:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}∼𝐾𝑠,𝛼𝛼(𝛼−1)/2 2
𝑡𝛼/2∑𝐶𝑗,𝑗
𝛼/2
𝑗∈𝐽1
as 𝑡→∞
Key points:
• Applies to Student-like distributions (heavy-tailed)
• Asymptotic behavior depends on positive diagonal elements of 𝐶
• Power law decay with exponent 𝛼/2
Proof Strategy
1. Change of variables: Transform the problem to work with standard normal distribution
2. Apply Theorem 5.1 (not provided in the excerpt) 3. Analyze the boundary of the
transformed set
Change of Variables
Define:
• 𝑌: Random vector with centered normal distribution (identity covariance)
• 𝛷(𝑦): Normal cumulative distribution function
• 𝑆𝛼(𝑦): Student-like cumulative distribution function
• 𝑆𝛼
←, 𝛷←: Inverse functions of 𝑆𝛼 and 𝛷 respectively
Key transformation:
𝑋=
𝑑𝑆𝛼
←∘𝛷(𝑌)
Define sets: 𝐴𝑡={𝑥∈ℝ𝑑:⟨𝐶𝑥,𝑥⟩>𝑡}=√𝑡𝐴1
𝐵𝑡={𝑦∈ℝ𝑑:⟨𝐶𝑆𝛼
←∘𝛷(𝑦),𝑆𝛼
←∘𝛷(𝑦)⟩>𝑡}=𝛷←∘𝑆𝛼(𝐴𝑡)
Important result:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}=∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥𝑖=∫ 𝑒−|𝑦|2/2
(2𝜋)𝑑/2
𝐵𝑡𝑑𝑦
Asymptotic Analysis
Asymptotic Equivalence
Key result:
(𝛷←∘𝑆𝛼(𝑥))2=2𝛼log𝑥−loglog𝑥−2log(𝐾𝑠,𝛼𝛼𝛼/2)−2log(2√𝜋)+𝑜(1)
as 𝑥→∞.
Boundary Parameterization
Lemma 8.2.2: Provides parameterizations for ∂𝐴𝑡 and ∂𝐵𝑡 near specific points.
For ∂𝐴𝑡 near 𝑝𝜖,𝑗,𝑡 =𝜖√𝑡/𝐶𝑗,𝑗𝑒𝑗:
𝑝(𝑣)=𝜖√𝑡
𝐶𝑗,𝑗[1− 1
2𝑡⟨𝐶𝑣,𝑣⟩+𝑂(⟨𝐶𝑣,𝑣⟩
𝑡2)]𝑒𝑗+𝑣
For ∂𝐵𝑡 near 𝑞𝜖,𝑗,𝑡 =𝛷←∘𝑆𝛼(𝑝𝜖,𝑗,𝑡):
𝑞(𝑣)=𝜖√𝛼log 𝑡
𝐶𝑗,𝑗−loglog√𝑡
2√𝛼log𝑡−log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋)
√𝛼log𝑡+𝑜( 1
√log𝑡)𝑒𝑗
+ ∑ 𝛷←
𝑑
𝑖=1,𝑖≠𝑗 ∘𝑆𝛼(𝑣𝑖)𝑒𝑖
Note: These parameterizations are crucial for applying Theorem 5.1 and deriving the
asymptotic behavior.
These lecture notes cover asymptotic analysis of quadratic forms for random vectors with
heavy-tailed distributions.
Asymptotic Behavior of Quadratic Forms
Main Theorem
Theorem 8.2.1: Let 𝑋 be a 𝑑-dimensional random vector with independent and identically
distributed components having a Student-like distribution. Let 𝐶 be a 𝑑×𝑑 matrix. Define
𝐽1={𝑗:𝐶𝑗,𝑗 >0}. If 𝐽1 is not empty, then:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}∼𝐾𝑠,𝛼𝛼(𝛼−1)/2 2
𝑡𝛼/2∑𝐶𝑗,𝑗
𝛼/2
𝑗∈𝐽1
as 𝑡→∞
Key points:
• Applies to Student-like distributions (heavy-tailed)
• Asymptotic behavior depends on positive diagonal elements of 𝐶
• Power law decay with exponent 𝛼/2
Proof Strategy
1. Change of variables: Transform the problem to work with standard normal distribution
2. Apply Theorem 5.1 (not provided in the excerpt) 3. Analyze the boundary of the
transformed set
Change of Variables
Define:
• 𝑌: Random vector with centered normal distribution (identity covariance)
• 𝛷(𝑦): Normal cumulative distribution function
• 𝑆𝛼(𝑦): Student-like cumulative distribution function
• 𝑆𝛼
←, 𝛷←: Inverse functions of 𝑆𝛼 and 𝛷 respectively
Key transformation:
𝑋=
𝑑𝑆𝛼
←∘𝛷(𝑌)
Define sets:
𝐴𝑡={𝑥∈ℝ𝑑:⟨𝐶𝑥,𝑥⟩>𝑡}=√𝑡𝐴1
𝐵𝑡={𝑦∈ℝ𝑑:⟨𝐶𝑆𝛼
←∘𝛷(𝑦),𝑆𝛼
←∘𝛷(𝑦)⟩>𝑡}=𝛷←∘𝑆𝛼(𝐴𝑡)
Important result:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}=∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥𝑖=∫ 𝑒−|𝑦|2/2
(2𝜋)𝑑/2
𝐵𝑡𝑑𝑦
Asymptotic Analysis
Asymptotic Equivalence
Key result:
(𝛷←∘𝑆𝛼(𝑥))2=2𝛼log𝑥−loglog𝑥−2log(𝐾𝑠,𝛼𝛼𝛼/2)−2log(2√𝜋)+𝑜(1)
as 𝑥→∞.
Boundary Parameterization
Lemma 8.2.2: Provides parameterizations for ∂𝐴𝑡 and ∂𝐵𝑡 near specific points.
For ∂𝐴𝑡 near 𝑝𝜖,𝑗,𝑡 =𝜖√𝑡/𝐶𝑗,𝑗𝑒𝑗:
𝑝(𝑣)=𝜖√𝑡
𝐶𝑗,𝑗[1− 1
2𝑡⟨𝐶𝑣,𝑣⟩+𝑂(⟨𝐶𝑣,𝑣⟩
𝑡2)]𝑒𝑗+𝑣
For ∂𝐵𝑡 near 𝑞𝜖,𝑗,𝑡 =𝛷←∘𝑆𝛼(𝑝𝜖,𝑗,𝑡):
𝑞(𝑣)=𝜖√𝛼log 𝑡
𝐶𝑗,𝑗−loglog√𝑡
2√𝛼log𝑡−log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋)
√𝛼log𝑡+𝑜( 1
√log𝑡)𝑒𝑗
+ ∑ 𝛷←
𝑑
𝑖=1,𝑖≠𝑗 ∘𝑆𝛼(𝑣𝑖)𝑒𝑖
Note: These parameterizations are crucial for applying Theorem 5.1 and deriving the
asymptotic behavior.
These lecture notes cover asymptotic analysis of quadratic forms for random vectors with
heavy-tailed distributions.
Asymptotic Behavior of Quadratic Forms
Main Theorem
Theorem 8.2.1: Let 𝑋 be a 𝑑-dimensional random vector with independent and identically
distributed components having a Student-like distribution. Let 𝐶 be a 𝑑×𝑑 matrix. Define
𝐽1={𝑗:𝐶𝑗,𝑗 >0}. If 𝐽1 is not empty, then:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}∼𝐾𝑠,𝛼𝛼(𝛼−1)/2 2
𝑡𝛼/2∑𝐶𝑗,𝑗
𝛼/2
𝑗∈𝐽1
as 𝑡→∞
Key points:
• Applies to Student-like distributions (heavy-tailed)
• Asymptotic behavior depends on positive diagonal elements of 𝐶
• Power law decay with exponent 𝛼/2
Proof Strategy
1. Change of variables: Transform the problem to work with standard normal distribution
2. Apply Theorem 5.1 (not provided in the excerpt) 3. Analyze the boundary of the
transformed set
Change of Variables
Define:
• 𝑌: Random vector with centered normal distribution (identity covariance)
• 𝛷(𝑦): Normal cumulative distribution function
• 𝑆𝛼(𝑦): Student-like cumulative distribution function
• 𝑆𝛼
←, 𝛷←: Inverse functions of 𝑆𝛼 and 𝛷 respectively
Key transformation:
𝑋=
𝑑𝑆𝛼
←∘𝛷(𝑌)
Define sets: 𝐴𝑡={𝑥∈ℝ𝑑:⟨𝐶𝑥,𝑥⟩>𝑡}=√𝑡𝐴1
𝐵𝑡={𝑦∈ℝ𝑑:⟨𝐶𝑆𝛼
←∘𝛷(𝑦),𝑆𝛼
←∘𝛷(𝑦)⟩>𝑡}=𝛷←∘𝑆𝛼(𝐴𝑡)
Important result:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}=∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥𝑖=∫ 𝑒−|𝑦|2/2
(2𝜋)𝑑/2
𝐵𝑡𝑑𝑦
Asymptotic Analysis
Asymptotic Equivalence
Key result:
(𝛷←∘𝑆𝛼(𝑥))2=2𝛼log𝑥−loglog𝑥−2log(𝐾𝑠,𝛼𝛼𝛼/2)−2log(2√𝜋)+𝑜(1)
as 𝑥→∞.
Boundary Parameterization
Lemma 8.2.2: Provides parameterizations for ∂𝐴𝑡 and ∂𝐵𝑡 near specific points.
For ∂𝐴𝑡 near 𝑝𝜖,𝑗,𝑡 =𝜖√𝑡/𝐶𝑗,𝑗𝑒𝑗:
𝑝(𝑣)=𝜖√𝑡
𝐶𝑗,𝑗[1− 1
2𝑡⟨𝐶𝑣,𝑣⟩+𝑂(⟨𝐶𝑣,𝑣⟩
𝑡2)]𝑒𝑗+𝑣
For ∂𝐵𝑡 near 𝑞𝜖,𝑗,𝑡 =𝛷←∘𝑆𝛼(𝑝𝜖,𝑗,𝑡):
𝑞(𝑣)=𝜖√𝛼log 𝑡
𝐶𝑗,𝑗−loglog√𝑡
2√𝛼log𝑡−log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋)
√𝛼log𝑡+𝑜( 1
√log𝑡)𝑒𝑗
+ ∑ 𝛷←
𝑑
𝑖=1,𝑖≠𝑗 ∘𝑆𝛼(𝑣𝑖)𝑒𝑖
Note: These parameterizations are crucial for applying Theorem 5.1 and deriving the
asymptotic behavior.
These lecture notes cover asymptotic analysis of quadratic forms for random vectors with
heavy-tailed distributions.
Asymptotic Behavior of Quadratic Forms
Main Theorem
Theorem 8.2.1: Let 𝑋 be a 𝑑-dimensional random vector with independent and identically
distributed components having a Student-like distribution. Let 𝐶 be a 𝑑×𝑑 matrix. Define
𝐽1={𝑗:𝐶𝑗,𝑗 >0}. If 𝐽1 is not empty, then:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}∼𝐾𝑠,𝛼𝛼(𝛼−1)/2 2
𝑡𝛼/2∑𝐶𝑗,𝑗
𝛼/2
𝑗∈𝐽1
as 𝑡→∞
Key points:
• Applies to Student-like distributions (heavy-tailed)
• Asymptotic behavior depends on positive diagonal elements of 𝐶
• Power law decay with exponent 𝛼/2
Proof Strategy
1. Change of variables: Transform the problem to work with standard normal distribution
2. Apply Theorem 5.1 (not provided in the excerpt) 3. Analyze the boundary of the
transformed set
Change of Variables
Define:
• 𝑌: Random vector with centered normal distribution (identity covariance)
• 𝛷(𝑦): Normal cumulative distribution function
• 𝑆𝛼(𝑦): Student-like cumulative distribution function
• 𝑆𝛼
←, 𝛷←: Inverse functions of 𝑆𝛼 and 𝛷 respectively
Key transformation:
𝑋=
𝑑𝑆𝛼
←∘𝛷(𝑌)
Define sets: 𝐴𝑡={𝑥∈ℝ𝑑:⟨𝐶𝑥,𝑥⟩>𝑡}=√𝑡𝐴1
𝐵𝑡={𝑦∈ℝ𝑑:⟨𝐶𝑆𝛼
←∘𝛷(𝑦),𝑆𝛼
←∘𝛷(𝑦)⟩>𝑡}=𝛷←∘𝑆𝛼(𝐴𝑡)
Important result:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}=∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥𝑖=∫ 𝑒−|𝑦|2/2
(2𝜋)𝑑/2
𝐵𝑡𝑑𝑦
Asymptotic Analysis
Asymptotic Equivalence
Key result:
(𝛷←∘𝑆𝛼(𝑥))2=2𝛼log𝑥−loglog𝑥−2log(𝐾𝑠,𝛼𝛼𝛼/2)−2log(2√𝜋)+𝑜(1)
as 𝑥→∞.
Boundary Parameterization
Lemma 8.2.2: Provides parameterizations for ∂𝐴𝑡 and ∂𝐵𝑡 near specific points.
For ∂𝐴𝑡 near 𝑝𝜖,𝑗,𝑡 =𝜖√𝑡/𝐶𝑗,𝑗𝑒𝑗:
𝑝(𝑣)=𝜖√𝑡
𝐶𝑗,𝑗[1− 1
2𝑡⟨𝐶𝑣,𝑣⟩+𝑂(⟨𝐶𝑣,𝑣⟩
𝑡2)]𝑒𝑗+𝑣
For ∂𝐵𝑡 near 𝑞𝜖,𝑗,𝑡 =𝛷←∘𝑆𝛼(𝑝𝜖,𝑗,𝑡):
𝑞(𝑣)=𝜖√𝛼log 𝑡
𝐶𝑗,𝑗−loglog√𝑡
2√𝛼log𝑡−log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋)
√𝛼log𝑡+𝑜( 1
√log𝑡)𝑒𝑗
+ ∑ 𝛷←
𝑑
𝑖=1,𝑖≠𝑗 ∘𝑆𝛼(𝑣𝑖)𝑒𝑖
Note: These parameterizations are crucial for applying Theorem 5.1 and deriving the
asymptotic behavior.
These lecture notes cover asymptotic analysis of quadratic forms for random vectors with
heavy-tailed distributions.
Asymptotic Behavior of Quadratic Forms
Main Theorem
Theorem 8.2.1: Let 𝑋 be a 𝑑-dimensional random vector with independent and identically
distributed components having a Student-like distribution. Let 𝐶 be a 𝑑×𝑑 matrix. Define
𝐽1={𝑗:𝐶𝑗,𝑗 >0}. If 𝐽1 is not empty, then:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}∼𝐾𝑠,𝛼𝛼(𝛼−1)/2 2
𝑡𝛼/2∑𝐶𝑗,𝑗
𝛼/2
𝑗∈𝐽1
as 𝑡→∞
Key points:
• Applies to Student-like distributions (heavy-tailed)
• Asymptotic behavior depends on positive diagonal elements of 𝐶
• Power law decay with exponent 𝛼/2
Proof Strategy
1. Change of variables: Transform the problem to work with standard normal distribution
2. Apply Theorem 5.1 (not provided in the excerpt) 3. Analyze the boundary of the
transformed set
Change of Variables
Define:
• 𝑌: Random vector with centered normal distribution (identity covariance)
• 𝛷(𝑦): Normal cumulative distribution function
• 𝑆𝛼(𝑦): Student-like cumulative distribution function
• 𝑆𝛼
←, 𝛷←: Inverse functions of 𝑆𝛼 and 𝛷 respectively
Key transformation:
𝑋=
𝑑𝑆𝛼
←∘𝛷(𝑌)
Define sets: 𝐴𝑡={𝑥∈ℝ𝑑:⟨𝐶𝑥,𝑥⟩>𝑡}=√𝑡𝐴1
𝐵𝑡={𝑦∈ℝ𝑑:⟨𝐶𝑆𝛼
←∘𝛷(𝑦),𝑆𝛼
←∘𝛷(𝑦)⟩>𝑡}=𝛷←∘𝑆𝛼(𝐴𝑡)
Important result:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}=∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥𝑖=∫ 𝑒−|𝑦|2/2
(2𝜋)𝑑/2
𝐵𝑡𝑑𝑦
Asymptotic Analysis
Asymptotic Equivalence
Key result:
(𝛷←∘𝑆𝛼(𝑥))2=2𝛼log𝑥−loglog𝑥−2log(𝐾𝑠,𝛼𝛼𝛼/2)−2log(2√𝜋)+𝑜(1)
as 𝑥→∞.
Boundary Parameterization
Lemma 8.2.2: Provides parameterizations for ∂𝐴𝑡 and ∂𝐵𝑡 near specific points.
For ∂𝐴𝑡 near 𝑝𝜖,𝑗,𝑡 =𝜖√𝑡/𝐶𝑗,𝑗𝑒𝑗:
𝑝(𝑣)=𝜖√𝑡
𝐶𝑗,𝑗[1− 1
2𝑡⟨𝐶𝑣,𝑣⟩+𝑂(⟨𝐶𝑣,𝑣⟩
𝑡2)]𝑒𝑗+𝑣
For ∂𝐵𝑡 near 𝑞𝜖,𝑗,𝑡 =𝛷←∘𝑆𝛼(𝑝𝜖,𝑗,𝑡):
𝑞(𝑣)=𝜖√𝛼log 𝑡
𝐶𝑗,𝑗−loglog√𝑡
2√𝛼log𝑡−log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋)
√𝛼log𝑡+𝑜( 1
√log𝑡)𝑒𝑗
+ ∑ 𝛷←
𝑑
𝑖=1,𝑖≠𝑗 ∘𝑆𝛼(𝑣𝑖)𝑒𝑖
Note: These parameterizations are crucial for applying Theorem 5.1 and deriving the
asymptotic behavior.
These lecture notes cover asymptotic analysis of quadratic forms for random vectors with
heavy-tailed distributions.
Asymptotic Behavior of Quadratic Forms
Main Theorem
Theorem 8.2.1: Let 𝑋 be a 𝑑-dimensional random vector with independent and identically
distributed components having a Student-like distribution. Let 𝐶 be a 𝑑×𝑑 matrix. Define
𝐽1={𝑗:𝐶𝑗,𝑗 >0}. If 𝐽1 is not empty, then:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}∼𝐾𝑠,𝛼𝛼(𝛼−1)/2 2
𝑡𝛼/2∑𝐶𝑗,𝑗
𝛼/2
𝑗∈𝐽1
as 𝑡→∞
Key points:
• Applies to Student-like distributions (heavy-tailed)
• Asymptotic behavior depends on positive diagonal elements of 𝐶
• Power law decay with exponent 𝛼/2
Proof Strategy
1. Change of variables: Transform the problem to work with standard normal distribution
2. Apply Theorem 5.1 (not provided in the excerpt) 3. Analyze the boundary of the
transformed set
Change of Variables
Define:
• 𝑌: Random vector with centered normal distribution (identity covariance)
• 𝛷(𝑦): Normal cumulative distribution function
• 𝑆𝛼(𝑦): Student-like cumulative distribution function
• 𝑆𝛼
←, 𝛷←: Inverse functions of 𝑆𝛼 and 𝛷 respectively
Key transformation:
𝑋=
𝑑𝑆𝛼
←∘𝛷(𝑌)
Define sets: 𝐴𝑡={𝑥∈ℝ𝑑:⟨𝐶𝑥,𝑥⟩>𝑡}=√𝑡𝐴1
𝐵𝑡={𝑦∈ℝ𝑑:⟨𝐶𝑆𝛼
←∘𝛷(𝑦),𝑆𝛼
←∘𝛷(𝑦)⟩>𝑡}=𝛷←∘𝑆𝛼(𝐴𝑡)
Important result:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}=∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥𝑖=∫ 𝑒−|𝑦|2/2
(2𝜋)𝑑/2
𝐵𝑡𝑑𝑦
Asymptotic Analysis
Asymptotic Equivalence
Key result:
(𝛷←∘𝑆𝛼(𝑥))2=2𝛼log𝑥−loglog𝑥−2log(𝐾𝑠,𝛼𝛼𝛼/2)−2log(2√𝜋)+𝑜(1)
as 𝑥→∞.
Boundary Parameterization
Lemma 8.2.2: Provides parameterizations for ∂𝐴𝑡 and ∂𝐵𝑡 near specific points.
For ∂𝐴𝑡 near 𝑝𝜖,𝑗,𝑡 =𝜖√𝑡/𝐶𝑗,𝑗𝑒𝑗:
𝑝(𝑣)=𝜖√𝑡
𝐶𝑗,𝑗[1− 1
2𝑡⟨𝐶𝑣,𝑣⟩+𝑂(⟨𝐶𝑣,𝑣⟩
𝑡2)]𝑒𝑗+𝑣
For ∂𝐵𝑡 near 𝑞𝜖,𝑗,𝑡 =𝛷←∘𝑆𝛼(𝑝𝜖,𝑗,𝑡):
𝑞(𝑣)=𝜖√𝛼log 𝑡
𝐶𝑗,𝑗−loglog√𝑡
2√𝛼log𝑡−log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋)
√𝛼log𝑡+𝑜( 1
√log𝑡)𝑒𝑗
+ ∑ 𝛷←
𝑑
𝑖=1,𝑖≠𝑗 ∘𝑆𝛼(𝑣𝑖)𝑒𝑖
Note: These parameterizations are crucial for applying Theorem 5.1 and deriving the
asymptotic behavior.
These lecture notes cover asymptotic analysis of quadratic forms for random vectors with
heavy-tailed distributions.
Asymptotic Behavior of Quadratic Forms
Main Theorem
Theorem 8.2.1: Let 𝑋 be a 𝑑-dimensional random vector with independent and identically
distributed components having a Student-like distribution. Let 𝐶 be a 𝑑×𝑑 matrix. Define
𝐽1={𝑗:𝐶𝑗,𝑗 >0}. If 𝐽1 is not empty, then:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}∼𝐾𝑠,𝛼𝛼(𝛼−1)/2 2
𝑡𝛼/2∑𝐶𝑗,𝑗
𝛼/2
𝑗∈𝐽1
as 𝑡→∞
Key points:
• Applies to Student-like distributions (heavy-tailed)
• Asymptotic behavior depends on positive diagonal elements of 𝐶
• Power law decay with exponent 𝛼/2
Proof Strategy
1. Change of variables: Transform the problem to work with standard normal distribution
2. Apply Theorem 5.1 (not provided in the excerpt) 3. Analyze the boundary of the
transformed set
Change of Variables
Define:
• 𝑌: Random vector with centered normal distribution (identity covariance)
• 𝛷(𝑦): Normal cumulative distribution function
• 𝑆𝛼(𝑦): Student-like cumulative distribution function
• 𝑆𝛼
←, 𝛷←: Inverse functions of 𝑆𝛼 and 𝛷 respectively
Key transformation:
𝑋=
𝑑𝑆𝛼
←∘𝛷(𝑌)
Define sets: 𝐴𝑡={𝑥∈ℝ𝑑:⟨𝐶𝑥,𝑥⟩>𝑡}=√𝑡𝐴1
𝐵𝑡={𝑦∈ℝ𝑑:⟨𝐶𝑆𝛼
←∘𝛷(𝑦),𝑆𝛼
←∘𝛷(𝑦)⟩>𝑡}=𝛷←∘𝑆𝛼(𝐴𝑡)
Important result:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}=∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥𝑖=∫ 𝑒−|𝑦|2/2
(2𝜋)𝑑/2
𝐵𝑡𝑑𝑦
Asymptotic Analysis
Asymptotic Equivalence
Key result:
(𝛷←∘𝑆𝛼(𝑥))2=2𝛼log𝑥−loglog𝑥−2log(𝐾𝑠,𝛼𝛼𝛼/2)−2log(2√𝜋)+𝑜(1)
as 𝑥→∞.
Boundary Parameterization
Lemma 8.2.2: Provides parameterizations for ∂𝐴𝑡 and ∂𝐵𝑡 near specific points.
For ∂𝐴𝑡 near 𝑝𝜖,𝑗,𝑡 =𝜖√𝑡/𝐶𝑗,𝑗𝑒𝑗:
𝑝(𝑣)=𝜖√𝑡
𝐶𝑗,𝑗[1− 1
2𝑡⟨𝐶𝑣,𝑣⟩+𝑂(⟨𝐶𝑣,𝑣⟩
𝑡2)]𝑒𝑗+𝑣
For ∂𝐵𝑡 near 𝑞𝜖,𝑗,𝑡 =𝛷←∘𝑆𝛼(𝑝𝜖,𝑗,𝑡):
𝑞(𝑣)=𝜖√𝛼log 𝑡
𝐶𝑗,𝑗−loglog√𝑡
2√𝛼log𝑡−log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋)
√𝛼log𝑡+𝑜( 1
√log𝑡)𝑒𝑗
+ ∑ 𝛷←
𝑑
𝑖=1,𝑖≠𝑗 ∘𝑆𝛼(𝑣𝑖)𝑒𝑖
Note: These parameterizations are crucial for applying Theorem 5.1 and deriving the
asymptotic behavior.
These lecture notes cover asymptotic analysis of quadratic forms for random vectors with
heavy-tailed distributions.
Asymptotic Behavior of Quadratic Forms
Main Theorem
Theorem 8.2.1: Let 𝑋 be a 𝑑-dimensional random vector with independent and identically
distributed components having a Student-like distribution. Let 𝐶 be a 𝑑×𝑑 matrix. Define
𝐽1={𝑗:𝐶𝑗,𝑗 >0}. If 𝐽1 is not empty, then:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}∼𝐾𝑠,𝛼𝛼(𝛼−1)/2 2
𝑡𝛼/2∑𝐶𝑗,𝑗
𝛼/2
𝑗∈𝐽1
as 𝑡→∞
Key points:
• Applies to Student-like distributions (heavy-tailed)
• Asymptotic behavior depends on positive diagonal elements of 𝐶
• Power law decay with exponent 𝛼/2
Proof Strategy
1. Change of variables: Transform the problem to work with standard normal distribution
2. Apply Theorem 5.1 (not provided in the excerpt) 3. Analyze the boundary of the
transformed set
Change of Variables
Define:
• 𝑌: Random vector with centered normal distribution (identity covariance)
• 𝛷(𝑦): Normal cumulative distribution function
• 𝑆𝛼(𝑦): Student-like cumulative distribution function
• 𝑆𝛼
←, 𝛷←: Inverse functions of 𝑆𝛼 and 𝛷 respectively
Key transformation:
𝑋=
𝑑𝑆𝛼
←∘𝛷(𝑌)
Define sets: 𝐴𝑡={𝑥∈ℝ𝑑:⟨𝐶𝑥,𝑥⟩>𝑡}=√𝑡𝐴1
𝐵𝑡={𝑦∈ℝ𝑑:⟨𝐶𝑆𝛼
←∘𝛷(𝑦),𝑆𝛼
←∘𝛷(𝑦)⟩>𝑡}=𝛷←∘𝑆𝛼(𝐴𝑡)
Important result:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}=∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥𝑖=∫ 𝑒−|𝑦|2/2
(2𝜋)𝑑/2
𝐵𝑡𝑑𝑦
Asymptotic Analysis
Asymptotic Equivalence
Key result:
(𝛷←∘𝑆𝛼(𝑥))2=2𝛼log𝑥−loglog𝑥−2log(𝐾𝑠,𝛼𝛼𝛼/2)−2log(2√𝜋)+𝑜(1)
as 𝑥→∞.
Boundary Parameterization
Lemma 8.2.2: Provides parameterizations for ∂𝐴𝑡 and ∂𝐵𝑡 near specific points.
For ∂𝐴𝑡 near 𝑝𝜖,𝑗,𝑡 =𝜖√𝑡/𝐶𝑗,𝑗𝑒𝑗:
𝑝(𝑣)=𝜖√𝑡
𝐶𝑗,𝑗[1− 1
2𝑡⟨𝐶𝑣,𝑣⟩+𝑂(⟨𝐶𝑣,𝑣⟩
𝑡2)]𝑒𝑗+𝑣
For ∂𝐵𝑡 near 𝑞𝜖,𝑗,𝑡 =𝛷←∘𝑆𝛼(𝑝𝜖,𝑗,𝑡):
𝑞(𝑣)=𝜖√𝛼log 𝑡
𝐶𝑗,𝑗−loglog√𝑡
2√𝛼log𝑡−log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋)
√𝛼log𝑡+𝑜( 1
√log𝑡)𝑒𝑗
+ ∑ 𝛷←
𝑑
𝑖=1,𝑖≠𝑗 ∘𝑆𝛼(𝑣𝑖)𝑒𝑖
Note: These parameterizations are crucial for applying Theorem 5.1 and deriving the
asymptotic behavior.
These lecture notes cover asymptotic analysis of quadratic forms for random vectors with
heavy-tailed distributions.
Asymptotic Behavior of Quadratic Forms
Main Theorem
Theorem 8.2.1: Let 𝑋 be a 𝑑-dimensional random vector with independent and identically
distributed components having a Student-like distribution. Let 𝐶 be a 𝑑×𝑑 matrix. Define
𝐽1={𝑗:𝐶𝑗,𝑗 >0}. If 𝐽1 is not empty, then:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}∼𝐾𝑠,𝛼𝛼(𝛼−1)/2 2
𝑡𝛼/2∑𝐶𝑗,𝑗
𝛼/2
𝑗∈𝐽1
as 𝑡→∞
Key points:
• Applies to Student-like distributions (heavy-tailed)
• Asymptotic behavior depends on positive diagonal elements of 𝐶
• Power law decay with exponent 𝛼/2
Proof Strategy
1. Change of variables: Transform the problem to work with standard normal distribution
2. Apply Theorem 5.1 (not provided in the excerpt) 3. Analyze the boundary of the
transformed set
Change of Variables
Define:
• 𝑌: Random vector with centered normal distribution (identity covariance)
• 𝛷(𝑦): Normal cumulative distribution function
• 𝑆𝛼(𝑦): Student-like cumulative distribution function
• 𝑆𝛼
←, 𝛷←: Inverse functions of 𝑆𝛼 and 𝛷 respectively
Key transformation:
𝑋=
𝑑𝑆𝛼
←∘𝛷(𝑌)
Define sets:
𝐴𝑡={𝑥∈ℝ𝑑:⟨𝐶𝑥,𝑥⟩>𝑡}=√𝑡𝐴1
𝐵𝑡={𝑦∈ℝ𝑑:⟨𝐶𝑆𝛼
←∘𝛷(𝑦),𝑆𝛼
←∘𝛷(𝑦)⟩>𝑡}=𝛷←∘𝑆𝛼(𝐴𝑡)
Important result:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}=∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥𝑖=∫ 𝑒−|𝑦|2/2
(2𝜋)𝑑/2
𝐵𝑡𝑑𝑦
Asymptotic Analysis
Asymptotic Equivalence
Key result:
(𝛷←∘𝑆𝛼(𝑥))2=2𝛼log𝑥−loglog𝑥−2log(𝐾𝑠,𝛼𝛼𝛼/2)−2log(2√𝜋)+𝑜(1)
as 𝑥→∞.
Boundary Parameterization
Lemma 8.2.2: Provides parameterizations for ∂𝐴𝑡 and ∂𝐵𝑡 near specific points.
For ∂𝐴𝑡 near 𝑝𝜖,𝑗,𝑡 =𝜖√𝑡/𝐶𝑗,𝑗𝑒𝑗:
𝑝(𝑣)=𝜖√𝑡
𝐶𝑗,𝑗[1− 1
2𝑡⟨𝐶𝑣,𝑣⟩+𝑂(⟨𝐶𝑣,𝑣⟩
𝑡2)]𝑒𝑗+𝑣
For ∂𝐵𝑡 near 𝑞𝜖,𝑗,𝑡 =𝛷←∘𝑆𝛼(𝑝𝜖,𝑗,𝑡):
𝑞(𝑣)=𝜖√𝛼log 𝑡
𝐶𝑗,𝑗−loglog√𝑡
2√𝛼log𝑡−log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋)
√𝛼log𝑡+𝑜( 1
√log𝑡)𝑒𝑗
+ ∑ 𝛷←
𝑑
𝑖=1,𝑖≠𝑗 ∘𝑆𝛼(𝑣𝑖)𝑒𝑖
Note: These parameterizations are crucial for applying Theorem 5.1 and deriving the
asymptotic behavior.
These lecture notes cover asymptotic analysis of quadratic forms for random vectors with
heavy-tailed distributions.
Asymptotic Behavior of Quadratic Forms
Main Theorem
Theorem 8.2.1: Let 𝑋 be a 𝑑-dimensional random vector with independent and identically
distributed components having a Student-like distribution. Let 𝐶 be a 𝑑×𝑑 matrix. Define
𝐽1={𝑗:𝐶𝑗,𝑗 >0}. If 𝐽1 is not empty, then:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}∼𝐾𝑠,𝛼𝛼(𝛼−1)/2 2
𝑡𝛼/2∑𝐶𝑗,𝑗
𝛼/2
𝑗∈𝐽1
as 𝑡→∞
Key points:
• Applies to Student-like distributions (heavy-tailed)
• Asymptotic behavior depends on positive diagonal elements of 𝐶
• Power law decay with exponent 𝛼/2
Proof Strategy
1. Change of variables: Transform the problem to work with standard normal distribution
2. Apply Theorem 5.1 (not provided in the excerpt) 3. Analyze the boundary of the
transformed set
Change of Variables
Define:
• 𝑌: Random vector with centered normal distribution (identity covariance)
• 𝛷(𝑦): Normal cumulative distribution function
• 𝑆𝛼(𝑦): Student-like cumulative distribution function
• 𝑆𝛼
←, 𝛷←: Inverse functions of 𝑆𝛼 and 𝛷 respectively
Key transformation:
𝑋=
𝑑𝑆𝛼
←∘𝛷(𝑌)
Define sets: 𝐴𝑡={𝑥∈ℝ𝑑:⟨𝐶𝑥,𝑥⟩>𝑡}=√𝑡𝐴1
𝐵𝑡={𝑦∈ℝ𝑑:⟨𝐶𝑆𝛼
←∘𝛷(𝑦),𝑆𝛼
←∘𝛷(𝑦)⟩>𝑡}=𝛷←∘𝑆𝛼(𝐴𝑡)
Important result:
𝑃{⟨𝐶𝑋,𝑋⟩>𝑡}=∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥𝑖=∫ 𝑒−|𝑦|2/2
(2𝜋)𝑑/2
𝐵𝑡𝑑𝑦
Asymptotic Analysis
Asymptotic Equivalence
Key result:
(𝛷←∘𝑆𝛼(𝑥))2=2𝛼log𝑥−loglog𝑥−2log(𝐾𝑠,𝛼𝛼𝛼/2)−2log(2√𝜋)+𝑜(1)
as 𝑥→∞.
Boundary Parameterization
Lemma 8.2.2: Provides parameterizations for ∂𝐴𝑡 and ∂𝐵𝑡 near specific points.
For ∂𝐴𝑡 near 𝑝𝜖,𝑗,𝑡 =𝜖√𝑡/𝐶𝑗,𝑗𝑒𝑗:
𝑝(𝑣)=𝜖√𝑡
𝐶𝑗,𝑗[1− 1
2𝑡⟨𝐶𝑣,𝑣⟩+𝑂(⟨𝐶𝑣,𝑣⟩
𝑡2)]𝑒𝑗+𝑣
For ∂𝐵𝑡 near 𝑞𝜖,𝑗,𝑡 =𝛷←∘𝑆𝛼(𝑝𝜖,𝑗,𝑡):
𝑞(𝑣)=𝜖√𝛼log 𝑡
𝐶𝑗,𝑗−loglog√𝑡
2√𝛼log𝑡−log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋)
√𝛼log𝑡+𝑜( 1
√log𝑡)𝑒𝑗
+ ∑ 𝛷←
𝑑
𝑖=1,𝑖≠𝑗 ∘𝑆𝛼(𝑣𝑖)𝑒𝑖
Note: These parameterizations are crucial for applying Theorem 5.1 and deriving the
asymptotic behavior.
