Homothetic Sets, Homogeneous , and Laplace’s Method
Introduction
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
1. ∃ neighborhood of 0 not intersecting 𝐴1
2. 𝐼 is strictly convex and 𝛼-positively homogeneous
3. 𝛼 > 1 (to ensure strict convexity)
4. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
5. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
6. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
7. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
1. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
2. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
1. Guess the value of 𝑐𝐴𝑡,𝑀
2. Estimate the integral using Theorem 5.1
3. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
4. Use homogeneity properties to simplify expressions
5. Choose 𝑐(𝑡) to satisfy condition (5.4)
6. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
7. ∃ neighborhood of 0 not intersecting 𝐴1
8. 𝐼 is strictly convex and 𝛼-positively homogeneous
9. 𝛼 > 1 (to ensure strict convexity)
10. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
11. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
12. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
13. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
14. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
15. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
16. Guess the value of 𝑐𝐴𝑡,𝑀
17. Estimate the integral using Theorem 5.1
18. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
19. Use homogeneity properties to simplify expressions
20. Choose 𝑐(𝑡) to satisfy condition (5.4)
21. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
22. ∃ neighborhood of 0 not intersecting 𝐴1
23. 𝐼 is strictly convex and 𝛼-positively homogeneous
24. 𝛼 > 1 (to ensure strict convexity)
25. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
26. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
27. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
28. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
29. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
30. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
31. Guess the value of 𝑐𝐴𝑡,𝑀
32. Estimate the integral using Theorem 5.1
33. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
34. Use homogeneity properties to simplify expressions
35. Choose 𝑐(𝑡) to satisfy condition (5.4)
36. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
37. ∃ neighborhood of 0 not intersecting 𝐴1
38. 𝐼 is strictly convex and 𝛼-positively homogeneous
39. 𝛼 > 1 (to ensure strict convexity)
40. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
41. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
42. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
43. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
44. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
45. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
46. Guess the value of 𝑐𝐴𝑡,𝑀
47. Estimate the integral using Theorem 5.1
48. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
49. Use homogeneity properties to simplify expressions
50. Choose 𝑐(𝑡) to satisfy condition (5.4)
51. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
52. ∃ neighborhood of 0 not intersecting 𝐴1
53. 𝐼 is strictly convex and 𝛼-positively homogeneous
54. 𝛼 > 1 (to ensure strict convexity)
55. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
56. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
57. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
58. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
59. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
60. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
61. Guess the value of 𝑐𝐴𝑡,𝑀
62. Estimate the integral using Theorem 5.1
63. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
64. Use homogeneity properties to simplify expressions
65. Choose 𝑐(𝑡) to satisfy condition (5.4)
66. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
67. ∃ neighborhood of 0 not intersecting 𝐴1
68. 𝐼 is strictly convex and 𝛼-positively homogeneous
69. 𝛼 > 1 (to ensure strict convexity)
70. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
71. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
72. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
73. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
74. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
75. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
76. Guess the value of 𝑐𝐴𝑡,𝑀
77. Estimate the integral using Theorem 5.1
78. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
79. Use homogeneity properties to simplify expressions
80. Choose 𝑐(𝑡) to satisfy condition (5.4)
81. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
82. ∃ neighborhood of 0 not intersecting 𝐴1
83. 𝐼 is strictly convex and 𝛼-positively homogeneous
84. 𝛼 > 1 (to ensure strict convexity)
85. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
86. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
87. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
88. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
89. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
90. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
91. Guess the value of 𝑐𝐴𝑡,𝑀
92. Estimate the integral using Theorem 5.1
93. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
94. Use homogeneity properties to simplify expressions
95. Choose 𝑐(𝑡) to satisfy condition (5.4)
96. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
97. ∃ neighborhood of 0 not intersecting 𝐴1
98. 𝐼 is strictly convex and 𝛼-positively homogeneous
99. 𝛼 > 1 (to ensure strict convexity)
100. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
101. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
102. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
103. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
104. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
105. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
106. Guess the value of 𝑐𝐴𝑡,𝑀
107. Estimate the integral using Theorem 5.1
108. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
109. Use homogeneity properties to simplify expressions
110. Choose 𝑐(𝑡) to satisfy condition (5.4)
111. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
112. ∃ neighborhood of 0 not intersecting 𝐴1
113. 𝐼 is strictly convex and 𝛼-positively homogeneous
114. 𝛼 > 1 (to ensure strict convexity)
115. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
116. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
117. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
118. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
119. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
120. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
121. Guess the value of 𝑐𝐴𝑡,𝑀
122. Estimate the integral using Theorem 5.1
123. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
124. Use homogeneity properties to simplify expressions
125. Choose 𝑐(𝑡) to satisfy condition (5.4)
126. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
127. ∃ neighborhood of 0 not intersecting 𝐴1
128. 𝐼 is strictly convex and 𝛼-positively homogeneous
129. 𝛼 > 1 (to ensure strict convexity)
130. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
131. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
132. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
133. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
134. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
135. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
136. Guess the value of 𝑐𝐴𝑡,𝑀
137. Estimate the integral using Theorem 5.1
138. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
139. Use homogeneity properties to simplify expressions
140. Choose 𝑐(𝑡) to satisfy condition (5.4)
141. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
142. ∃ neighborhood of 0 not intersecting 𝐴1
143. 𝐼 is strictly convex and 𝛼-positively homogeneous
144. 𝛼 > 1 (to ensure strict convexity)
145. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
146. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
147. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
148. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
149. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
150. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
151. Guess the value of 𝑐𝐴𝑡,𝑀
152. Estimate the integral using Theorem 5.1
153. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
154. Use homogeneity properties to simplify expressions
155. Choose 𝑐(𝑡) to satisfy condition (5.4)
156. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
157. ∃ neighborhood of 0 not intersecting 𝐴1
158. 𝐼 is strictly convex and 𝛼-positively homogeneous
159. 𝛼 > 1 (to ensure strict convexity)
160. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
161. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
162. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
163. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
164. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
165. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
166. Guess the value of 𝑐𝐴𝑡,𝑀
167. Estimate the integral using Theorem 5.1
168. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
169. Use homogeneity properties to simplify expressions
170. Choose 𝑐(𝑡) to satisfy condition (5.4)
171. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
172. ∃ neighborhood of 0 not intersecting 𝐴1
173. 𝐼 is strictly convex and 𝛼-positively homogeneous
174. 𝛼 > 1 (to ensure strict convexity)
175. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
176. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
177. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
178. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
179. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
180. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
181. Guess the value of 𝑐𝐴𝑡,𝑀
182. Estimate the integral using Theorem 5.1
183. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
184. Use homogeneity properties to simplify expressions
185. Choose 𝑐(𝑡) to satisfy condition (5.4)
186. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
187. ∃ neighborhood of 0 not intersecting 𝐴1
188. 𝐼 is strictly convex and 𝛼-positively homogeneous
189. 𝛼 > 1 (to ensure strict convexity)
190. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
191. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
192. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
193. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
194. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
195. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
196. Guess the value of 𝑐𝐴𝑡,𝑀
197. Estimate the integral using Theorem 5.1
198. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
199. Use homogeneity properties to simplify expressions
200. Choose 𝑐(𝑡) to satisfy condition (5.4)
201. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
202. ∃ neighborhood of 0 not intersecting 𝐴1
203. 𝐼 is strictly convex and 𝛼-positively homogeneous
204. 𝛼 > 1 (to ensure strict convexity)
205. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
206. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
207. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
208. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
209. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
210. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
211. Guess the value of 𝑐𝐴𝑡,𝑀
212. Estimate the integral using Theorem 5.1
213. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
214. Use homogeneity properties to simplify expressions
215. Choose 𝑐(𝑡) to satisfy condition (5.4)
216. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
217. ∃ neighborhood of 0 not intersecting 𝐴1
218. 𝐼 is strictly convex and 𝛼-positively homogeneous
219. 𝛼 > 1 (to ensure strict convexity)
220. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
221. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
222. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
223. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
224. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
225. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
226. Guess the value of 𝑐𝐴𝑡,𝑀
227. Estimate the integral using Theorem 5.1
228. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
229. Use homogeneity properties to simplify expressions
230. Choose 𝑐(𝑡) to satisfy condition (5.4)
231. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
232. ∃ neighborhood of 0 not intersecting 𝐴1
233. 𝐼 is strictly convex and 𝛼-positively homogeneous
234. 𝛼 > 1 (to ensure strict convexity)
235. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
236. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
237. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
238. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
239. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
240. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
241. Guess the value of 𝑐𝐴𝑡,𝑀
242. Estimate the integral using Theorem 5.1
243. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
244. Use homogeneity properties to simplify expressions
245. Choose 𝑐(𝑡) to satisfy condition (5.4)
246. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
247. ∃ neighborhood of 0 not intersecting 𝐴1
248. 𝐼 is strictly convex and 𝛼-positively homogeneous
249. 𝛼 > 1 (to ensure strict convexity)
250. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
251. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
252. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
253. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
254. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
255. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
256. Guess the value of 𝑐𝐴𝑡,𝑀
257. Estimate the integral using Theorem 5.1
258. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
259. Use homogeneity properties to simplify expressions
260. Choose 𝑐(𝑡) to satisfy condition (5.4)
261. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
262. ∃ neighborhood of 0 not intersecting 𝐴1
263. 𝐼 is strictly convex and 𝛼-positively homogeneous
264. 𝛼 > 1 (to ensure strict convexity)
265. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
266. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
267. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
268. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
269. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
270. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
271. Guess the value of 𝑐𝐴𝑡,𝑀
272. Estimate the integral using Theorem 5.1
273. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
274. Use homogeneity properties to simplify expressions
275. Choose 𝑐(𝑡) to satisfy condition (5.4)
276. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
277. ∃ neighborhood of 0 not intersecting 𝐴1
278. 𝐼 is strictly convex and 𝛼-positively homogeneous
279. 𝛼 > 1 (to ensure strict convexity)
280. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
281. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
282. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
283. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
284. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
285. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
286. Guess the value of 𝑐𝐴𝑡,𝑀
287. Estimate the integral using Theorem 5.1
288. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
289. Use homogeneity properties to simplify expressions
290. Choose 𝑐(𝑡) to satisfy condition (5.4)
291. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
292. ∃ neighborhood of 0 not intersecting 𝐴1
293. 𝐼 is strictly convex and 𝛼-positively homogeneous
294. 𝛼 > 1 (to ensure strict convexity)
295. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
296. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
297. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
298. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
299. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
300. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
301. Guess the value of 𝑐𝐴𝑡,𝑀
302. Estimate the integral using Theorem 5.1
303. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
304. Use homogeneity properties to simplify expressions
305. Choose 𝑐(𝑡) to satisfy condition (5.4)
306. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
307. ∃ neighborhood of 0 not intersecting 𝐴1
308. 𝐼 is strictly convex and 𝛼-positively homogeneous
309. 𝛼 > 1 (to ensure strict convexity)
310. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
311. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
312. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
313. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
314. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
315. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
316. Guess the value of 𝑐𝐴𝑡,𝑀
317. Estimate the integral using Theorem 5.1
318. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
319. Use homogeneity properties to simplify expressions
320. Choose 𝑐(𝑡) to satisfy condition (5.4)
321. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
322. ∃ neighborhood of 0 not intersecting 𝐴1
323. 𝐼 is strictly convex and 𝛼-positively homogeneous
324. 𝛼 > 1 (to ensure strict convexity)
325. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
326. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
327. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
328. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
329. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
330. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
331. Guess the value of 𝑐𝐴𝑡,𝑀
332. Estimate the integral using Theorem 5.1
333. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
334. Use homogeneity properties to simplify expressions
335. Choose 𝑐(𝑡) to satisfy condition (5.4)
336. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
337. ∃ neighborhood of 0 not intersecting 𝐴1
338. 𝐼 is strictly convex and 𝛼-positively homogeneous
339. 𝛼 > 1 (to ensure strict convexity)
340. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
341. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
342. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
343. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
344. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
345. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
346. Guess the value of 𝑐𝐴𝑡,𝑀
347. Estimate the integral using Theorem 5.1
348. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
349. Use homogeneity properties to simplify expressions
350. Choose 𝑐(𝑡) to satisfy condition (5.4)
351. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
352. ∃ neighborhood of 0 not intersecting 𝐴1
353. 𝐼 is strictly convex and 𝛼-positively homogeneous
354. 𝛼 > 1 (to ensure strict convexity)
355. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
356. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
357. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
358. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
359. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
360. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
361. Guess the value of 𝑐𝐴𝑡,𝑀
362. Estimate the integral using Theorem 5.1
363. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
364. Use homogeneity properties to simplify expressions
365. Choose 𝑐(𝑡) to satisfy condition (5.4)
366. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
367. ∃ neighborhood of 0 not intersecting 𝐴1
368. 𝐼 is strictly convex and 𝛼-positively homogeneous
369. 𝛼 > 1 (to ensure strict convexity)
370. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
371. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
372. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
373. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
374. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
375. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
376. Guess the value of 𝑐𝐴𝑡,𝑀
377. Estimate the integral using Theorem 5.1
378. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
379. Use homogeneity properties to simplify expressions
380. Choose 𝑐(𝑡) to satisfy condition (5.4)
381. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
382. ∃ neighborhood of 0 not intersecting 𝐴1
383. 𝐼 is strictly convex and 𝛼-positively homogeneous
384. 𝛼 > 1 (to ensure strict convexity)
385. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
386. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
387. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
388. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
389. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
390. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
391. Guess the value of 𝑐𝐴𝑡,𝑀
392. Estimate the integral using Theorem 5.1
393. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
394. Use homogeneity properties to simplify expressions
395. Choose 𝑐(𝑡) to satisfy condition (5.4)
396. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
397. ∃ neighborhood of 0 not intersecting 𝐴1
398. 𝐼 is strictly convex and 𝛼-positively homogeneous
399. 𝛼 > 1 (to ensure strict convexity)
400. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
401. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
402. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
403. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
404. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
405. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
406. Guess the value of 𝑐𝐴𝑡,𝑀
407. Estimate the integral using Theorem 5.1
408. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
409. Use homogeneity properties to simplify expressions
410. Choose 𝑐(𝑡) to satisfy condition (5.4)
411. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
412. ∃ neighborhood of 0 not intersecting 𝐴1
413. 𝐼 is strictly convex and 𝛼-positively homogeneous
414. 𝛼 > 1 (to ensure strict convexity)
415. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
416. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
417. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
418. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
419. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
420. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
421. Guess the value of 𝑐𝐴𝑡,𝑀
422. Estimate the integral using Theorem 5.1
423. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
424. Use homogeneity properties to simplify expressions
425. Choose 𝑐(𝑡) to satisfy condition (5.4)
426. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
427. ∃ neighborhood of 0 not intersecting 𝐴1
428. 𝐼 is strictly convex and 𝛼-positively homogeneous
429. 𝛼 > 1 (to ensure strict convexity)
430. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
431. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
432. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
433. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
434. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
435. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
436. Guess the value of 𝑐𝐴𝑡,𝑀
437. Estimate the integral using Theorem 5.1
438. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
439. Use homogeneity properties to simplify expressions
440. Choose 𝑐(𝑡) to satisfy condition (5.4)
441. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
442. ∃ neighborhood of 0 not intersecting 𝐴1
443. 𝐼 is strictly convex and 𝛼-positively homogeneous
444. 𝛼 > 1 (to ensure strict convexity)
445. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
446. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
447. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
448. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
449. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
450. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
451. Guess the value of 𝑐𝐴𝑡,𝑀
452. Estimate the integral using Theorem 5.1
453. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
454. Use homogeneity properties to simplify expressions
455. Choose 𝑐(𝑡) to satisfy condition (5.4)
456. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
457. ∃ neighborhood of 0 not intersecting 𝐴1
458. 𝐼 is strictly convex and 𝛼-positively homogeneous
459. 𝛼 > 1 (to ensure strict convexity)
460. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
461. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
462. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
463. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
464. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
465. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
466. Guess the value of 𝑐𝐴𝑡,𝑀
467. Estimate the integral using Theorem 5.1
468. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
469. Use homogeneity properties to simplify expressions
470. Choose 𝑐(𝑡) to satisfy condition (5.4)
471. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
472. ∃ neighborhood of 0 not intersecting 𝐴1
473. 𝐼 is strictly convex and 𝛼-positively homogeneous
474. 𝛼 > 1 (to ensure strict convexity)
475. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
476. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
477. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
478. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
479. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
480. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
481. Guess the value of 𝑐𝐴𝑡,𝑀
482. Estimate the integral using Theorem 5.1
483. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
484. Use homogeneity properties to simplify expressions
485. Choose 𝑐(𝑡) to satisfy condition (5.4)
486. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
487. ∃ neighborhood of 0 not intersecting 𝐴1
488. 𝐼 is strictly convex and 𝛼-positively homogeneous
489. 𝛼 > 1 (to ensure strict convexity)
490. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
491. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
492. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
493. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
494. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
495. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
496. Guess the value of 𝑐𝐴𝑡,𝑀
497. Estimate the integral using Theorem 5.1
498. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
499. Use homogeneity properties to simplify expressions
500. Choose 𝑐(𝑡) to satisfy condition (5.4)
501. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
502. ∃ neighborhood of 0 not intersecting 𝐴1
503. 𝐼 is strictly convex and 𝛼-positively homogeneous
504. 𝛼 > 1 (to ensure strict convexity)
505. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
506. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
507. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
508. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
509. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
510. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
511. Guess the value of 𝑐𝐴𝑡,𝑀
512. Estimate the integral using Theorem 5.1
513. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
514. Use homogeneity properties to simplify expressions
515. Choose 𝑐(𝑡) to satisfy condition (5.4)
516. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
517. ∃ neighborhood of 0 not intersecting 𝐴1
518. 𝐼 is strictly convex and 𝛼-positively homogeneous
519. 𝛼 > 1 (to ensure strict convexity)
520. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
521. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
522. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
523. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
524. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
525. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
526. Guess the value of 𝑐𝐴𝑡,𝑀
527. Estimate the integral using Theorem 5.1
528. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
529. Use homogeneity properties to simplify expressions
530. Choose 𝑐(𝑡) to satisfy condition (5.4)
531. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
532. ∃ neighborhood of 0 not intersecting 𝐴1
533. 𝐼 is strictly convex and 𝛼-positively homogeneous
534. 𝛼 > 1 (to ensure strict convexity)
535. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
536. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
537. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
538. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
539. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
540. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
541. Guess the value of 𝑐𝐴𝑡,𝑀
542. Estimate the integral using Theorem 5.1
543. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
544. Use homogeneity properties to simplify expressions
545. Choose 𝑐(𝑡) to satisfy condition (5.4)
546. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
547. ∃ neighborhood of 0 not intersecting 𝐴1
548. 𝐼 is strictly convex and 𝛼-positively homogeneous
549. 𝛼 > 1 (to ensure strict convexity)
550. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
551. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
552. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
553. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
554. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
555. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
556. Guess the value of 𝑐𝐴𝑡,𝑀
557. Estimate the integral using Theorem 5.1
558. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
559. Use homogeneity properties to simplify expressions
560. Choose 𝑐(𝑡) to satisfy condition (5.4)
561. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
562. ∃ neighborhood of 0 not intersecting 𝐴1
563. 𝐼 is strictly convex and 𝛼-positively homogeneous
564. 𝛼 > 1 (to ensure strict convexity)
565. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
566. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
567. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
568. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
569. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
570. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
571. Guess the value of 𝑐𝐴𝑡,𝑀
572. Estimate the integral using Theorem 5.1
573. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
574. Use homogeneity properties to simplify expressions
575. Choose 𝑐(𝑡) to satisfy condition (5.4)
576. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
577. ∃ neighborhood of 0 not intersecting 𝐴1
578. 𝐼 is strictly convex and 𝛼-positively homogeneous
579. 𝛼 > 1 (to ensure strict convexity)
580. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
581. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
582. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
583. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
584. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
585. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
586. Guess the value of 𝑐𝐴𝑡,𝑀
587. Estimate the integral using Theorem 5.1
588. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
589. Use homogeneity properties to simplify expressions
590. Choose 𝑐(𝑡) to satisfy condition (5.4)
591. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
592. ∃ neighborhood of 0 not intersecting 𝐴1
593. 𝐼 is strictly convex and 𝛼-positively homogeneous
594. 𝛼 > 1 (to ensure strict convexity)
595. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
596. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
597. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
598. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
599. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
600. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
601. Guess the value of 𝑐𝐴𝑡,𝑀
602. Estimate the integral using Theorem 5.1
603. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
604. Use homogeneity properties to simplify expressions
605. Choose 𝑐(𝑡) to satisfy condition (5.4)
606. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
607. ∃ neighborhood of 0 not intersecting 𝐴1
608. 𝐼 is strictly convex and 𝛼-positively homogeneous
609. 𝛼 > 1 (to ensure strict convexity)
610. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
611. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
612. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
613. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
614. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
615. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
616. Guess the value of 𝑐𝐴𝑡,𝑀
617. Estimate the integral using Theorem 5.1
618. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
619. Use homogeneity properties to simplify expressions
620. Choose 𝑐(𝑡) to satisfy condition (5.4)
621. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
622. ∃ neighborhood of 0 not intersecting 𝐴1
623. 𝐼 is strictly convex and 𝛼-positively homogeneous
624. 𝛼 > 1 (to ensure strict convexity)
625. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
626. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
627. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
628. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
629. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
630. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
631. Guess the value of 𝑐𝐴𝑡,𝑀
632. Estimate the integral using Theorem 5.1
633. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
634. Use homogeneity properties to simplify expressions
635. Choose 𝑐(𝑡) to satisfy condition (5.4)
636. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
637. ∃ neighborhood of 0 not intersecting 𝐴1
638. 𝐼 is strictly convex and 𝛼-positively homogeneous
639. 𝛼 > 1 (to ensure strict convexity)
640. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
641. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
642. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
643. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
644. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
645. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
646. Guess the value of 𝑐𝐴𝑡,𝑀
647. Estimate the integral using Theorem 5.1
648. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
649. Use homogeneity properties to simplify expressions
650. Choose 𝑐(𝑡) to satisfy condition (5.4)
651. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
652. ∃ neighborhood of 0 not intersecting 𝐴1
653. 𝐼 is strictly convex and 𝛼-positively homogeneous
654. 𝛼 > 1 (to ensure strict convexity)
655. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
656. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
657. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
658. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
659. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
660. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
661. Guess the value of 𝑐𝐴𝑡,𝑀
662. Estimate the integral using Theorem 5.1
663. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
664. Use homogeneity properties to simplify expressions
665. Choose 𝑐(𝑡) to satisfy condition (5.4)
666. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
667. ∃ neighborhood of 0 not intersecting 𝐴1
668. 𝐼 is strictly convex and 𝛼-positively homogeneous
669. 𝛼 > 1 (to ensure strict convexity)
670. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
671. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
672. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
673. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
674. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
675. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
676. Guess the value of 𝑐𝐴𝑡,𝑀
677. Estimate the integral using Theorem 5.1
678. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
679. Use homogeneity properties to simplify expressions
680. Choose 𝑐(𝑡) to satisfy condition (5.4)
681. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
682. ∃ neighborhood of 0 not intersecting 𝐴1
683. 𝐼 is strictly convex and 𝛼-positively homogeneous
684. 𝛼 > 1 (to ensure strict convexity)
685. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
686. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
687. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
688. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
689. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
690. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
691. Guess the value of 𝑐𝐴𝑡,𝑀
692. Estimate the integral using Theorem 5.1
693. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
694. Use homogeneity properties to simplify expressions
695. Choose 𝑐(𝑡) to satisfy condition (5.4)
696. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
697. ∃ neighborhood of 0 not intersecting 𝐴1
698. 𝐼 is strictly convex and 𝛼-positively homogeneous
699. 𝛼 > 1 (to ensure strict convexity)
700. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
701. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
702. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
703. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
704. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
705. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
706. Guess the value of 𝑐𝐴𝑡,𝑀
707. Estimate the integral using Theorem 5.1
708. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
709. Use homogeneity properties to simplify expressions
710. Choose 𝑐(𝑡) to satisfy condition (5.4)
711. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
712. ∃ neighborhood of 0 not intersecting 𝐴1
713. 𝐼 is strictly convex and 𝛼-positively homogeneous
714. 𝛼 > 1 (to ensure strict convexity)
715. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
716. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
717. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
718. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
719. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
720. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
721. Guess the value of 𝑐𝐴𝑡,𝑀
722. Estimate the integral using Theorem 5.1
723. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
724. Use homogeneity properties to simplify expressions
725. Choose 𝑐(𝑡) to satisfy condition (5.4)
726. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
727. ∃ neighborhood of 0 not intersecting 𝐴1
728. 𝐼 is strictly convex and 𝛼-positively homogeneous
729. 𝛼 > 1 (to ensure strict convexity)
730. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
731. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
732. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
733. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
734. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
735. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
736. Guess the value of 𝑐𝐴𝑡,𝑀
737. Estimate the integral using Theorem 5.1
738. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
739. Use homogeneity properties to simplify expressions
740. Choose 𝑐(𝑡) to satisfy condition (5.4)
741. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
742. ∃ neighborhood of 0 not intersecting 𝐴1
743. 𝐼 is strictly convex and 𝛼-positively homogeneous
744. 𝛼 > 1 (to ensure strict convexity)
745. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
746. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
747. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
748. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
749. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
750. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
751. Guess the value of 𝑐𝐴𝑡,𝑀
752. Estimate the integral using Theorem 5.1
753. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
754. Use homogeneity properties to simplify expressions
755. Choose 𝑐(𝑡) to satisfy condition (5.4)
756. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
757. ∃ neighborhood of 0 not intersecting 𝐴1
758. 𝐼 is strictly convex and 𝛼-positively homogeneous
759. 𝛼 > 1 (to ensure strict convexity)
760. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
761. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
762. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
763. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
764. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
765. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
766. Guess the value of 𝑐𝐴𝑡,𝑀
767. Estimate the integral using Theorem 5.1
768. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
769. Use homogeneity properties to simplify expressions
770. Choose 𝑐(𝑡) to satisfy condition (5.4)
771. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
772. ∃ neighborhood of 0 not intersecting 𝐴1
773. 𝐼 is strictly convex and 𝛼-positively homogeneous
774. 𝛼 > 1 (to ensure strict convexity)
775. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
776. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
777. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
778. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
779. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
780. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
781. Guess the value of 𝑐𝐴𝑡,𝑀
782. Estimate the integral using Theorem 5.1
783. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
784. Use homogeneity properties to simplify expressions
785. Choose 𝑐(𝑡) to satisfy condition (5.4)
786. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
787. ∃ neighborhood of 0 not intersecting 𝐴1
788. 𝐼 is strictly convex and 𝛼-positively homogeneous
789. 𝛼 > 1 (to ensure strict convexity)
790. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
791. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
792. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
793. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
794. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
795. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
796. Guess the value of 𝑐𝐴𝑡,𝑀
797. Estimate the integral using Theorem 5.1
798. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
799. Use homogeneity properties to simplify expressions
800. Choose 𝑐(𝑡) to satisfy condition (5.4)
801. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
802. ∃ neighborhood of 0 not intersecting 𝐴1
803. 𝐼 is strictly convex and 𝛼-positively homogeneous
804. 𝛼 > 1 (to ensure strict convexity)
805. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
806. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
807. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
808. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
809. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
810. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
811. Guess the value of 𝑐𝐴𝑡,𝑀
812. Estimate the integral using Theorem 5.1
813. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
814. Use homogeneity properties to simplify expressions
815. Choose 𝑐(𝑡) to satisfy condition (5.4)
816. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
817. ∃ neighborhood of 0 not intersecting 𝐴1
818. 𝐼 is strictly convex and 𝛼-positively homogeneous
819. 𝛼 > 1 (to ensure strict convexity)
820. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
821. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
822. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
823. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
824. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
825. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
826. Guess the value of 𝑐𝐴𝑡,𝑀
827. Estimate the integral using Theorem 5.1
828. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
829. Use homogeneity properties to simplify expressions
830. Choose 𝑐(𝑡) to satisfy condition (5.4)
831. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
832. ∃ neighborhood of 0 not intersecting 𝐴1
833. 𝐼 is strictly convex and 𝛼-positively homogeneous
834. 𝛼 > 1 (to ensure strict convexity)
835. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
836. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
837. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
838. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
839. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
840. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
841. Guess the value of 𝑐𝐴𝑡,𝑀
842. Estimate the integral using Theorem 5.1
843. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
844. Use homogeneity properties to simplify expressions
845. Choose 𝑐(𝑡) to satisfy condition (5.4)
846. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
847. ∃ neighborhood of 0 not intersecting 𝐴1
848. 𝐼 is strictly convex and 𝛼-positively homogeneous
849. 𝛼 > 1 (to ensure strict convexity)
850. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
851. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
852. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
853. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
854. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
855. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
856. Guess the value of 𝑐𝐴𝑡,𝑀
857. Estimate the integral using Theorem 5.1
858. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
859. Use homogeneity properties to simplify expressions
860. Choose 𝑐(𝑡) to satisfy condition (5.4)
861. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
862. ∃ neighborhood of 0 not intersecting 𝐴1
863. 𝐼 is strictly convex and 𝛼-positively homogeneous
864. 𝛼 > 1 (to ensure strict convexity)
865. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
866. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
867. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
868. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
869. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
870. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
871. Guess the value of 𝑐𝐴𝑡,𝑀
872. Estimate the integral using Theorem 5.1
873. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
874. Use homogeneity properties to simplify expressions
875. Choose 𝑐(𝑡) to satisfy condition (5.4)
876. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
877. ∃ neighborhood of 0 not intersecting 𝐴1
878. 𝐼 is strictly convex and 𝛼-positively homogeneous
879. 𝛼 > 1 (to ensure strict convexity)
880. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
881. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
882. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
883. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
884. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
885. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
886. Guess the value of 𝑐𝐴𝑡,𝑀
887. Estimate the integral using Theorem 5.1
888. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
889. Use homogeneity properties to simplify expressions
890. Choose 𝑐(𝑡) to satisfy condition (5.4)
891. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
892. ∃ neighborhood of 0 not intersecting 𝐴1
893. 𝐼 is strictly convex and 𝛼-positively homogeneous
894. 𝛼 > 1 (to ensure strict convexity)
895. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
896. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
897. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
898. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
899. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
900. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
901. Guess the value of 𝑐𝐴𝑡,𝑀
902. Estimate the integral using Theorem 5.1
903. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
904. Use homogeneity properties to simplify expressions
905. Choose 𝑐(𝑡) to satisfy condition (5.4)
906. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
907. ∃ neighborhood of 0 not intersecting 𝐴1
908. 𝐼 is strictly convex and 𝛼-positively homogeneous
909. 𝛼 > 1 (to ensure strict convexity)
910. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
911. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
912. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
913. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
914. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
915. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
916. Guess the value of 𝑐𝐴𝑡,𝑀
917. Estimate the integral using Theorem 5.1
918. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
919. Use homogeneity properties to simplify expressions
920. Choose 𝑐(𝑡) to satisfy condition (5.4)
921. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
922. ∃ neighborhood of 0 not intersecting 𝐴1
923. 𝐼 is strictly convex and 𝛼-positively homogeneous
924. 𝛼 > 1 (to ensure strict convexity)
925. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
926. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
927. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
928. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
929. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
930. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
931. Guess the value of 𝑐𝐴𝑡,𝑀
932. Estimate the integral using Theorem 5.1
933. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
934. Use homogeneity properties to simplify expressions
935. Choose 𝑐(𝑡) to satisfy condition (5.4)
936. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
937. ∃ neighborhood of 0 not intersecting 𝐴1
938. 𝐼 is strictly convex and 𝛼-positively homogeneous
939. 𝛼 > 1 (to ensure strict convexity)
940. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
941. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
942. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
943. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
944. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
945. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
946. Guess the value of 𝑐𝐴𝑡,𝑀
947. Estimate the integral using Theorem 5.1
948. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
949. Use homogeneity properties to simplify expressions
950. Choose 𝑐(𝑡) to satisfy condition (5.4)
951. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
952. ∃ neighborhood of 0 not intersecting 𝐴1
953. 𝐼 is strictly convex and 𝛼-positively homogeneous
954. 𝛼 > 1 (to ensure strict convexity)
955. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
956. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
957. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
958. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
959. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
960. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
961. Guess the value of 𝑐𝐴𝑡,𝑀
962. Estimate the integral using Theorem 5.1
963. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
964. Use homogeneity properties to simplify expressions
965. Choose 𝑐(𝑡) to satisfy condition (5.4)
966. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
967. ∃ neighborhood of 0 not intersecting 𝐴1
968. 𝐼 is strictly convex and 𝛼-positively homogeneous
969. 𝛼 > 1 (to ensure strict convexity)
970. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
971. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
972. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
973. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
974. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
975. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
976. Guess the value of 𝑐𝐴𝑡,𝑀
977. Estimate the integral using Theorem 5.1
978. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
979. Use homogeneity properties to simplify expressions
980. Choose 𝑐(𝑡) to satisfy condition (5.4)
981. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
982. ∃ neighborhood of 0 not intersecting 𝐴1
983. 𝐼 is strictly convex and 𝛼-positively homogeneous
984. 𝛼 > 1 (to ensure strict convexity)
985. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
986. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
987. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
988. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
989. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
990. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
991. Guess the value of 𝑐𝐴𝑡,𝑀
992. Estimate the integral using Theorem 5.1
993. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
994. Use homogeneity properties to simplify expressions
995. Choose 𝑐(𝑡) to satisfy condition (5.4)
996. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
997. ∃ neighborhood of 0 not intersecting 𝐴1
998. 𝐼 is strictly convex and 𝛼-positively homogeneous
999. 𝛼 > 1 (to ensure strict convexity)
1000. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
1001. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
1002. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
1003. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
1004. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
1005. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
1006. Guess the value of 𝑐𝐴𝑡,𝑀
1007. Estimate the integral using Theorem 5.1
1008. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
1009. Use homogeneity properties to simplify expressions
1010. Choose 𝑐(𝑡) to satisfy condition (5.4)
1011. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
1012. ∃ neighborhood of 0 not intersecting 𝐴1
1013. 𝐼 is strictly convex and 𝛼-positively homogeneous
1014. 𝛼 > 1 (to ensure strict convexity)
1015. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
1016. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
1017. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
1018. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
1019. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
1020. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
1021. Guess the value of 𝑐𝐴𝑡,𝑀
1022. Estimate the integral using Theorem 5.1
1023. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
1024. Use homogeneity properties to simplify expressions
1025. Choose 𝑐(𝑡) to satisfy condition (5.4)
1026. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
1027. ∃ neighborhood of 0 not intersecting 𝐴1
1028. 𝐼 is strictly convex and 𝛼-positively homogeneous
1029. 𝛼 > 1 (to ensure strict convexity)
1030. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
1031. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
1032. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
1033. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
1034. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
1035. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
1036. Guess the value of 𝑐𝐴𝑡,𝑀
1037. Estimate the integral using Theorem 5.1
1038. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
1039. Use homogeneity properties to simplify expressions
1040. Choose 𝑐(𝑡) to satisfy condition (5.4)
1041. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
1042. ∃ neighborhood of 0 not intersecting 𝐴1
1043. 𝐼 is strictly convex and 𝛼-positively homogeneous
1044. 𝛼 > 1 (to ensure strict convexity)
1045. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
1046. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
1047. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
1048. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
1049. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
1050. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
1051. Guess the value of 𝑐𝐴𝑡,𝑀
1052. Estimate the integral using Theorem 5.1
1053. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
1054. Use homogeneity properties to simplify expressions
1055. Choose 𝑐(𝑡) to satisfy condition (5.4)
1056. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
1057. ∃ neighborhood of 0 not intersecting 𝐴1
1058. 𝐼 is strictly convex and 𝛼-positively homogeneous
1059. 𝛼 > 1 (to ensure strict convexity)
1060. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
1061. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
1062. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
1063. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
1064. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
1065. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
1066. Guess the value of 𝑐𝐴𝑡,𝑀
1067. Estimate the integral using Theorem 5.1
1068. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
1069. Use homogeneity properties to simplify expressions
1070. Choose 𝑐(𝑡) to satisfy condition (5.4)
1071. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
1072. ∃ neighborhood of 0 not intersecting 𝐴1
1073. 𝐼 is strictly convex and 𝛼-positively homogeneous
1074. 𝛼 > 1 (to ensure strict convexity)
1075. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
1076. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
1077. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
1078. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
1079. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
1080. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
1081. Guess the value of 𝑐𝐴𝑡,𝑀
1082. Estimate the integral using Theorem 5.1
1083. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
1084. Use homogeneity properties to simplify expressions
1085. Choose 𝑐(𝑡) to satisfy condition (5.4)
1086. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
1087. ∃ neighborhood of 0 not intersecting 𝐴1
1088. 𝐼 is strictly convex and 𝛼-positively homogeneous
1089. 𝛼 > 1 (to ensure strict convexity)
1090. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
1091. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
1092. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
1093. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
1094. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
1095. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
1096. Guess the value of 𝑐𝐴𝑡,𝑀
1097. Estimate the integral using Theorem 5.1
1098. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
1099. Use homogeneity properties to simplify expressions
1100. Choose 𝑐(𝑡) to satisfy condition (5.4)
1101. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
1102. ∃ neighborhood of 0 not intersecting 𝐴1
1103. 𝐼 is strictly convex and 𝛼-positively homogeneous
1104. 𝛼 > 1 (to ensure strict convexity)
1105. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
1106. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
1107. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
1108. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
1109. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
1110. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
1111. Guess the value of 𝑐𝐴𝑡,𝑀
1112. Estimate the integral using Theorem 5.1
1113. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
1114. Use homogeneity properties to simplify expressions
1115. Choose 𝑐(𝑡) to satisfy condition (5.4)
1116. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
1117. ∃ neighborhood of 0 not intersecting 𝐴1
1118. 𝐼 is strictly convex and 𝛼-positively homogeneous
1119. 𝛼 > 1 (to ensure strict convexity)
1120. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
1121. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
1122. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
1123. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
1124. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
1125. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
1126. Guess the value of 𝑐𝐴𝑡,𝑀
1127. Estimate the integral using Theorem 5.1
1128. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
1129. Use homogeneity properties to simplify expressions
1130. Choose 𝑐(𝑡) to satisfy condition (5.4)
1131. Verify all assumptions needed for Theorem 5.1
These lecture notes cover homothetic sets, homogeneous functions, and their application in
Laplace’s method for asymptotic analysis.
Key Concepts
• Set 𝐴1: A set with a neighborhood of 0 not intersecting 𝐴1.
• Homothetic sets: 𝐴𝑡= 𝑡𝐴1, moving to infinity as 𝑡 → ∞.
• Function 𝐼: Strictly convex, 𝛼-positively homogeneous function.
• 𝐷𝐴1: Defined as 𝐴1∩𝐼(𝐴1).
Important Assumptions
1132. ∃ neighborhood of 0 not intersecting 𝐴1
1133. 𝐼 is strictly convex and 𝛼-positively homogeneous
1134. 𝛼 > 1 (to ensure strict convexity)
1135. ∂𝐴1, 𝐷𝐴1, and 𝑐 are smooth (twice continuously differentiable) manifolds
1136. 𝐴1 separates from 𝐼(𝐴1) with contact of order 1
1137. det𝐺𝐴1(𝑝)≠ 0 for any 𝑝 ∈ 𝐷𝐴1
1138. Local thickness assumption near 𝐷𝐴1
Key Equation
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Main Theorem (Theorem 7.1)
Under assumptions (7.1)-(7.5), and if 𝐷𝐴1 is a base manifold for 𝐴1, then:
∫ 𝑒−𝐼(𝑥)
𝐴𝑡𝑑𝑥 ∼ 𝑐1𝑒−𝑡𝛼𝐼(𝐴1)𝑡𝑘−(𝛼−2)𝑑−𝑘
2−𝛼
2 as 𝑡 → ∞
where
𝑐1=(2𝜋)𝑑−𝑘−1
2∫𝑑𝑀𝐷𝐴1
|𝐷𝐼|𝑑−𝑘+1
2(det𝐺𝐴1)1
2
𝐷𝐴1
Note: For 𝛼 = 2, the polynomial term in 𝑡 has exponent 𝑘−1, independent of the ambient
space dimension 𝑑.
Lemma 7.2
If 𝐼 is 𝛼-homogeneous, then:
1139. 𝜓(𝑝,𝑠)= 𝑡−1𝜓(𝑡𝑝,𝑡𝛼𝑠)
1140. 𝜏𝑡𝐴1(𝑡𝑝)= 𝑡𝛼𝜏𝐴1(𝑝)
Proof Outline of Theorem 7.1
1141. Guess the value of 𝑐𝐴𝑡,𝑀
1142. Estimate the integral using Theorem 5.1
1143. Define 𝐷𝐴𝑡=𝑡𝐷𝐴1
1144. Use homogeneity properties to simplify expressions
1145. Choose 𝑐(𝑡) to satisfy condition (5.4)
1146. Verify all assumptions needed for Theorem 5.1
Conclusion
This approach demonstrates how Theorem 5.1 can be used as a generalization of Laplace’s
method for asymptotic analysis of integrals over homothetic sets with homogeneous
functions.
