Advanced Asymptotics and Conditional Distributions
Conditional Distributions: Example 4
Product of Cauchy Variables
Consider 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
Key points:
• Set 𝐴𝑡= {(𝑥,𝑦):𝑥𝑦 > 𝑡}
• Unique contact point between ∂𝐴𝑡 and maximal level set of density
• Decay of density crucial
• Shape of domain affects results
Asymptotic Behavior
𝑃{𝑋𝑌 > 𝑡} ∼ 2
𝜋2log𝑡
𝑡 as 𝑡 → ∞
For 𝛼 ∈ (0,1):
𝑃{𝑋 > 𝑡𝛼|𝑋𝑌 > 𝑡} ∼ 1−𝛼
2
Result: (𝑋,𝑌)/𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to 𝛿(0,0)
Logarithmic Scale Analysis
Writing 𝑋 = 𝜖1𝑒𝑈, 𝑌 = 𝜖2𝑒𝑉:
Key result: Distribution of (𝑈,𝑉)/log𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to uniform
distribution over {(𝑠,1−𝑠):𝑠 ∈ [0,1]}
Motivation for Systematic Methods
• Specific arguments for simple cases not generalizable
• Higher dimensions make visualization impossible
• Need for systematic procedures independent of intuition
Connection to Laplace’s Method
Theorem (Laplace’s Approximation): For ∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 with suitable conditions:
∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 ∼ (2𝜋)𝑑/2
√det𝐷2𝐼(𝑎∗)𝜆−𝑑/2𝑒−𝜆𝐼(𝑎∗) as 𝜆 → ∞
Relation to Our Problem
For 𝐴 = 𝑡𝐴1 and 𝐼 homogeneous of degree 𝛼:
∫ 𝑒−𝐼(𝑥)
𝑡𝐴1𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Key insight: Our method generalizes Laplace’s method with an infinite-dimensional
parameter
Challenges and Approach
• 𝐼 may not have a unique minimum on 𝐴
• Minimum may not be in the interior of 𝐴
• 𝐼 attains minimum on a set 𝐷𝐴 (potentially a k-dimensional manifold)
Approach:
• Apply Laplace’s approximation along fibers orthogonal to 𝐷𝐴
• Integrate approximations over ∂𝐴
• Careful change of variables and control of Jacobian
Scope of the Study
• Focus on sets with smooth boundary
• Assume 𝐼 is convex (can often transform problem to this case)
• Consider points where 𝐼(𝑥)−𝐼(𝐴) is bounded or not too large
Overview of Upcoming Chapters
• Chapters 2-5: Proof of main theorem for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Chapter 6: Special case of translated sets
• Chapter 7: Scaled sets and homogeneous 𝐼
• Chapter 8: Tail probabilities of quadratic forms
• Chapter 9: Random linear forms
• Chapter 10: Random matrices
• Chapter 11: Applications to time series analysis
• Chapter 12: Suprema of stochastic processes
Key Distribution Families
Symmetric Weibull-like Distribution
𝑤𝛼(𝑥)= 𝐾𝑤,𝛼𝑒−|𝑥|𝛼/𝛼,𝑥 ∈ ℝ
Student-like Distribution
Tail behavior:
𝑆𝛼(−𝑥)∼ 1−𝑆𝛼(𝑥)∼𝐾𝑠,𝛼𝛼(𝛼−1)/2
𝑥𝛼 as 𝑥 → ∞
Importance:
• Weibull-like: Includes Gaussian (𝛼 = 2), shows discontinuities at 𝛼 = 2
• Student-like: Representative of heavy-tailed distributions, includes symmetric
stable distributions
Product of Cauchy Variables
Consider 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
Key points:
• Set 𝐴𝑡= {(𝑥,𝑦):𝑥𝑦 > 𝑡}
• Unique contact point between ∂𝐴𝑡 and maximal level set of density
• Decay of density crucial
• Shape of domain affects results
Asymptotic Behavior
𝑃{𝑋𝑌 > 𝑡} ∼ 2
𝜋2log𝑡
𝑡 as 𝑡 → ∞
For 𝛼 ∈ (0,1):
𝑃{𝑋 > 𝑡𝛼|𝑋𝑌 > 𝑡} ∼ 1−𝛼
2
Result: (𝑋,𝑌)/𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to 𝛿(0,0)
Logarithmic Scale Analysis
Writing 𝑋 = 𝜖1𝑒𝑈, 𝑌 = 𝜖2𝑒𝑉:
Key result: Distribution of (𝑈,𝑉)/log𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to uniform
distribution over {(𝑠,1−𝑠):𝑠 ∈ [0,1]}
Motivation for Systematic Methods
• Specific arguments for simple cases not generalizable
• Higher dimensions make visualization impossible
• Need for systematic procedures independent of intuition
Connection to Laplace’s Method
Theorem (Laplace’s Approximation): For ∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 with suitable conditions:
∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 ∼ (2𝜋)𝑑/2
√det𝐷2𝐼(𝑎∗)𝜆−𝑑/2𝑒−𝜆𝐼(𝑎∗) as 𝜆 → ∞
Relation to Our Problem
For 𝐴 = 𝑡𝐴1 and 𝐼 homogeneous of degree 𝛼:
∫ 𝑒−𝐼(𝑥)
𝑡𝐴1𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Key insight: Our method generalizes Laplace’s method with an infinite-dimensional
parameter
Challenges and Approach
• 𝐼 may not have a unique minimum on 𝐴
• Minimum may not be in the interior of 𝐴
• 𝐼 attains minimum on a set 𝐷𝐴 (potentially a k-dimensional manifold)
Approach:
• Apply Laplace’s approximation along fibers orthogonal to 𝐷𝐴
• Integrate approximations over ∂𝐴
• Careful change of variables and control of Jacobian
Scope of the Study
• Focus on sets with smooth boundary
• Assume 𝐼 is convex (can often transform problem to this case)
• Consider points where 𝐼(𝑥)−𝐼(𝐴) is bounded or not too large
Overview of Upcoming Chapters
• Chapters 2-5: Proof of main theorem for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Chapter 6: Special case of translated sets
• Chapter 7: Scaled sets and homogeneous 𝐼
• Chapter 8: Tail probabilities of quadratic forms
• Chapter 9: Random linear forms
• Chapter 10: Random matrices
• Chapter 11: Applications to time series analysis
• Chapter 12: Suprema of stochastic processes
Key Distribution Families
Symmetric Weibull-like Distribution
𝑤𝛼(𝑥)= 𝐾𝑤,𝛼𝑒−|𝑥|𝛼/𝛼,𝑥 ∈ ℝ
Student-like Distribution
Tail behavior:
𝑆𝛼(−𝑥)∼ 1−𝑆𝛼(𝑥)∼𝐾𝑠,𝛼𝛼(𝛼−1)/2
𝑥𝛼 as 𝑥 → ∞
Importance:
• Weibull-like: Includes Gaussian (𝛼 = 2), shows discontinuities at 𝛼 = 2
• Student-like: Representative of heavy-tailed distributions, includes symmetric
stable distributions
Product of Cauchy Variables
Consider 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
Key points:
• Set 𝐴𝑡= {(𝑥,𝑦):𝑥𝑦 > 𝑡}
• Unique contact point between ∂𝐴𝑡 and maximal level set of density
• Decay of density crucial
• Shape of domain affects results
Asymptotic Behavior
𝑃{𝑋𝑌 > 𝑡} ∼ 2
𝜋2log𝑡
𝑡 as 𝑡 → ∞
For 𝛼 ∈ (0,1):
𝑃{𝑋 > 𝑡𝛼|𝑋𝑌 > 𝑡} ∼ 1−𝛼
2
Result: (𝑋,𝑌)/𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to 𝛿(0,0)
Logarithmic Scale Analysis
Writing 𝑋 = 𝜖1𝑒𝑈, 𝑌 = 𝜖2𝑒𝑉:
Key result: Distribution of (𝑈,𝑉)/log𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to uniform
distribution over {(𝑠,1−𝑠):𝑠 ∈ [0,1]}
Motivation for Systematic Methods
• Specific arguments for simple cases not generalizable
• Higher dimensions make visualization impossible
• Need for systematic procedures independent of intuition
Connection to Laplace’s Method
Theorem (Laplace’s Approximation): For ∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 with suitable conditions:
∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 ∼ (2𝜋)𝑑/2
√det𝐷2𝐼(𝑎∗)𝜆−𝑑/2𝑒−𝜆𝐼(𝑎∗) as 𝜆 → ∞
Relation to Our Problem
For 𝐴 = 𝑡𝐴1 and 𝐼 homogeneous of degree 𝛼:
∫ 𝑒−𝐼(𝑥)
𝑡𝐴1𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Key insight: Our method generalizes Laplace’s method with an infinite-dimensional
parameter
Challenges and Approach
• 𝐼 may not have a unique minimum on 𝐴
• Minimum may not be in the interior of 𝐴
• 𝐼 attains minimum on a set 𝐷𝐴 (potentially a k-dimensional manifold)
Approach:
• Apply Laplace’s approximation along fibers orthogonal to 𝐷𝐴
• Integrate approximations over ∂𝐴
• Careful change of variables and control of Jacobian
Scope of the Study
• Focus on sets with smooth boundary
• Assume 𝐼 is convex (can often transform problem to this case)
• Consider points where 𝐼(𝑥)−𝐼(𝐴) is bounded or not too large
Overview of Upcoming Chapters
• Chapters 2-5: Proof of main theorem for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Chapter 6: Special case of translated sets
• Chapter 7: Scaled sets and homogeneous 𝐼
• Chapter 8: Tail probabilities of quadratic forms
• Chapter 9: Random linear forms
• Chapter 10: Random matrices
• Chapter 11: Applications to time series analysis
• Chapter 12: Suprema of stochastic processes
Key Distribution Families
Symmetric Weibull-like Distribution
𝑤𝛼(𝑥)= 𝐾𝑤,𝛼𝑒−|𝑥|𝛼/𝛼,𝑥 ∈ ℝ
Student-like Distribution
Tail behavior:
𝑆𝛼(−𝑥)∼ 1−𝑆𝛼(𝑥)∼𝐾𝑠,𝛼𝛼(𝛼−1)/2
𝑥𝛼 as 𝑥 → ∞
Importance:
• Weibull-like: Includes Gaussian (𝛼 = 2), shows discontinuities at 𝛼 = 2
• Student-like: Representative of heavy-tailed distributions, includes symmetric
stable distributions
Product of Cauchy Variables
Consider 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
Key points:
• Set 𝐴𝑡= {(𝑥,𝑦):𝑥𝑦 > 𝑡}
• Unique contact point between ∂𝐴𝑡 and maximal level set of density
• Decay of density crucial
• Shape of domain affects results
Asymptotic Behavior
𝑃{𝑋𝑌 > 𝑡} ∼ 2
𝜋2log𝑡
𝑡 as 𝑡 → ∞
For 𝛼 ∈ (0,1):
𝑃{𝑋 > 𝑡𝛼|𝑋𝑌 > 𝑡} ∼ 1−𝛼
2
Result: (𝑋,𝑌)/𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to 𝛿(0,0)
Logarithmic Scale Analysis
Writing 𝑋 = 𝜖1𝑒𝑈, 𝑌 = 𝜖2𝑒𝑉:
Key result: Distribution of (𝑈,𝑉)/log𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to uniform
distribution over {(𝑠,1−𝑠):𝑠 ∈ [0,1]}
Motivation for Systematic Methods
• Specific arguments for simple cases not generalizable
• Higher dimensions make visualization impossible
• Need for systematic procedures independent of intuition
Connection to Laplace’s Method
Theorem (Laplace’s Approximation): For ∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 with suitable conditions:
∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 ∼ (2𝜋)𝑑/2
√det𝐷2𝐼(𝑎∗)𝜆−𝑑/2𝑒−𝜆𝐼(𝑎∗) as 𝜆 → ∞
Relation to Our Problem
For 𝐴 = 𝑡𝐴1 and 𝐼 homogeneous of degree 𝛼:
∫ 𝑒−𝐼(𝑥)
𝑡𝐴1𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Key insight: Our method generalizes Laplace’s method with an infinite-dimensional
parameter
Challenges and Approach
• 𝐼 may not have a unique minimum on 𝐴
• Minimum may not be in the interior of 𝐴
• 𝐼 attains minimum on a set 𝐷𝐴 (potentially a k-dimensional manifold)
Approach:
• Apply Laplace’s approximation along fibers orthogonal to 𝐷𝐴
• Integrate approximations over ∂𝐴
• Careful change of variables and control of Jacobian
Scope of the Study
• Focus on sets with smooth boundary
• Assume 𝐼 is convex (can often transform problem to this case)
• Consider points where 𝐼(𝑥)−𝐼(𝐴) is bounded or not too large
Overview of Upcoming Chapters
• Chapters 2-5: Proof of main theorem for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Chapter 6: Special case of translated sets
• Chapter 7: Scaled sets and homogeneous 𝐼
• Chapter 8: Tail probabilities of quadratic forms
• Chapter 9: Random linear forms
• Chapter 10: Random matrices
• Chapter 11: Applications to time series analysis
• Chapter 12: Suprema of stochastic processes
Key Distribution Families
Symmetric Weibull-like Distribution
𝑤𝛼(𝑥)= 𝐾𝑤,𝛼𝑒−|𝑥|𝛼/𝛼,𝑥 ∈ ℝ
Student-like Distribution
Tail behavior:
𝑆𝛼(−𝑥)∼ 1−𝑆𝛼(𝑥)∼𝐾𝑠,𝛼𝛼(𝛼−1)/2
𝑥𝛼 as 𝑥 → ∞
Importance:
• Weibull-like: Includes Gaussian (𝛼 = 2), shows discontinuities at 𝛼 = 2
• Student-like: Representative of heavy-tailed distributions, includes symmetric
stable distributions
Product of Cauchy Variables
Consider 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
Key points:
• Set 𝐴𝑡= {(𝑥,𝑦):𝑥𝑦 > 𝑡}
• Unique contact point between ∂𝐴𝑡 and maximal level set of density
• Decay of density crucial
• Shape of domain affects results
Asymptotic Behavior
𝑃{𝑋𝑌 > 𝑡} ∼ 2
𝜋2log𝑡
𝑡 as 𝑡 → ∞
For 𝛼 ∈ (0,1):
𝑃{𝑋 > 𝑡𝛼|𝑋𝑌 > 𝑡} ∼ 1−𝛼
2
Result: (𝑋,𝑌)/𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to 𝛿(0,0)
Logarithmic Scale Analysis
Writing 𝑋 = 𝜖1𝑒𝑈, 𝑌 = 𝜖2𝑒𝑉:
Key result: Distribution of (𝑈,𝑉)/log𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to uniform
distribution over {(𝑠,1−𝑠):𝑠 ∈ [0,1]}
Motivation for Systematic Methods
• Specific arguments for simple cases not generalizable
• Higher dimensions make visualization impossible
• Need for systematic procedures independent of intuition
Connection to Laplace’s Method
Theorem (Laplace’s Approximation): For ∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 with suitable conditions:
∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 ∼ (2𝜋)𝑑/2
√det𝐷2𝐼(𝑎∗)𝜆−𝑑/2𝑒−𝜆𝐼(𝑎∗) as 𝜆 → ∞
Relation to Our Problem
For 𝐴 = 𝑡𝐴1 and 𝐼 homogeneous of degree 𝛼:
∫ 𝑒−𝐼(𝑥)
𝑡𝐴1𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Key insight: Our method generalizes Laplace’s method with an infinite-dimensional
parameter
Challenges and Approach
• 𝐼 may not have a unique minimum on 𝐴
• Minimum may not be in the interior of 𝐴
• 𝐼 attains minimum on a set 𝐷𝐴 (potentially a k-dimensional manifold)
Approach:
• Apply Laplace’s approximation along fibers orthogonal to 𝐷𝐴
• Integrate approximations over ∂𝐴
• Careful change of variables and control of Jacobian
Scope of the Study
• Focus on sets with smooth boundary
• Assume 𝐼 is convex (can often transform problem to this case)
• Consider points where 𝐼(𝑥)−𝐼(𝐴) is bounded or not too large
Overview of Upcoming Chapters
• Chapters 2-5: Proof of main theorem for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Chapter 6: Special case of translated sets
• Chapter 7: Scaled sets and homogeneous 𝐼
• Chapter 8: Tail probabilities of quadratic forms
• Chapter 9: Random linear forms
• Chapter 10: Random matrices
• Chapter 11: Applications to time series analysis
• Chapter 12: Suprema of stochastic processes
Key Distribution Families
Symmetric Weibull-like Distribution
𝑤𝛼(𝑥)= 𝐾𝑤,𝛼𝑒−|𝑥|𝛼/𝛼,𝑥 ∈ ℝ
Student-like Distribution
Tail behavior:
𝑆𝛼(−𝑥)∼ 1−𝑆𝛼(𝑥)∼𝐾𝑠,𝛼𝛼(𝛼−1)/2
𝑥𝛼 as 𝑥 → ∞
Importance:
• Weibull-like: Includes Gaussian (𝛼 = 2), shows discontinuities at 𝛼 = 2
• Student-like: Representative of heavy-tailed distributions, includes symmetric
stable distributions
Product of Cauchy Variables
Consider 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
Key points:
• Set 𝐴𝑡= {(𝑥,𝑦):𝑥𝑦 > 𝑡}
• Unique contact point between ∂𝐴𝑡 and maximal level set of density
• Decay of density crucial
• Shape of domain affects results
Asymptotic Behavior
𝑃{𝑋𝑌 > 𝑡} ∼ 2
𝜋2log𝑡
𝑡 as 𝑡 → ∞
For 𝛼 ∈ (0,1):
𝑃{𝑋 > 𝑡𝛼|𝑋𝑌 > 𝑡} ∼ 1−𝛼
2
Result: (𝑋,𝑌)/𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to 𝛿(0,0)
Logarithmic Scale Analysis
Writing 𝑋 = 𝜖1𝑒𝑈, 𝑌 = 𝜖2𝑒𝑉:
Key result: Distribution of (𝑈,𝑉)/log𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to uniform
distribution over {(𝑠,1−𝑠):𝑠 ∈ [0,1]}
Motivation for Systematic Methods
• Specific arguments for simple cases not generalizable
• Higher dimensions make visualization impossible
• Need for systematic procedures independent of intuition
Connection to Laplace’s Method
Theorem (Laplace’s Approximation): For ∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 with suitable conditions:
∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 ∼ (2𝜋)𝑑/2
√det𝐷2𝐼(𝑎∗)𝜆−𝑑/2𝑒−𝜆𝐼(𝑎∗) as 𝜆 → ∞
Relation to Our Problem
For 𝐴 = 𝑡𝐴1 and 𝐼 homogeneous of degree 𝛼:
∫ 𝑒−𝐼(𝑥)
𝑡𝐴1𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Key insight: Our method generalizes Laplace’s method with an infinite-dimensional
parameter
Challenges and Approach
• 𝐼 may not have a unique minimum on 𝐴
• Minimum may not be in the interior of 𝐴
• 𝐼 attains minimum on a set 𝐷𝐴 (potentially a k-dimensional manifold)
Approach:
• Apply Laplace’s approximation along fibers orthogonal to 𝐷𝐴
• Integrate approximations over ∂𝐴
• Careful change of variables and control of Jacobian
Scope of the Study
• Focus on sets with smooth boundary
• Assume 𝐼 is convex (can often transform problem to this case)
• Consider points where 𝐼(𝑥)−𝐼(𝐴) is bounded or not too large
Overview of Upcoming Chapters
• Chapters 2-5: Proof of main theorem for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Chapter 6: Special case of translated sets
• Chapter 7: Scaled sets and homogeneous 𝐼
• Chapter 8: Tail probabilities of quadratic forms
• Chapter 9: Random linear forms
• Chapter 10: Random matrices
• Chapter 11: Applications to time series analysis
• Chapter 12: Suprema of stochastic processes
Key Distribution Families
Symmetric Weibull-like Distribution
𝑤𝛼(𝑥)= 𝐾𝑤,𝛼𝑒−|𝑥|𝛼/𝛼,𝑥 ∈ ℝ
Student-like Distribution
Tail behavior:
𝑆𝛼(−𝑥)∼ 1−𝑆𝛼(𝑥)∼𝐾𝑠,𝛼𝛼(𝛼−1)/2
𝑥𝛼 as 𝑥 → ∞
Importance:
• Weibull-like: Includes Gaussian (𝛼 = 2), shows discontinuities at 𝛼 = 2
• Student-like: Representative of heavy-tailed distributions, includes symmetric
stable distributions
Product of Cauchy Variables
Consider 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
Key points:
• Set 𝐴𝑡= {(𝑥,𝑦):𝑥𝑦 > 𝑡}
• Unique contact point between ∂𝐴𝑡 and maximal level set of density
• Decay of density crucial
• Shape of domain affects results
Asymptotic Behavior
𝑃{𝑋𝑌 > 𝑡} ∼ 2
𝜋2log𝑡
𝑡 as 𝑡 → ∞
For 𝛼 ∈ (0,1):
𝑃{𝑋 > 𝑡𝛼|𝑋𝑌 > 𝑡} ∼ 1−𝛼
2
Result: (𝑋,𝑌)/𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to 𝛿(0,0)
Logarithmic Scale Analysis
Writing 𝑋 = 𝜖1𝑒𝑈, 𝑌 = 𝜖2𝑒𝑉:
Key result: Distribution of (𝑈,𝑉)/log𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to uniform
distribution over {(𝑠,1−𝑠):𝑠 ∈ [0,1]}
Motivation for Systematic Methods
• Specific arguments for simple cases not generalizable
• Higher dimensions make visualization impossible
• Need for systematic procedures independent of intuition
Connection to Laplace’s Method
Theorem (Laplace’s Approximation): For ∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 with suitable conditions:
∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 ∼ (2𝜋)𝑑/2
√det𝐷2𝐼(𝑎∗)𝜆−𝑑/2𝑒−𝜆𝐼(𝑎∗) as 𝜆 → ∞
Relation to Our Problem
For 𝐴 = 𝑡𝐴1 and 𝐼 homogeneous of degree 𝛼:
∫ 𝑒−𝐼(𝑥)
𝑡𝐴1𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Key insight: Our method generalizes Laplace’s method with an infinite-dimensional
parameter
Challenges and Approach
• 𝐼 may not have a unique minimum on 𝐴
• Minimum may not be in the interior of 𝐴
• 𝐼 attains minimum on a set 𝐷𝐴 (potentially a k-dimensional manifold)
Approach:
• Apply Laplace’s approximation along fibers orthogonal to 𝐷𝐴
• Integrate approximations over ∂𝐴
• Careful change of variables and control of Jacobian
Scope of the Study
• Focus on sets with smooth boundary
• Assume 𝐼 is convex (can often transform problem to this case)
• Consider points where 𝐼(𝑥)−𝐼(𝐴) is bounded or not too large
Overview of Upcoming Chapters
• Chapters 2-5: Proof of main theorem for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Chapter 6: Special case of translated sets
• Chapter 7: Scaled sets and homogeneous 𝐼
• Chapter 8: Tail probabilities of quadratic forms
• Chapter 9: Random linear forms
• Chapter 10: Random matrices
• Chapter 11: Applications to time series analysis
• Chapter 12: Suprema of stochastic processes
Key Distribution Families
Symmetric Weibull-like Distribution
𝑤𝛼(𝑥)= 𝐾𝑤,𝛼𝑒−|𝑥|𝛼/𝛼,𝑥 ∈ ℝ
Student-like Distribution
Tail behavior:
𝑆𝛼(−𝑥)∼ 1−𝑆𝛼(𝑥)∼𝐾𝑠,𝛼𝛼(𝛼−1)/2
𝑥𝛼 as 𝑥 → ∞
Importance:
• Weibull-like: Includes Gaussian (𝛼 = 2), shows discontinuities at 𝛼 = 2
• Student-like: Representative of heavy-tailed distributions, includes symmetric
stable distributions
Product of Cauchy Variables
Consider 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
Key points:
• Set 𝐴𝑡= {(𝑥,𝑦):𝑥𝑦 > 𝑡}
• Unique contact point between ∂𝐴𝑡 and maximal level set of density
• Decay of density crucial
• Shape of domain affects results
Asymptotic Behavior
𝑃{𝑋𝑌 > 𝑡} ∼ 2
𝜋2log𝑡
𝑡 as 𝑡 → ∞
For 𝛼 ∈ (0,1):
𝑃{𝑋 > 𝑡𝛼|𝑋𝑌 > 𝑡} ∼ 1−𝛼
2
Result: (𝑋,𝑌)/𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to 𝛿(0,0)
Logarithmic Scale Analysis
Writing 𝑋 = 𝜖1𝑒𝑈, 𝑌 = 𝜖2𝑒𝑉:
Key result: Distribution of (𝑈,𝑉)/log𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to uniform
distribution over {(𝑠,1−𝑠):𝑠 ∈ [0,1]}
Motivation for Systematic Methods
• Specific arguments for simple cases not generalizable
• Higher dimensions make visualization impossible
• Need for systematic procedures independent of intuition
Connection to Laplace’s Method
Theorem (Laplace’s Approximation): For ∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 with suitable conditions:
∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 ∼ (2𝜋)𝑑/2
√det𝐷2𝐼(𝑎∗)𝜆−𝑑/2𝑒−𝜆𝐼(𝑎∗) as 𝜆 → ∞
Relation to Our Problem
For 𝐴 = 𝑡𝐴1 and 𝐼 homogeneous of degree 𝛼:
∫ 𝑒−𝐼(𝑥)
𝑡𝐴1𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Key insight: Our method generalizes Laplace’s method with an infinite-dimensional
parameter
Challenges and Approach
• 𝐼 may not have a unique minimum on 𝐴
• Minimum may not be in the interior of 𝐴
• 𝐼 attains minimum on a set 𝐷𝐴 (potentially a k-dimensional manifold)
Approach:
• Apply Laplace’s approximation along fibers orthogonal to 𝐷𝐴
• Integrate approximations over ∂𝐴
• Careful change of variables and control of Jacobian
Scope of the Study
• Focus on sets with smooth boundary
• Assume 𝐼 is convex (can often transform problem to this case)
• Consider points where 𝐼(𝑥)−𝐼(𝐴) is bounded or not too large
Overview of Upcoming Chapters
• Chapters 2-5: Proof of main theorem for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Chapter 6: Special case of translated sets
• Chapter 7: Scaled sets and homogeneous 𝐼
• Chapter 8: Tail probabilities of quadratic forms
• Chapter 9: Random linear forms
• Chapter 10: Random matrices
• Chapter 11: Applications to time series analysis
• Chapter 12: Suprema of stochastic processes
Key Distribution Families
Symmetric Weibull-like Distribution
𝑤𝛼(𝑥)= 𝐾𝑤,𝛼𝑒−|𝑥|𝛼/𝛼,𝑥 ∈ ℝ
Student-like Distribution
Tail behavior:
𝑆𝛼(−𝑥)∼ 1−𝑆𝛼(𝑥)∼𝐾𝑠,𝛼𝛼(𝛼−1)/2
𝑥𝛼 as 𝑥 → ∞
Importance:
• Weibull-like: Includes Gaussian (𝛼 = 2), shows discontinuities at 𝛼 = 2
• Student-like: Representative of heavy-tailed distributions, includes symmetric
stable distributions
Product of Cauchy Variables
Consider 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
Key points:
• Set 𝐴𝑡= {(𝑥,𝑦):𝑥𝑦 > 𝑡}
• Unique contact point between ∂𝐴𝑡 and maximal level set of density
• Decay of density crucial
• Shape of domain affects results
Asymptotic Behavior
𝑃{𝑋𝑌 > 𝑡} ∼ 2
𝜋2log𝑡
𝑡 as 𝑡 → ∞
For 𝛼 ∈ (0,1):
𝑃{𝑋 > 𝑡𝛼|𝑋𝑌 > 𝑡} ∼ 1−𝛼
2
Result: (𝑋,𝑌)/𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to 𝛿(0,0)
Logarithmic Scale Analysis
Writing 𝑋 = 𝜖1𝑒𝑈, 𝑌 = 𝜖2𝑒𝑉:
Key result: Distribution of (𝑈,𝑉)/log𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to uniform
distribution over {(𝑠,1−𝑠):𝑠 ∈ [0,1]}
Motivation for Systematic Methods
• Specific arguments for simple cases not generalizable
• Higher dimensions make visualization impossible
• Need for systematic procedures independent of intuition
Connection to Laplace’s Method
Theorem (Laplace’s Approximation): For ∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 with suitable conditions:
∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 ∼ (2𝜋)𝑑/2
√det𝐷2𝐼(𝑎∗)𝜆−𝑑/2𝑒−𝜆𝐼(𝑎∗) as 𝜆 → ∞
Relation to Our Problem
For 𝐴 = 𝑡𝐴1 and 𝐼 homogeneous of degree 𝛼:
∫ 𝑒−𝐼(𝑥)
𝑡𝐴1𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Key insight: Our method generalizes Laplace’s method with an infinite-dimensional
parameter
Challenges and Approach
• 𝐼 may not have a unique minimum on 𝐴
• Minimum may not be in the interior of 𝐴
• 𝐼 attains minimum on a set 𝐷𝐴 (potentially a k-dimensional manifold)
Approach:
• Apply Laplace’s approximation along fibers orthogonal to 𝐷𝐴
• Integrate approximations over ∂𝐴
• Careful change of variables and control of Jacobian
Scope of the Study
• Focus on sets with smooth boundary
• Assume 𝐼 is convex (can often transform problem to this case)
• Consider points where 𝐼(𝑥)−𝐼(𝐴) is bounded or not too large
Overview of Upcoming Chapters
• Chapters 2-5: Proof of main theorem for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Chapter 6: Special case of translated sets
• Chapter 7: Scaled sets and homogeneous 𝐼
• Chapter 8: Tail probabilities of quadratic forms
• Chapter 9: Random linear forms
• Chapter 10: Random matrices
• Chapter 11: Applications to time series analysis
• Chapter 12: Suprema of stochastic processes
Key Distribution Families
Symmetric Weibull-like Distribution
𝑤𝛼(𝑥)= 𝐾𝑤,𝛼𝑒−|𝑥|𝛼/𝛼,𝑥 ∈ ℝ
Student-like Distribution
Tail behavior:
𝑆𝛼(−𝑥)∼ 1−𝑆𝛼(𝑥)∼𝐾𝑠,𝛼𝛼(𝛼−1)/2
𝑥𝛼 as 𝑥 → ∞
Importance:
• Weibull-like: Includes Gaussian (𝛼 = 2), shows discontinuities at 𝛼 = 2
• Student-like: Representative of heavy-tailed distributions, includes symmetric
stable distributions
Product of Cauchy Variables
Consider 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
Key points:
• Set 𝐴𝑡= {(𝑥,𝑦):𝑥𝑦 > 𝑡}
• Unique contact point between ∂𝐴𝑡 and maximal level set of density
• Decay of density crucial
• Shape of domain affects results
Asymptotic Behavior
𝑃{𝑋𝑌 > 𝑡} ∼ 2
𝜋2log𝑡
𝑡 as 𝑡 → ∞
For 𝛼 ∈ (0,1):
𝑃{𝑋 > 𝑡𝛼|𝑋𝑌 > 𝑡} ∼ 1−𝛼
2
Result: (𝑋,𝑌)/𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to 𝛿(0,0)
Logarithmic Scale Analysis
Writing 𝑋 = 𝜖1𝑒𝑈, 𝑌 = 𝜖2𝑒𝑉:
Key result: Distribution of (𝑈,𝑉)/log𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to uniform
distribution over {(𝑠,1−𝑠):𝑠 ∈ [0,1]}
Motivation for Systematic Methods
• Specific arguments for simple cases not generalizable
• Higher dimensions make visualization impossible
• Need for systematic procedures independent of intuition
Connection to Laplace’s Method
Theorem (Laplace’s Approximation): For ∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 with suitable conditions:
∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 ∼ (2𝜋)𝑑/2
√det𝐷2𝐼(𝑎∗)𝜆−𝑑/2𝑒−𝜆𝐼(𝑎∗) as 𝜆 → ∞
Relation to Our Problem
For 𝐴 = 𝑡𝐴1 and 𝐼 homogeneous of degree 𝛼:
∫ 𝑒−𝐼(𝑥)
𝑡𝐴1𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Key insight: Our method generalizes Laplace’s method with an infinite-dimensional
parameter
Challenges and Approach
• 𝐼 may not have a unique minimum on 𝐴
• Minimum may not be in the interior of 𝐴
• 𝐼 attains minimum on a set 𝐷𝐴 (potentially a k-dimensional manifold)
Approach:
• Apply Laplace’s approximation along fibers orthogonal to 𝐷𝐴
• Integrate approximations over ∂𝐴
• Careful change of variables and control of Jacobian
Scope of the Study
• Focus on sets with smooth boundary
• Assume 𝐼 is convex (can often transform problem to this case)
• Consider points where 𝐼(𝑥)−𝐼(𝐴) is bounded or not too large
Overview of Upcoming Chapters
• Chapters 2-5: Proof of main theorem for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Chapter 6: Special case of translated sets
• Chapter 7: Scaled sets and homogeneous 𝐼
• Chapter 8: Tail probabilities of quadratic forms
• Chapter 9: Random linear forms
• Chapter 10: Random matrices
• Chapter 11: Applications to time series analysis
• Chapter 12: Suprema of stochastic processes
Key Distribution Families
Symmetric Weibull-like Distribution
𝑤𝛼(𝑥)= 𝐾𝑤,𝛼𝑒−|𝑥|𝛼/𝛼,𝑥 ∈ ℝ
Student-like Distribution
Tail behavior:
𝑆𝛼(−𝑥)∼ 1−𝑆𝛼(𝑥)∼𝐾𝑠,𝛼𝛼(𝛼−1)/2
𝑥𝛼 as 𝑥 → ∞
Importance:
• Weibull-like: Includes Gaussian (𝛼 = 2), shows discontinuities at 𝛼 = 2
• Student-like: Representative of heavy-tailed distributions, includes symmetric
stable distributions
Product of Cauchy Variables
Consider 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
Key points:
• Set 𝐴𝑡= {(𝑥,𝑦):𝑥𝑦 > 𝑡}
• Unique contact point between ∂𝐴𝑡 and maximal level set of density
• Decay of density crucial
• Shape of domain affects results
Asymptotic Behavior
𝑃{𝑋𝑌 > 𝑡} ∼ 2
𝜋2log𝑡
𝑡 as 𝑡 → ∞
For 𝛼 ∈ (0,1):
𝑃{𝑋 > 𝑡𝛼|𝑋𝑌 > 𝑡} ∼ 1−𝛼
2
Result: (𝑋,𝑌)/𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to 𝛿(0,0)
Logarithmic Scale Analysis
Writing 𝑋 = 𝜖1𝑒𝑈, 𝑌 = 𝜖2𝑒𝑉:
Key result: Distribution of (𝑈,𝑉)/log𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to uniform
distribution over {(𝑠,1−𝑠):𝑠 ∈ [0,1]}
Motivation for Systematic Methods
• Specific arguments for simple cases not generalizable
• Higher dimensions make visualization impossible
• Need for systematic procedures independent of intuition
Connection to Laplace’s Method
Theorem (Laplace’s Approximation): For ∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 with suitable conditions:
∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 ∼ (2𝜋)𝑑/2
√det𝐷2𝐼(𝑎∗)𝜆−𝑑/2𝑒−𝜆𝐼(𝑎∗) as 𝜆 → ∞
Relation to Our Problem
For 𝐴 = 𝑡𝐴1 and 𝐼 homogeneous of degree 𝛼:
∫ 𝑒−𝐼(𝑥)
𝑡𝐴1𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Key insight: Our method generalizes Laplace’s method with an infinite-dimensional
parameter
Challenges and Approach
• 𝐼 may not have a unique minimum on 𝐴
• Minimum may not be in the interior of 𝐴
• 𝐼 attains minimum on a set 𝐷𝐴 (potentially a k-dimensional manifold)
Approach:
• Apply Laplace’s approximation along fibers orthogonal to 𝐷𝐴
• Integrate approximations over ∂𝐴
• Careful change of variables and control of Jacobian
Scope of the Study
• Focus on sets with smooth boundary
• Assume 𝐼 is convex (can often transform problem to this case)
• Consider points where 𝐼(𝑥)−𝐼(𝐴) is bounded or not too large
Overview of Upcoming Chapters
• Chapters 2-5: Proof of main theorem for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Chapter 6: Special case of translated sets
• Chapter 7: Scaled sets and homogeneous 𝐼
• Chapter 8: Tail probabilities of quadratic forms
• Chapter 9: Random linear forms
• Chapter 10: Random matrices
• Chapter 11: Applications to time series analysis
• Chapter 12: Suprema of stochastic processes
Key Distribution Families
Symmetric Weibull-like Distribution
𝑤𝛼(𝑥)= 𝐾𝑤,𝛼𝑒−|𝑥|𝛼/𝛼,𝑥 ∈ ℝ
Student-like Distribution
Tail behavior:
𝑆𝛼(−𝑥)∼ 1−𝑆𝛼(𝑥)∼𝐾𝑠,𝛼𝛼(𝛼−1)/2
𝑥𝛼 as 𝑥 → ∞
Importance:
• Weibull-like: Includes Gaussian (𝛼 = 2), shows discontinuities at 𝛼 = 2
• Student-like: Representative of heavy-tailed distributions, includes symmetric
stable distributions
Product of Cauchy Variables
Consider 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
Key points:
• Set 𝐴𝑡= {(𝑥,𝑦):𝑥𝑦 > 𝑡}
• Unique contact point between ∂𝐴𝑡 and maximal level set of density
• Decay of density crucial
• Shape of domain affects results
Asymptotic Behavior
𝑃{𝑋𝑌 > 𝑡} ∼ 2
𝜋2log𝑡
𝑡 as 𝑡 → ∞
For 𝛼 ∈ (0,1):
𝑃{𝑋 > 𝑡𝛼|𝑋𝑌 > 𝑡} ∼ 1−𝛼
2
Result: (𝑋,𝑌)/𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to 𝛿(0,0)
Logarithmic Scale Analysis
Writing 𝑋 = 𝜖1𝑒𝑈, 𝑌 = 𝜖2𝑒𝑉:
Key result: Distribution of (𝑈,𝑉)/log𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to uniform
distribution over {(𝑠,1−𝑠):𝑠 ∈ [0,1]}
Motivation for Systematic Methods
• Specific arguments for simple cases not generalizable
• Higher dimensions make visualization impossible
• Need for systematic procedures independent of intuition
Connection to Laplace’s Method
Theorem (Laplace’s Approximation): For ∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 with suitable conditions:
∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 ∼ (2𝜋)𝑑/2
√det𝐷2𝐼(𝑎∗)𝜆−𝑑/2𝑒−𝜆𝐼(𝑎∗) as 𝜆 → ∞
Relation to Our Problem
For 𝐴 = 𝑡𝐴1 and 𝐼 homogeneous of degree 𝛼:
∫ 𝑒−𝐼(𝑥)
𝑡𝐴1𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Key insight: Our method generalizes Laplace’s method with an infinite-dimensional
parameter
Challenges and Approach
• 𝐼 may not have a unique minimum on 𝐴
• Minimum may not be in the interior of 𝐴
• 𝐼 attains minimum on a set 𝐷𝐴 (potentially a k-dimensional manifold)
Approach:
• Apply Laplace’s approximation along fibers orthogonal to 𝐷𝐴
• Integrate approximations over ∂𝐴
• Careful change of variables and control of Jacobian
Scope of the Study
• Focus on sets with smooth boundary
• Assume 𝐼 is convex (can often transform problem to this case)
• Consider points where 𝐼(𝑥)−𝐼(𝐴) is bounded or not too large
Overview of Upcoming Chapters
• Chapters 2-5: Proof of main theorem for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Chapter 6: Special case of translated sets
• Chapter 7: Scaled sets and homogeneous 𝐼
• Chapter 8: Tail probabilities of quadratic forms
• Chapter 9: Random linear forms
• Chapter 10: Random matrices
• Chapter 11: Applications to time series analysis
• Chapter 12: Suprema of stochastic processes
Key Distribution Families
Symmetric Weibull-like Distribution
𝑤𝛼(𝑥)= 𝐾𝑤,𝛼𝑒−|𝑥|𝛼/𝛼,𝑥 ∈ ℝ
Student-like Distribution
Tail behavior:
𝑆𝛼(−𝑥)∼ 1−𝑆𝛼(𝑥)∼𝐾𝑠,𝛼𝛼(𝛼−1)/2
𝑥𝛼 as 𝑥 → ∞
Importance:
• Weibull-like: Includes Gaussian (𝛼 = 2), shows discontinuities at 𝛼 = 2
• Student-like: Representative of heavy-tailed distributions, includes symmetric
stable distributions
Product of Cauchy Variables
Consider 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
Key points:
• Set 𝐴𝑡= {(𝑥,𝑦):𝑥𝑦 > 𝑡}
• Unique contact point between ∂𝐴𝑡 and maximal level set of density
• Decay of density crucial
• Shape of domain affects results
Asymptotic Behavior
𝑃{𝑋𝑌 > 𝑡} ∼ 2
𝜋2log𝑡
𝑡 as 𝑡 → ∞
For 𝛼 ∈ (0,1):
𝑃{𝑋 > 𝑡𝛼|𝑋𝑌 > 𝑡} ∼ 1−𝛼
2
Result: (𝑋,𝑌)/𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to 𝛿(0,0)
Logarithmic Scale Analysis
Writing 𝑋 = 𝜖1𝑒𝑈, 𝑌 = 𝜖2𝑒𝑉:
Key result: Distribution of (𝑈,𝑉)/log𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to uniform
distribution over {(𝑠,1−𝑠):𝑠 ∈ [0,1]}
Motivation for Systematic Methods
• Specific arguments for simple cases not generalizable
• Higher dimensions make visualization impossible
• Need for systematic procedures independent of intuition
Connection to Laplace’s Method
Theorem (Laplace’s Approximation): For ∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 with suitable conditions:
∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 ∼ (2𝜋)𝑑/2
√det𝐷2𝐼(𝑎∗)𝜆−𝑑/2𝑒−𝜆𝐼(𝑎∗) as 𝜆 → ∞
Relation to Our Problem
For 𝐴 = 𝑡𝐴1 and 𝐼 homogeneous of degree 𝛼:
∫ 𝑒−𝐼(𝑥)
𝑡𝐴1𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Key insight: Our method generalizes Laplace’s method with an infinite-dimensional
parameter
Challenges and Approach
• 𝐼 may not have a unique minimum on 𝐴
• Minimum may not be in the interior of 𝐴
• 𝐼 attains minimum on a set 𝐷𝐴 (potentially a k-dimensional manifold)
Approach:
• Apply Laplace’s approximation along fibers orthogonal to 𝐷𝐴
• Integrate approximations over ∂𝐴
• Careful change of variables and control of Jacobian
Scope of the Study
• Focus on sets with smooth boundary
• Assume 𝐼 is convex (can often transform problem to this case)
• Consider points where 𝐼(𝑥)−𝐼(𝐴) is bounded or not too large
Overview of Upcoming Chapters
• Chapters 2-5: Proof of main theorem for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Chapter 6: Special case of translated sets
• Chapter 7: Scaled sets and homogeneous 𝐼
• Chapter 8: Tail probabilities of quadratic forms
• Chapter 9: Random linear forms
• Chapter 10: Random matrices
• Chapter 11: Applications to time series analysis
• Chapter 12: Suprema of stochastic processes
Key Distribution Families
Symmetric Weibull-like Distribution
𝑤𝛼(𝑥)= 𝐾𝑤,𝛼𝑒−|𝑥|𝛼/𝛼,𝑥 ∈ ℝ
Student-like Distribution
Tail behavior:
𝑆𝛼(−𝑥)∼ 1−𝑆𝛼(𝑥)∼𝐾𝑠,𝛼𝛼(𝛼−1)/2
𝑥𝛼 as 𝑥 → ∞
Importance:
• Weibull-like: Includes Gaussian (𝛼 = 2), shows discontinuities at 𝛼 = 2
• Student-like: Representative of heavy-tailed distributions, includes symmetric
stable distributions
Product of Cauchy Variables
Consider 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
Key points:
• Set 𝐴𝑡= {(𝑥,𝑦):𝑥𝑦 > 𝑡}
• Unique contact point between ∂𝐴𝑡 and maximal level set of density
• Decay of density crucial
• Shape of domain affects results
Asymptotic Behavior
𝑃{𝑋𝑌 > 𝑡} ∼ 2
𝜋2log𝑡
𝑡 as 𝑡 → ∞
For 𝛼 ∈ (0,1):
𝑃{𝑋 > 𝑡𝛼|𝑋𝑌 > 𝑡} ∼ 1−𝛼
2
Result: (𝑋,𝑌)/𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to 𝛿(0,0)
Logarithmic Scale Analysis
Writing 𝑋 = 𝜖1𝑒𝑈, 𝑌 = 𝜖2𝑒𝑉:
Key result: Distribution of (𝑈,𝑉)/log𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to uniform
distribution over {(𝑠,1−𝑠):𝑠 ∈ [0,1]}
Motivation for Systematic Methods
• Specific arguments for simple cases not generalizable
• Higher dimensions make visualization impossible
• Need for systematic procedures independent of intuition
Connection to Laplace’s Method
Theorem (Laplace’s Approximation): For ∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 with suitable conditions:
∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 ∼ (2𝜋)𝑑/2
√det𝐷2𝐼(𝑎∗)𝜆−𝑑/2𝑒−𝜆𝐼(𝑎∗) as 𝜆 → ∞
Relation to Our Problem
For 𝐴 = 𝑡𝐴1 and 𝐼 homogeneous of degree 𝛼:
∫ 𝑒−𝐼(𝑥)
𝑡𝐴1𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Key insight: Our method generalizes Laplace’s method with an infinite-dimensional
parameter
Challenges and Approach
• 𝐼 may not have a unique minimum on 𝐴
• Minimum may not be in the interior of 𝐴
• 𝐼 attains minimum on a set 𝐷𝐴 (potentially a k-dimensional manifold)
Approach:
• Apply Laplace’s approximation along fibers orthogonal to 𝐷𝐴
• Integrate approximations over ∂𝐴
• Careful change of variables and control of Jacobian
Scope of the Study
• Focus on sets with smooth boundary
• Assume 𝐼 is convex (can often transform problem to this case)
• Consider points where 𝐼(𝑥)−𝐼(𝐴) is bounded or not too large
Overview of Upcoming Chapters
• Chapters 2-5: Proof of main theorem for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Chapter 6: Special case of translated sets
• Chapter 7: Scaled sets and homogeneous 𝐼
• Chapter 8: Tail probabilities of quadratic forms
• Chapter 9: Random linear forms
• Chapter 10: Random matrices
• Chapter 11: Applications to time series analysis
• Chapter 12: Suprema of stochastic processes
Key Distribution Families
Symmetric Weibull-like Distribution
𝑤𝛼(𝑥)= 𝐾𝑤,𝛼𝑒−|𝑥|𝛼/𝛼,𝑥 ∈ ℝ
Student-like Distribution
Tail behavior:
𝑆𝛼(−𝑥)∼ 1−𝑆𝛼(𝑥)∼𝐾𝑠,𝛼𝛼(𝛼−1)/2
𝑥𝛼 as 𝑥 → ∞
Importance:
• Weibull-like: Includes Gaussian (𝛼 = 2), shows discontinuities at 𝛼 = 2
• Student-like: Representative of heavy-tailed distributions, includes symmetric
stable distributions
Product of Cauchy Variables
Consider 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
Key points:
• Set 𝐴𝑡= {(𝑥,𝑦):𝑥𝑦 > 𝑡}
• Unique contact point between ∂𝐴𝑡 and maximal level set of density
• Decay of density crucial
• Shape of domain affects results
Asymptotic Behavior
𝑃{𝑋𝑌 > 𝑡} ∼ 2
𝜋2log𝑡
𝑡 as 𝑡 → ∞
For 𝛼 ∈ (0,1):
𝑃{𝑋 > 𝑡𝛼|𝑋𝑌 > 𝑡} ∼ 1−𝛼
2
Result: (𝑋,𝑌)/𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to 𝛿(0,0)
Logarithmic Scale Analysis
Writing 𝑋 = 𝜖1𝑒𝑈, 𝑌 = 𝜖2𝑒𝑉:
Key result: Distribution of (𝑈,𝑉)/log𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to uniform
distribution over {(𝑠,1−𝑠):𝑠 ∈ [0,1]}
Motivation for Systematic Methods
• Specific arguments for simple cases not generalizable
• Higher dimensions make visualization impossible
• Need for systematic procedures independent of intuition
Connection to Laplace’s Method
Theorem (Laplace’s Approximation): For ∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 with suitable conditions:
∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 ∼ (2𝜋)𝑑/2
√det𝐷2𝐼(𝑎∗)𝜆−𝑑/2𝑒−𝜆𝐼(𝑎∗) as 𝜆 → ∞
Relation to Our Problem
For 𝐴 = 𝑡𝐴1 and 𝐼 homogeneous of degree 𝛼:
∫ 𝑒−𝐼(𝑥)
𝑡𝐴1𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Key insight: Our method generalizes Laplace’s method with an infinite-dimensional
parameter
Challenges and Approach
• 𝐼 may not have a unique minimum on 𝐴
• Minimum may not be in the interior of 𝐴
• 𝐼 attains minimum on a set 𝐷𝐴 (potentially a k-dimensional manifold)
Approach:
• Apply Laplace’s approximation along fibers orthogonal to 𝐷𝐴
• Integrate approximations over ∂𝐴
• Careful change of variables and control of Jacobian
Scope of the Study
• Focus on sets with smooth boundary
• Assume 𝐼 is convex (can often transform problem to this case)
• Consider points where 𝐼(𝑥)−𝐼(𝐴) is bounded or not too large
Overview of Upcoming Chapters
• Chapters 2-5: Proof of main theorem for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Chapter 6: Special case of translated sets
• Chapter 7: Scaled sets and homogeneous 𝐼
• Chapter 8: Tail probabilities of quadratic forms
• Chapter 9: Random linear forms
• Chapter 10: Random matrices
• Chapter 11: Applications to time series analysis
• Chapter 12: Suprema of stochastic processes
Key Distribution Families
Symmetric Weibull-like Distribution
𝑤𝛼(𝑥)= 𝐾𝑤,𝛼𝑒−|𝑥|𝛼/𝛼,𝑥 ∈ ℝ
Student-like Distribution
Tail behavior:
𝑆𝛼(−𝑥)∼ 1−𝑆𝛼(𝑥)∼𝐾𝑠,𝛼𝛼(𝛼−1)/2
𝑥𝛼 as 𝑥 → ∞
Importance:
• Weibull-like: Includes Gaussian (𝛼 = 2), shows discontinuities at 𝛼 = 2
• Student-like: Representative of heavy-tailed distributions, includes symmetric
stable distributions
Product of Cauchy Variables
Consider 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
Key points:
• Set 𝐴𝑡= {(𝑥,𝑦):𝑥𝑦 > 𝑡}
• Unique contact point between ∂𝐴𝑡 and maximal level set of density
• Decay of density crucial
• Shape of domain affects results
Asymptotic Behavior
𝑃{𝑋𝑌 > 𝑡} ∼ 2
𝜋2log𝑡
𝑡 as 𝑡 → ∞
For 𝛼 ∈ (0,1):
𝑃{𝑋 > 𝑡𝛼|𝑋𝑌 > 𝑡} ∼ 1−𝛼
2
Result: (𝑋,𝑌)/𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to 𝛿(0,0)
Logarithmic Scale Analysis
Writing 𝑋 = 𝜖1𝑒𝑈, 𝑌 = 𝜖2𝑒𝑉:
Key result: Distribution of (𝑈,𝑉)/log𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to uniform
distribution over {(𝑠,1−𝑠):𝑠 ∈ [0,1]}
Motivation for Systematic Methods
• Specific arguments for simple cases not generalizable
• Higher dimensions make visualization impossible
• Need for systematic procedures independent of intuition
Connection to Laplace’s Method
Theorem (Laplace’s Approximation): For ∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 with suitable conditions:
∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 ∼ (2𝜋)𝑑/2
√det𝐷2𝐼(𝑎∗)𝜆−𝑑/2𝑒−𝜆𝐼(𝑎∗) as 𝜆 → ∞
Relation to Our Problem
For 𝐴 = 𝑡𝐴1 and 𝐼 homogeneous of degree 𝛼:
∫ 𝑒−𝐼(𝑥)
𝑡𝐴1𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Key insight: Our method generalizes Laplace’s method with an infinite-dimensional
parameter
Challenges and Approach
• 𝐼 may not have a unique minimum on 𝐴
• Minimum may not be in the interior of 𝐴
• 𝐼 attains minimum on a set 𝐷𝐴 (potentially a k-dimensional manifold)
Approach:
• Apply Laplace’s approximation along fibers orthogonal to 𝐷𝐴
• Integrate approximations over ∂𝐴
• Careful change of variables and control of Jacobian
Scope of the Study
• Focus on sets with smooth boundary
• Assume 𝐼 is convex (can often transform problem to this case)
• Consider points where 𝐼(𝑥)−𝐼(𝐴) is bounded or not too large
Overview of Upcoming Chapters
• Chapters 2-5: Proof of main theorem for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Chapter 6: Special case of translated sets
• Chapter 7: Scaled sets and homogeneous 𝐼
• Chapter 8: Tail probabilities of quadratic forms
• Chapter 9: Random linear forms
• Chapter 10: Random matrices
• Chapter 11: Applications to time series analysis
• Chapter 12: Suprema of stochastic processes
Key Distribution Families
Symmetric Weibull-like Distribution
𝑤𝛼(𝑥)= 𝐾𝑤,𝛼𝑒−|𝑥|𝛼/𝛼,𝑥 ∈ ℝ
Student-like Distribution
Tail behavior:
𝑆𝛼(−𝑥)∼ 1−𝑆𝛼(𝑥)∼𝐾𝑠,𝛼𝛼(𝛼−1)/2
𝑥𝛼 as 𝑥 → ∞
Importance:
• Weibull-like: Includes Gaussian (𝛼 = 2), shows discontinuities at 𝛼 = 2
• Student-like: Representative of heavy-tailed distributions, includes symmetric
stable distributions
Product of Cauchy Variables
Consider 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
Key points:
• Set 𝐴𝑡= {(𝑥,𝑦):𝑥𝑦 > 𝑡}
• Unique contact point between ∂𝐴𝑡 and maximal level set of density
• Decay of density crucial
• Shape of domain affects results
Asymptotic Behavior
𝑃{𝑋𝑌 > 𝑡} ∼ 2
𝜋2log𝑡
𝑡 as 𝑡 → ∞
For 𝛼 ∈ (0,1):
𝑃{𝑋 > 𝑡𝛼|𝑋𝑌 > 𝑡} ∼ 1−𝛼
2
Result: (𝑋,𝑌)/𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to 𝛿(0,0)
Logarithmic Scale Analysis
Writing 𝑋 = 𝜖1𝑒𝑈, 𝑌 = 𝜖2𝑒𝑉:
Key result: Distribution of (𝑈,𝑉)/log𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to uniform
distribution over {(𝑠,1−𝑠):𝑠 ∈ [0,1]}
Motivation for Systematic Methods
• Specific arguments for simple cases not generalizable
• Higher dimensions make visualization impossible
• Need for systematic procedures independent of intuition
Connection to Laplace’s Method
Theorem (Laplace’s Approximation): For ∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 with suitable conditions:
∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 ∼ (2𝜋)𝑑/2
√det𝐷2𝐼(𝑎∗)𝜆−𝑑/2𝑒−𝜆𝐼(𝑎∗) as 𝜆 → ∞
Relation to Our Problem
For 𝐴 = 𝑡𝐴1 and 𝐼 homogeneous of degree 𝛼:
∫ 𝑒−𝐼(𝑥)
𝑡𝐴1𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Key insight: Our method generalizes Laplace’s method with an infinite-dimensional
parameter
Challenges and Approach
• 𝐼 may not have a unique minimum on 𝐴
• Minimum may not be in the interior of 𝐴
• 𝐼 attains minimum on a set 𝐷𝐴 (potentially a k-dimensional manifold)
Approach:
• Apply Laplace’s approximation along fibers orthogonal to 𝐷𝐴
• Integrate approximations over ∂𝐴
• Careful change of variables and control of Jacobian
Scope of the Study
• Focus on sets with smooth boundary
• Assume 𝐼 is convex (can often transform problem to this case)
• Consider points where 𝐼(𝑥)−𝐼(𝐴) is bounded or not too large
Overview of Upcoming Chapters
• Chapters 2-5: Proof of main theorem for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Chapter 6: Special case of translated sets
• Chapter 7: Scaled sets and homogeneous 𝐼
• Chapter 8: Tail probabilities of quadratic forms
• Chapter 9: Random linear forms
• Chapter 10: Random matrices
• Chapter 11: Applications to time series analysis
• Chapter 12: Suprema of stochastic processes
Key Distribution Families
Symmetric Weibull-like Distribution
𝑤𝛼(𝑥)= 𝐾𝑤,𝛼𝑒−|𝑥|𝛼/𝛼,𝑥 ∈ ℝ
Student-like Distribution
Tail behavior:
𝑆𝛼(−𝑥)∼ 1−𝑆𝛼(𝑥)∼𝐾𝑠,𝛼𝛼(𝛼−1)/2
𝑥𝛼 as 𝑥 → ∞
Importance:
• Weibull-like: Includes Gaussian (𝛼 = 2), shows discontinuities at 𝛼 = 2
• Student-like: Representative of heavy-tailed distributions, includes symmetric
stable distributions
Product of Cauchy Variables
Consider 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
Key points:
• Set 𝐴𝑡= {(𝑥,𝑦):𝑥𝑦 > 𝑡}
• Unique contact point between ∂𝐴𝑡 and maximal level set of density
• Decay of density crucial
• Shape of domain affects results
Asymptotic Behavior
𝑃{𝑋𝑌 > 𝑡} ∼ 2
𝜋2log𝑡
𝑡 as 𝑡 → ∞
For 𝛼 ∈ (0,1):
𝑃{𝑋 > 𝑡𝛼|𝑋𝑌 > 𝑡} ∼ 1−𝛼
2
Result: (𝑋,𝑌)/𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to 𝛿(0,0)
Logarithmic Scale Analysis
Writing 𝑋 = 𝜖1𝑒𝑈, 𝑌 = 𝜖2𝑒𝑉:
Key result: Distribution of (𝑈,𝑉)/log𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to uniform
distribution over {(𝑠,1−𝑠):𝑠 ∈ [0,1]}
Motivation for Systematic Methods
• Specific arguments for simple cases not generalizable
• Higher dimensions make visualization impossible
• Need for systematic procedures independent of intuition
Connection to Laplace’s Method
Theorem (Laplace’s Approximation): For ∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 with suitable conditions:
∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 ∼ (2𝜋)𝑑/2
√det𝐷2𝐼(𝑎∗)𝜆−𝑑/2𝑒−𝜆𝐼(𝑎∗) as 𝜆 → ∞
Relation to Our Problem
For 𝐴 = 𝑡𝐴1 and 𝐼 homogeneous of degree 𝛼:
∫ 𝑒−𝐼(𝑥)
𝑡𝐴1𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Key insight: Our method generalizes Laplace’s method with an infinite-dimensional
parameter
Challenges and Approach
• 𝐼 may not have a unique minimum on 𝐴
• Minimum may not be in the interior of 𝐴
• 𝐼 attains minimum on a set 𝐷𝐴 (potentially a k-dimensional manifold)
Approach:
• Apply Laplace’s approximation along fibers orthogonal to 𝐷𝐴
• Integrate approximations over ∂𝐴
• Careful change of variables and control of Jacobian
Scope of the Study
• Focus on sets with smooth boundary
• Assume 𝐼 is convex (can often transform problem to this case)
• Consider points where 𝐼(𝑥)−𝐼(𝐴) is bounded or not too large
Overview of Upcoming Chapters
• Chapters 2-5: Proof of main theorem for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Chapter 6: Special case of translated sets
• Chapter 7: Scaled sets and homogeneous 𝐼
• Chapter 8: Tail probabilities of quadratic forms
• Chapter 9: Random linear forms
• Chapter 10: Random matrices
• Chapter 11: Applications to time series analysis
• Chapter 12: Suprema of stochastic processes
Key Distribution Families
Symmetric Weibull-like Distribution
𝑤𝛼(𝑥)= 𝐾𝑤,𝛼𝑒−|𝑥|𝛼/𝛼,𝑥 ∈ ℝ
Student-like Distribution
Tail behavior:
𝑆𝛼(−𝑥)∼ 1−𝑆𝛼(𝑥)∼𝐾𝑠,𝛼𝛼(𝛼−1)/2
𝑥𝛼 as 𝑥 → ∞
Importance:
• Weibull-like: Includes Gaussian (𝛼 = 2), shows discontinuities at 𝛼 = 2
• Student-like: Representative of heavy-tailed distributions, includes symmetric
stable distributions
Product of Cauchy Variables
Consider 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
Key points:
• Set 𝐴𝑡= {(𝑥,𝑦):𝑥𝑦 > 𝑡}
• Unique contact point between ∂𝐴𝑡 and maximal level set of density
• Decay of density crucial
• Shape of domain affects results
Asymptotic Behavior
𝑃{𝑋𝑌 > 𝑡} ∼ 2
𝜋2log𝑡
𝑡 as 𝑡 → ∞
For 𝛼 ∈ (0,1):
𝑃{𝑋 > 𝑡𝛼|𝑋𝑌 > 𝑡} ∼ 1−𝛼
2
Result: (𝑋,𝑌)/𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to 𝛿(0,0)
Logarithmic Scale Analysis
Writing 𝑋 = 𝜖1𝑒𝑈, 𝑌 = 𝜖2𝑒𝑉:
Key result: Distribution of (𝑈,𝑉)/log𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to uniform
distribution over {(𝑠,1−𝑠):𝑠 ∈ [0,1]}
Motivation for Systematic Methods
• Specific arguments for simple cases not generalizable
• Higher dimensions make visualization impossible
• Need for systematic procedures independent of intuition
Connection to Laplace’s Method
Theorem (Laplace’s Approximation): For ∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 with suitable conditions:
∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 ∼ (2𝜋)𝑑/2
√det𝐷2𝐼(𝑎∗)𝜆−𝑑/2𝑒−𝜆𝐼(𝑎∗) as 𝜆 → ∞
Relation to Our Problem
For 𝐴 = 𝑡𝐴1 and 𝐼 homogeneous of degree 𝛼:
∫ 𝑒−𝐼(𝑥)
𝑡𝐴1𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Key insight: Our method generalizes Laplace’s method with an infinite-dimensional
parameter
Challenges and Approach
• 𝐼 may not have a unique minimum on 𝐴
• Minimum may not be in the interior of 𝐴
• 𝐼 attains minimum on a set 𝐷𝐴 (potentially a k-dimensional manifold)
Approach:
• Apply Laplace’s approximation along fibers orthogonal to 𝐷𝐴
• Integrate approximations over ∂𝐴
• Careful change of variables and control of Jacobian
Scope of the Study
• Focus on sets with smooth boundary
• Assume 𝐼 is convex (can often transform problem to this case)
• Consider points where 𝐼(𝑥)−𝐼(𝐴) is bounded or not too large
Overview of Upcoming Chapters
• Chapters 2-5: Proof of main theorem for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Chapter 6: Special case of translated sets
• Chapter 7: Scaled sets and homogeneous 𝐼
• Chapter 8: Tail probabilities of quadratic forms
• Chapter 9: Random linear forms
• Chapter 10: Random matrices
• Chapter 11: Applications to time series analysis
• Chapter 12: Suprema of stochastic processes
Key Distribution Families
Symmetric Weibull-like Distribution
𝑤𝛼(𝑥)= 𝐾𝑤,𝛼𝑒−|𝑥|𝛼/𝛼,𝑥 ∈ ℝ
Student-like Distribution
Tail behavior:
𝑆𝛼(−𝑥)∼ 1−𝑆𝛼(𝑥)∼𝐾𝑠,𝛼𝛼(𝛼−1)/2
𝑥𝛼 as 𝑥 → ∞
Importance:
• Weibull-like: Includes Gaussian (𝛼 = 2), shows discontinuities at 𝛼 = 2
• Student-like: Representative of heavy-tailed distributions, includes symmetric
stable distributions
Product of Cauchy Variables
Consider 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
Key points:
• Set 𝐴𝑡= {(𝑥,𝑦):𝑥𝑦 > 𝑡}
• Unique contact point between ∂𝐴𝑡 and maximal level set of density
• Decay of density crucial
• Shape of domain affects results
Asymptotic Behavior
𝑃{𝑋𝑌 > 𝑡} ∼ 2
𝜋2log𝑡
𝑡 as 𝑡 → ∞
For 𝛼 ∈ (0,1):
𝑃{𝑋 > 𝑡𝛼|𝑋𝑌 > 𝑡} ∼ 1−𝛼
2
Result: (𝑋,𝑌)/𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to 𝛿(0,0)
Logarithmic Scale Analysis
Writing 𝑋 = 𝜖1𝑒𝑈, 𝑌 = 𝜖2𝑒𝑉:
Key result: Distribution of (𝑈,𝑉)/log𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to uniform
distribution over {(𝑠,1−𝑠):𝑠 ∈ [0,1]}
Motivation for Systematic Methods
• Specific arguments for simple cases not generalizable
• Higher dimensions make visualization impossible
• Need for systematic procedures independent of intuition
Connection to Laplace’s Method
Theorem (Laplace’s Approximation): For ∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 with suitable conditions:
∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 ∼ (2𝜋)𝑑/2
√det𝐷2𝐼(𝑎∗)𝜆−𝑑/2𝑒−𝜆𝐼(𝑎∗) as 𝜆 → ∞
Relation to Our Problem
For 𝐴 = 𝑡𝐴1 and 𝐼 homogeneous of degree 𝛼:
∫ 𝑒−𝐼(𝑥)
𝑡𝐴1𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Key insight: Our method generalizes Laplace’s method with an infinite-dimensional
parameter
Challenges and Approach
• 𝐼 may not have a unique minimum on 𝐴
• Minimum may not be in the interior of 𝐴
• 𝐼 attains minimum on a set 𝐷𝐴 (potentially a k-dimensional manifold)
Approach:
• Apply Laplace’s approximation along fibers orthogonal to 𝐷𝐴
• Integrate approximations over ∂𝐴
• Careful change of variables and control of Jacobian
Scope of the Study
• Focus on sets with smooth boundary
• Assume 𝐼 is convex (can often transform problem to this case)
• Consider points where 𝐼(𝑥)−𝐼(𝐴) is bounded or not too large
Overview of Upcoming Chapters
• Chapters 2-5: Proof of main theorem for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Chapter 6: Special case of translated sets
• Chapter 7: Scaled sets and homogeneous 𝐼
• Chapter 8: Tail probabilities of quadratic forms
• Chapter 9: Random linear forms
• Chapter 10: Random matrices
• Chapter 11: Applications to time series analysis
• Chapter 12: Suprema of stochastic processes
Key Distribution Families
Symmetric Weibull-like Distribution
𝑤𝛼(𝑥)= 𝐾𝑤,𝛼𝑒−|𝑥|𝛼/𝛼,𝑥 ∈ ℝ
Student-like Distribution
Tail behavior:
𝑆𝛼(−𝑥)∼ 1−𝑆𝛼(𝑥)∼𝐾𝑠,𝛼𝛼(𝛼−1)/2
𝑥𝛼 as 𝑥 → ∞
Importance:
• Weibull-like: Includes Gaussian (𝛼 = 2), shows discontinuities at 𝛼 = 2
• Student-like: Representative of heavy-tailed distributions, includes symmetric
stable distributions
Product of Cauchy Variables
Consider 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
Key points:
• Set 𝐴𝑡= {(𝑥,𝑦):𝑥𝑦 > 𝑡}
• Unique contact point between ∂𝐴𝑡 and maximal level set of density
• Decay of density crucial
• Shape of domain affects results
Asymptotic Behavior
𝑃{𝑋𝑌 > 𝑡} ∼ 2
𝜋2log𝑡
𝑡 as 𝑡 → ∞
For 𝛼 ∈ (0,1):
𝑃{𝑋 > 𝑡𝛼|𝑋𝑌 > 𝑡} ∼ 1−𝛼
2
Result: (𝑋,𝑌)/𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to 𝛿(0,0)
Logarithmic Scale Analysis
Writing 𝑋 = 𝜖1𝑒𝑈, 𝑌 = 𝜖2𝑒𝑉:
Key result: Distribution of (𝑈,𝑉)/log𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to uniform
distribution over {(𝑠,1−𝑠):𝑠 ∈ [0,1]}
Motivation for Systematic Methods
• Specific arguments for simple cases not generalizable
• Higher dimensions make visualization impossible
• Need for systematic procedures independent of intuition
Connection to Laplace’s Method
Theorem (Laplace’s Approximation): For ∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 with suitable conditions:
∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 ∼ (2𝜋)𝑑/2
√det𝐷2𝐼(𝑎∗)𝜆−𝑑/2𝑒−𝜆𝐼(𝑎∗) as 𝜆 → ∞
Relation to Our Problem
For 𝐴 = 𝑡𝐴1 and 𝐼 homogeneous of degree 𝛼:
∫ 𝑒−𝐼(𝑥)
𝑡𝐴1𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Key insight: Our method generalizes Laplace’s method with an infinite-dimensional
parameter
Challenges and Approach
• 𝐼 may not have a unique minimum on 𝐴
• Minimum may not be in the interior of 𝐴
• 𝐼 attains minimum on a set 𝐷𝐴 (potentially a k-dimensional manifold)
Approach:
• Apply Laplace’s approximation along fibers orthogonal to 𝐷𝐴
• Integrate approximations over ∂𝐴
• Careful change of variables and control of Jacobian
Scope of the Study
• Focus on sets with smooth boundary
• Assume 𝐼 is convex (can often transform problem to this case)
• Consider points where 𝐼(𝑥)−𝐼(𝐴) is bounded or not too large
Overview of Upcoming Chapters
• Chapters 2-5: Proof of main theorem for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Chapter 6: Special case of translated sets
• Chapter 7: Scaled sets and homogeneous 𝐼
• Chapter 8: Tail probabilities of quadratic forms
• Chapter 9: Random linear forms
• Chapter 10: Random matrices
• Chapter 11: Applications to time series analysis
• Chapter 12: Suprema of stochastic processes
Key Distribution Families
Symmetric Weibull-like Distribution
𝑤𝛼(𝑥)= 𝐾𝑤,𝛼𝑒−|𝑥|𝛼/𝛼,𝑥 ∈ ℝ
Student-like Distribution
Tail behavior:
𝑆𝛼(−𝑥)∼ 1−𝑆𝛼(𝑥)∼𝐾𝑠,𝛼𝛼(𝛼−1)/2
𝑥𝛼 as 𝑥 → ∞
Importance:
• Weibull-like: Includes Gaussian (𝛼 = 2), shows discontinuities at 𝛼 = 2
• Student-like: Representative of heavy-tailed distributions, includes symmetric
stable distributions
Product of Cauchy Variables
Consider 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
Key points:
• Set 𝐴𝑡= {(𝑥,𝑦):𝑥𝑦 > 𝑡}
• Unique contact point between ∂𝐴𝑡 and maximal level set of density
• Decay of density crucial
• Shape of domain affects results
Asymptotic Behavior
𝑃{𝑋𝑌 > 𝑡} ∼ 2
𝜋2log𝑡
𝑡 as 𝑡 → ∞
For 𝛼 ∈ (0,1):
𝑃{𝑋 > 𝑡𝛼|𝑋𝑌 > 𝑡} ∼ 1−𝛼
2
Result: (𝑋,𝑌)/𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to 𝛿(0,0)
Logarithmic Scale Analysis
Writing 𝑋 = 𝜖1𝑒𝑈, 𝑌 = 𝜖2𝑒𝑉:
Key result: Distribution of (𝑈,𝑉)/log𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to uniform
distribution over {(𝑠,1−𝑠):𝑠 ∈ [0,1]}
Motivation for Systematic Methods
• Specific arguments for simple cases not generalizable
• Higher dimensions make visualization impossible
• Need for systematic procedures independent of intuition
Connection to Laplace’s Method
Theorem (Laplace’s Approximation): For ∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 with suitable conditions:
∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 ∼ (2𝜋)𝑑/2
√det𝐷2𝐼(𝑎∗)𝜆−𝑑/2𝑒−𝜆𝐼(𝑎∗) as 𝜆 → ∞
Relation to Our Problem
For 𝐴 = 𝑡𝐴1 and 𝐼 homogeneous of degree 𝛼:
∫ 𝑒−𝐼(𝑥)
𝑡𝐴1𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Key insight: Our method generalizes Laplace’s method with an infinite-dimensional
parameter
Challenges and Approach
• 𝐼 may not have a unique minimum on 𝐴
• Minimum may not be in the interior of 𝐴
• 𝐼 attains minimum on a set 𝐷𝐴 (potentially a k-dimensional manifold)
Approach:
• Apply Laplace’s approximation along fibers orthogonal to 𝐷𝐴
• Integrate approximations over ∂𝐴
• Careful change of variables and control of Jacobian
Scope of the Study
• Focus on sets with smooth boundary
• Assume 𝐼 is convex (can often transform problem to this case)
• Consider points where 𝐼(𝑥)−𝐼(𝐴) is bounded or not too large
Overview of Upcoming Chapters
• Chapters 2-5: Proof of main theorem for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Chapter 6: Special case of translated sets
• Chapter 7: Scaled sets and homogeneous 𝐼
• Chapter 8: Tail probabilities of quadratic forms
• Chapter 9: Random linear forms
• Chapter 10: Random matrices
• Chapter 11: Applications to time series analysis
• Chapter 12: Suprema of stochastic processes
Key Distribution Families
Symmetric Weibull-like Distribution
𝑤𝛼(𝑥)= 𝐾𝑤,𝛼𝑒−|𝑥|𝛼/𝛼,𝑥 ∈ ℝ
Student-like Distribution
Tail behavior:
𝑆𝛼(−𝑥)∼ 1−𝑆𝛼(𝑥)∼𝐾𝑠,𝛼𝛼(𝛼−1)/2
𝑥𝛼 as 𝑥 → ∞
Importance:
• Weibull-like: Includes Gaussian (𝛼 = 2), shows discontinuities at 𝛼 = 2
• Student-like: Representative of heavy-tailed distributions, includes symmetric
stable distributions
Product of Cauchy Variables
Consider 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
Key points:
• Set 𝐴𝑡= {(𝑥,𝑦):𝑥𝑦 > 𝑡}
• Unique contact point between ∂𝐴𝑡 and maximal level set of density
• Decay of density crucial
• Shape of domain affects results
Asymptotic Behavior
𝑃{𝑋𝑌 > 𝑡} ∼ 2
𝜋2log𝑡
𝑡 as 𝑡 → ∞
For 𝛼 ∈ (0,1):
𝑃{𝑋 > 𝑡𝛼|𝑋𝑌 > 𝑡} ∼ 1−𝛼
2
Result: (𝑋,𝑌)/𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to 𝛿(0,0)
Logarithmic Scale Analysis
Writing 𝑋 = 𝜖1𝑒𝑈, 𝑌 = 𝜖2𝑒𝑉:
Key result: Distribution of (𝑈,𝑉)/log𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to uniform
distribution over {(𝑠,1−𝑠):𝑠 ∈ [0,1]}
Motivation for Systematic Methods
• Specific arguments for simple cases not generalizable
• Higher dimensions make visualization impossible
• Need for systematic procedures independent of intuition
Connection to Laplace’s Method
Theorem (Laplace’s Approximation): For ∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 with suitable conditions:
∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 ∼ (2𝜋)𝑑/2
√det𝐷2𝐼(𝑎∗)𝜆−𝑑/2𝑒−𝜆𝐼(𝑎∗) as 𝜆 → ∞
Relation to Our Problem
For 𝐴 = 𝑡𝐴1 and 𝐼 homogeneous of degree 𝛼:
∫ 𝑒−𝐼(𝑥)
𝑡𝐴1𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Key insight: Our method generalizes Laplace’s method with an infinite-dimensional
parameter
Challenges and Approach
• 𝐼 may not have a unique minimum on 𝐴
• Minimum may not be in the interior of 𝐴
• 𝐼 attains minimum on a set 𝐷𝐴 (potentially a k-dimensional manifold)
Approach:
• Apply Laplace’s approximation along fibers orthogonal to 𝐷𝐴
• Integrate approximations over ∂𝐴
• Careful change of variables and control of Jacobian
Scope of the Study
• Focus on sets with smooth boundary
• Assume 𝐼 is convex (can often transform problem to this case)
• Consider points where 𝐼(𝑥)−𝐼(𝐴) is bounded or not too large
Overview of Upcoming Chapters
• Chapters 2-5: Proof of main theorem for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Chapter 6: Special case of translated sets
• Chapter 7: Scaled sets and homogeneous 𝐼
• Chapter 8: Tail probabilities of quadratic forms
• Chapter 9: Random linear forms
• Chapter 10: Random matrices
• Chapter 11: Applications to time series analysis
• Chapter 12: Suprema of stochastic processes
Key Distribution Families
Symmetric Weibull-like Distribution
𝑤𝛼(𝑥)= 𝐾𝑤,𝛼𝑒−|𝑥|𝛼/𝛼,𝑥 ∈ ℝ
Student-like Distribution
Tail behavior:
𝑆𝛼(−𝑥)∼ 1−𝑆𝛼(𝑥)∼𝐾𝑠,𝛼𝛼(𝛼−1)/2
𝑥𝛼 as 𝑥 → ∞
Importance:
• Weibull-like: Includes Gaussian (𝛼 = 2), shows discontinuities at 𝛼 = 2
• Student-like: Representative of heavy-tailed distributions, includes symmetric
stable distributions
Product of Cauchy Variables
Consider 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
Key points:
• Set 𝐴𝑡= {(𝑥,𝑦):𝑥𝑦 > 𝑡}
• Unique contact point between ∂𝐴𝑡 and maximal level set of density
• Decay of density crucial
• Shape of domain affects results
Asymptotic Behavior
𝑃{𝑋𝑌 > 𝑡} ∼ 2
𝜋2log𝑡
𝑡 as 𝑡 → ∞
For 𝛼 ∈ (0,1):
𝑃{𝑋 > 𝑡𝛼|𝑋𝑌 > 𝑡} ∼ 1−𝛼
2
Result: (𝑋,𝑌)/𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to 𝛿(0,0)
Logarithmic Scale Analysis
Writing 𝑋 = 𝜖1𝑒𝑈, 𝑌 = 𝜖2𝑒𝑉:
Key result: Distribution of (𝑈,𝑉)/log𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to uniform
distribution over {(𝑠,1−𝑠):𝑠 ∈ [0,1]}
Motivation for Systematic Methods
• Specific arguments for simple cases not generalizable
• Higher dimensions make visualization impossible
• Need for systematic procedures independent of intuition
Connection to Laplace’s Method
Theorem (Laplace’s Approximation): For ∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 with suitable conditions:
∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 ∼ (2𝜋)𝑑/2
√det𝐷2𝐼(𝑎∗)𝜆−𝑑/2𝑒−𝜆𝐼(𝑎∗) as 𝜆 → ∞
Relation to Our Problem
For 𝐴 = 𝑡𝐴1 and 𝐼 homogeneous of degree 𝛼:
∫ 𝑒−𝐼(𝑥)
𝑡𝐴1𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Key insight: Our method generalizes Laplace’s method with an infinite-dimensional
parameter
Challenges and Approach
• 𝐼 may not have a unique minimum on 𝐴
• Minimum may not be in the interior of 𝐴
• 𝐼 attains minimum on a set 𝐷𝐴 (potentially a k-dimensional manifold)
Approach:
• Apply Laplace’s approximation along fibers orthogonal to 𝐷𝐴
• Integrate approximations over ∂𝐴
• Careful change of variables and control of Jacobian
Scope of the Study
• Focus on sets with smooth boundary
• Assume 𝐼 is convex (can often transform problem to this case)
• Consider points where 𝐼(𝑥)−𝐼(𝐴) is bounded or not too large
Overview of Upcoming Chapters
• Chapters 2-5: Proof of main theorem for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Chapter 6: Special case of translated sets
• Chapter 7: Scaled sets and homogeneous 𝐼
• Chapter 8: Tail probabilities of quadratic forms
• Chapter 9: Random linear forms
• Chapter 10: Random matrices
• Chapter 11: Applications to time series analysis
• Chapter 12: Suprema of stochastic processes
Key Distribution Families
Symmetric Weibull-like Distribution
𝑤𝛼(𝑥)= 𝐾𝑤,𝛼𝑒−|𝑥|𝛼/𝛼,𝑥 ∈ ℝ
Student-like Distribution
Tail behavior:
𝑆𝛼(−𝑥)∼ 1−𝑆𝛼(𝑥)∼𝐾𝑠,𝛼𝛼(𝛼−1)/2
𝑥𝛼 as 𝑥 → ∞
Importance:
• Weibull-like: Includes Gaussian (𝛼 = 2), shows discontinuities at 𝛼 = 2
• Student-like: Representative of heavy-tailed distributions, includes symmetric
stable distributions
Product of Cauchy Variables
Consider 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
Key points:
• Set 𝐴𝑡= {(𝑥,𝑦):𝑥𝑦 > 𝑡}
• Unique contact point between ∂𝐴𝑡 and maximal level set of density
• Decay of density crucial
• Shape of domain affects results
Asymptotic Behavior
𝑃{𝑋𝑌 > 𝑡} ∼ 2
𝜋2log𝑡
𝑡 as 𝑡 → ∞
For 𝛼 ∈ (0,1):
𝑃{𝑋 > 𝑡𝛼|𝑋𝑌 > 𝑡} ∼ 1−𝛼
2
Result: (𝑋,𝑌)/𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to 𝛿(0,0)
Logarithmic Scale Analysis
Writing 𝑋 = 𝜖1𝑒𝑈, 𝑌 = 𝜖2𝑒𝑉:
Key result: Distribution of (𝑈,𝑉)/log𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to uniform
distribution over {(𝑠,1−𝑠):𝑠 ∈ [0,1]}
Motivation for Systematic Methods
• Specific arguments for simple cases not generalizable
• Higher dimensions make visualization impossible
• Need for systematic procedures independent of intuition
Connection to Laplace’s Method
Theorem (Laplace’s Approximation): For ∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 with suitable conditions:
∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 ∼ (2𝜋)𝑑/2
√det𝐷2𝐼(𝑎∗)𝜆−𝑑/2𝑒−𝜆𝐼(𝑎∗) as 𝜆 → ∞
Relation to Our Problem
For 𝐴 = 𝑡𝐴1 and 𝐼 homogeneous of degree 𝛼:
∫ 𝑒−𝐼(𝑥)
𝑡𝐴1𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Key insight: Our method generalizes Laplace’s method with an infinite-dimensional
parameter
Challenges and Approach
• 𝐼 may not have a unique minimum on 𝐴
• Minimum may not be in the interior of 𝐴
• 𝐼 attains minimum on a set 𝐷𝐴 (potentially a k-dimensional manifold)
Approach:
• Apply Laplace’s approximation along fibers orthogonal to 𝐷𝐴
• Integrate approximations over ∂𝐴
• Careful change of variables and control of Jacobian
Scope of the Study
• Focus on sets with smooth boundary
• Assume 𝐼 is convex (can often transform problem to this case)
• Consider points where 𝐼(𝑥)−𝐼(𝐴) is bounded or not too large
Overview of Upcoming Chapters
• Chapters 2-5: Proof of main theorem for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Chapter 6: Special case of translated sets
• Chapter 7: Scaled sets and homogeneous 𝐼
• Chapter 8: Tail probabilities of quadratic forms
• Chapter 9: Random linear forms
• Chapter 10: Random matrices
• Chapter 11: Applications to time series analysis
• Chapter 12: Suprema of stochastic processes
Key Distribution Families
Symmetric Weibull-like Distribution
𝑤𝛼(𝑥)= 𝐾𝑤,𝛼𝑒−|𝑥|𝛼/𝛼,𝑥 ∈ ℝ
Student-like Distribution
Tail behavior:
𝑆𝛼(−𝑥)∼ 1−𝑆𝛼(𝑥)∼𝐾𝑠,𝛼𝛼(𝛼−1)/2
𝑥𝛼 as 𝑥 → ∞
Importance:
• Weibull-like: Includes Gaussian (𝛼 = 2), shows discontinuities at 𝛼 = 2
• Student-like: Representative of heavy-tailed distributions, includes symmetric
stable distributions
Product of Cauchy Variables
Consider 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
Key points:
• Set 𝐴𝑡= {(𝑥,𝑦):𝑥𝑦 > 𝑡}
• Unique contact point between ∂𝐴𝑡 and maximal level set of density
• Decay of density crucial
• Shape of domain affects results
Asymptotic Behavior
𝑃{𝑋𝑌 > 𝑡} ∼ 2
𝜋2log𝑡
𝑡 as 𝑡 → ∞
For 𝛼 ∈ (0,1):
𝑃{𝑋 > 𝑡𝛼|𝑋𝑌 > 𝑡} ∼ 1−𝛼
2
Result: (𝑋,𝑌)/𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to 𝛿(0,0)
Logarithmic Scale Analysis
Writing 𝑋 = 𝜖1𝑒𝑈, 𝑌 = 𝜖2𝑒𝑉:
Key result: Distribution of (𝑈,𝑉)/log𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to uniform
distribution over {(𝑠,1−𝑠):𝑠 ∈ [0,1]}
Motivation for Systematic Methods
• Specific arguments for simple cases not generalizable
• Higher dimensions make visualization impossible
• Need for systematic procedures independent of intuition
Connection to Laplace’s Method
Theorem (Laplace’s Approximation): For ∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 with suitable conditions:
∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 ∼ (2𝜋)𝑑/2
√det𝐷2𝐼(𝑎∗)𝜆−𝑑/2𝑒−𝜆𝐼(𝑎∗) as 𝜆 → ∞
Relation to Our Problem
For 𝐴 = 𝑡𝐴1 and 𝐼 homogeneous of degree 𝛼:
∫ 𝑒−𝐼(𝑥)
𝑡𝐴1𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Key insight: Our method generalizes Laplace’s method with an infinite-dimensional
parameter
Challenges and Approach
• 𝐼 may not have a unique minimum on 𝐴
• Minimum may not be in the interior of 𝐴
• 𝐼 attains minimum on a set 𝐷𝐴 (potentially a k-dimensional manifold)
Approach:
• Apply Laplace’s approximation along fibers orthogonal to 𝐷𝐴
• Integrate approximations over ∂𝐴
• Careful change of variables and control of Jacobian
Scope of the Study
• Focus on sets with smooth boundary
• Assume 𝐼 is convex (can often transform problem to this case)
• Consider points where 𝐼(𝑥)−𝐼(𝐴) is bounded or not too large
Overview of Upcoming Chapters
• Chapters 2-5: Proof of main theorem for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Chapter 6: Special case of translated sets
• Chapter 7: Scaled sets and homogeneous 𝐼
• Chapter 8: Tail probabilities of quadratic forms
• Chapter 9: Random linear forms
• Chapter 10: Random matrices
• Chapter 11: Applications to time series analysis
• Chapter 12: Suprema of stochastic processes
Key Distribution Families
Symmetric Weibull-like Distribution
𝑤𝛼(𝑥)= 𝐾𝑤,𝛼𝑒−|𝑥|𝛼/𝛼,𝑥 ∈ ℝ
Student-like Distribution
Tail behavior:
𝑆𝛼(−𝑥)∼ 1−𝑆𝛼(𝑥)∼𝐾𝑠,𝛼𝛼(𝛼−1)/2
𝑥𝛼 as 𝑥 → ∞
Importance:
• Weibull-like: Includes Gaussian (𝛼 = 2), shows discontinuities at 𝛼 = 2
• Student-like: Representative of heavy-tailed distributions, includes symmetric
stable distributions
Product of Cauchy Variables
Consider 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
Key points:
• Set 𝐴𝑡= {(𝑥,𝑦):𝑥𝑦 > 𝑡}
• Unique contact point between ∂𝐴𝑡 and maximal level set of density
• Decay of density crucial
• Shape of domain affects results
Asymptotic Behavior
𝑃{𝑋𝑌 > 𝑡} ∼ 2
𝜋2log𝑡
𝑡 as 𝑡 → ∞
For 𝛼 ∈ (0,1):
𝑃{𝑋 > 𝑡𝛼|𝑋𝑌 > 𝑡} ∼ 1−𝛼
2
Result: (𝑋,𝑌)/𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to 𝛿(0,0)
Logarithmic Scale Analysis
Writing 𝑋 = 𝜖1𝑒𝑈, 𝑌 = 𝜖2𝑒𝑉:
Key result: Distribution of (𝑈,𝑉)/log𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to uniform
distribution over {(𝑠,1−𝑠):𝑠 ∈ [0,1]}
Motivation for Systematic Methods
• Specific arguments for simple cases not generalizable
• Higher dimensions make visualization impossible
• Need for systematic procedures independent of intuition
Connection to Laplace’s Method
Theorem (Laplace’s Approximation): For ∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 with suitable conditions:
∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 ∼ (2𝜋)𝑑/2
√det𝐷2𝐼(𝑎∗)𝜆−𝑑/2𝑒−𝜆𝐼(𝑎∗) as 𝜆 → ∞
Relation to Our Problem
For 𝐴 = 𝑡𝐴1 and 𝐼 homogeneous of degree 𝛼:
∫ 𝑒−𝐼(𝑥)
𝑡𝐴1𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Key insight: Our method generalizes Laplace’s method with an infinite-dimensional
parameter
Challenges and Approach
• 𝐼 may not have a unique minimum on 𝐴
• Minimum may not be in the interior of 𝐴
• 𝐼 attains minimum on a set 𝐷𝐴 (potentially a k-dimensional manifold)
Approach:
• Apply Laplace’s approximation along fibers orthogonal to 𝐷𝐴
• Integrate approximations over ∂𝐴
• Careful change of variables and control of Jacobian
Scope of the Study
• Focus on sets with smooth boundary
• Assume 𝐼 is convex (can often transform problem to this case)
• Consider points where 𝐼(𝑥)−𝐼(𝐴) is bounded or not too large
Overview of Upcoming Chapters
• Chapters 2-5: Proof of main theorem for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Chapter 6: Special case of translated sets
• Chapter 7: Scaled sets and homogeneous 𝐼
• Chapter 8: Tail probabilities of quadratic forms
• Chapter 9: Random linear forms
• Chapter 10: Random matrices
• Chapter 11: Applications to time series analysis
• Chapter 12: Suprema of stochastic processes
Key Distribution Families
Symmetric Weibull-like Distribution
𝑤𝛼(𝑥)= 𝐾𝑤,𝛼𝑒−|𝑥|𝛼/𝛼,𝑥 ∈ ℝ
Student-like Distribution
Tail behavior:
𝑆𝛼(−𝑥)∼ 1−𝑆𝛼(𝑥)∼𝐾𝑠,𝛼𝛼(𝛼−1)/2
𝑥𝛼 as 𝑥 → ∞
Importance:
• Weibull-like: Includes Gaussian (𝛼 = 2), shows discontinuities at 𝛼 = 2
• Student-like: Representative of heavy-tailed distributions, includes symmetric
stable distributions
Product of Cauchy Variables
Consider 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
Key points:
• Set 𝐴𝑡= {(𝑥,𝑦):𝑥𝑦 > 𝑡}
• Unique contact point between ∂𝐴𝑡 and maximal level set of density
• Decay of density crucial
• Shape of domain affects results
Asymptotic Behavior
𝑃{𝑋𝑌 > 𝑡} ∼ 2
𝜋2log𝑡
𝑡 as 𝑡 → ∞
For 𝛼 ∈ (0,1):
𝑃{𝑋 > 𝑡𝛼|𝑋𝑌 > 𝑡} ∼ 1−𝛼
2
Result: (𝑋,𝑌)/𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to 𝛿(0,0)
Logarithmic Scale Analysis
Writing 𝑋 = 𝜖1𝑒𝑈, 𝑌 = 𝜖2𝑒𝑉:
Key result: Distribution of (𝑈,𝑉)/log𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to uniform
distribution over {(𝑠,1−𝑠):𝑠 ∈ [0,1]}
Motivation for Systematic Methods
• Specific arguments for simple cases not generalizable
• Higher dimensions make visualization impossible
• Need for systematic procedures independent of intuition
Connection to Laplace’s Method
Theorem (Laplace’s Approximation): For ∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 with suitable conditions:
∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 ∼ (2𝜋)𝑑/2
√det𝐷2𝐼(𝑎∗)𝜆−𝑑/2𝑒−𝜆𝐼(𝑎∗) as 𝜆 → ∞
Relation to Our Problem
For 𝐴 = 𝑡𝐴1 and 𝐼 homogeneous of degree 𝛼:
∫ 𝑒−𝐼(𝑥)
𝑡𝐴1𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Key insight: Our method generalizes Laplace’s method with an infinite-dimensional
parameter
Challenges and Approach
• 𝐼 may not have a unique minimum on 𝐴
• Minimum may not be in the interior of 𝐴
• 𝐼 attains minimum on a set 𝐷𝐴 (potentially a k-dimensional manifold)
Approach:
• Apply Laplace’s approximation along fibers orthogonal to 𝐷𝐴
• Integrate approximations over ∂𝐴
• Careful change of variables and control of Jacobian
Scope of the Study
• Focus on sets with smooth boundary
• Assume 𝐼 is convex (can often transform problem to this case)
• Consider points where 𝐼(𝑥)−𝐼(𝐴) is bounded or not too large
Overview of Upcoming Chapters
• Chapters 2-5: Proof of main theorem for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Chapter 6: Special case of translated sets
• Chapter 7: Scaled sets and homogeneous 𝐼
• Chapter 8: Tail probabilities of quadratic forms
• Chapter 9: Random linear forms
• Chapter 10: Random matrices
• Chapter 11: Applications to time series analysis
• Chapter 12: Suprema of stochastic processes
Key Distribution Families
Symmetric Weibull-like Distribution
𝑤𝛼(𝑥)= 𝐾𝑤,𝛼𝑒−|𝑥|𝛼/𝛼,𝑥 ∈ ℝ
Student-like Distribution
Tail behavior:
𝑆𝛼(−𝑥)∼ 1−𝑆𝛼(𝑥)∼𝐾𝑠,𝛼𝛼(𝛼−1)/2
𝑥𝛼 as 𝑥 → ∞
Importance:
• Weibull-like: Includes Gaussian (𝛼 = 2), shows discontinuities at 𝛼 = 2
• Student-like: Representative of heavy-tailed distributions, includes symmetric
stable distributions
Product of Cauchy Variables
Consider 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
Key points:
• Set 𝐴𝑡= {(𝑥,𝑦):𝑥𝑦 > 𝑡}
• Unique contact point between ∂𝐴𝑡 and maximal level set of density
• Decay of density crucial
• Shape of domain affects results
Asymptotic Behavior
𝑃{𝑋𝑌 > 𝑡} ∼ 2
𝜋2log𝑡
𝑡 as 𝑡 → ∞
For 𝛼 ∈ (0,1):
𝑃{𝑋 > 𝑡𝛼|𝑋𝑌 > 𝑡} ∼ 1−𝛼
2
Result: (𝑋,𝑌)/𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to 𝛿(0,0)
Logarithmic Scale Analysis
Writing 𝑋 = 𝜖1𝑒𝑈, 𝑌 = 𝜖2𝑒𝑉:
Key result: Distribution of (𝑈,𝑉)/log𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to uniform
distribution over {(𝑠,1−𝑠):𝑠 ∈ [0,1]}
Motivation for Systematic Methods
• Specific arguments for simple cases not generalizable
• Higher dimensions make visualization impossible
• Need for systematic procedures independent of intuition
Connection to Laplace’s Method
Theorem (Laplace’s Approximation): For ∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 with suitable conditions:
∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 ∼ (2𝜋)𝑑/2
√det𝐷2𝐼(𝑎∗)𝜆−𝑑/2𝑒−𝜆𝐼(𝑎∗) as 𝜆 → ∞
Relation to Our Problem
For 𝐴 = 𝑡𝐴1 and 𝐼 homogeneous of degree 𝛼:
∫ 𝑒−𝐼(𝑥)
𝑡𝐴1𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Key insight: Our method generalizes Laplace’s method with an infinite-dimensional
parameter
Challenges and Approach
• 𝐼 may not have a unique minimum on 𝐴
• Minimum may not be in the interior of 𝐴
• 𝐼 attains minimum on a set 𝐷𝐴 (potentially a k-dimensional manifold)
Approach:
• Apply Laplace’s approximation along fibers orthogonal to 𝐷𝐴
• Integrate approximations over ∂𝐴
• Careful change of variables and control of Jacobian
Scope of the Study
• Focus on sets with smooth boundary
• Assume 𝐼 is convex (can often transform problem to this case)
• Consider points where 𝐼(𝑥)−𝐼(𝐴) is bounded or not too large
Overview of Upcoming Chapters
• Chapters 2-5: Proof of main theorem for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Chapter 6: Special case of translated sets
• Chapter 7: Scaled sets and homogeneous 𝐼
• Chapter 8: Tail probabilities of quadratic forms
• Chapter 9: Random linear forms
• Chapter 10: Random matrices
• Chapter 11: Applications to time series analysis
• Chapter 12: Suprema of stochastic processes
Key Distribution Families
Symmetric Weibull-like Distribution
𝑤𝛼(𝑥)= 𝐾𝑤,𝛼𝑒−|𝑥|𝛼/𝛼,𝑥 ∈ ℝ
Student-like Distribution
Tail behavior:
𝑆𝛼(−𝑥)∼ 1−𝑆𝛼(𝑥)∼𝐾𝑠,𝛼𝛼(𝛼−1)/2
𝑥𝛼 as 𝑥 → ∞
Importance:
• Weibull-like: Includes Gaussian (𝛼 = 2), shows discontinuities at 𝛼 = 2
• Student-like: Representative of heavy-tailed distributions, includes symmetric
stable distributions
Product of Cauchy Variables
Consider 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
Key points:
• Set 𝐴𝑡= {(𝑥,𝑦):𝑥𝑦 > 𝑡}
• Unique contact point between ∂𝐴𝑡 and maximal level set of density
• Decay of density crucial
• Shape of domain affects results
Asymptotic Behavior
𝑃{𝑋𝑌 > 𝑡} ∼ 2
𝜋2log𝑡
𝑡 as 𝑡 → ∞
For 𝛼 ∈ (0,1):
𝑃{𝑋 > 𝑡𝛼|𝑋𝑌 > 𝑡} ∼ 1−𝛼
2
Result: (𝑋,𝑌)/𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to 𝛿(0,0)
Logarithmic Scale Analysis
Writing 𝑋 = 𝜖1𝑒𝑈, 𝑌 = 𝜖2𝑒𝑉:
Key result: Distribution of (𝑈,𝑉)/log𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to uniform
distribution over {(𝑠,1−𝑠):𝑠 ∈ [0,1]}
Motivation for Systematic Methods
• Specific arguments for simple cases not generalizable
• Higher dimensions make visualization impossible
• Need for systematic procedures independent of intuition
Connection to Laplace’s Method
Theorem (Laplace’s Approximation): For ∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 with suitable conditions:
∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 ∼ (2𝜋)𝑑/2
√det𝐷2𝐼(𝑎∗)𝜆−𝑑/2𝑒−𝜆𝐼(𝑎∗) as 𝜆 → ∞
Relation to Our Problem
For 𝐴 = 𝑡𝐴1 and 𝐼 homogeneous of degree 𝛼:
∫ 𝑒−𝐼(𝑥)
𝑡𝐴1𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Key insight: Our method generalizes Laplace’s method with an infinite-dimensional
parameter
Challenges and Approach
• 𝐼 may not have a unique minimum on 𝐴
• Minimum may not be in the interior of 𝐴
• 𝐼 attains minimum on a set 𝐷𝐴 (potentially a k-dimensional manifold)
Approach:
• Apply Laplace’s approximation along fibers orthogonal to 𝐷𝐴
• Integrate approximations over ∂𝐴
• Careful change of variables and control of Jacobian
Scope of the Study
• Focus on sets with smooth boundary
• Assume 𝐼 is convex (can often transform problem to this case)
• Consider points where 𝐼(𝑥)−𝐼(𝐴) is bounded or not too large
Overview of Upcoming Chapters
• Chapters 2-5: Proof of main theorem for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Chapter 6: Special case of translated sets
• Chapter 7: Scaled sets and homogeneous 𝐼
• Chapter 8: Tail probabilities of quadratic forms
• Chapter 9: Random linear forms
• Chapter 10: Random matrices
• Chapter 11: Applications to time series analysis
• Chapter 12: Suprema of stochastic processes
Key Distribution Families
Symmetric Weibull-like Distribution
𝑤𝛼(𝑥)= 𝐾𝑤,𝛼𝑒−|𝑥|𝛼/𝛼,𝑥 ∈ ℝ
Student-like Distribution
Tail behavior:
𝑆𝛼(−𝑥)∼ 1−𝑆𝛼(𝑥)∼𝐾𝑠,𝛼𝛼(𝛼−1)/2
𝑥𝛼 as 𝑥 → ∞
Importance:
• Weibull-like: Includes Gaussian (𝛼 = 2), shows discontinuities at 𝛼 = 2
• Student-like: Representative of heavy-tailed distributions, includes symmetric
stable distributions
Product of Cauchy Variables
Consider 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
Key points:
• Set 𝐴𝑡= {(𝑥,𝑦):𝑥𝑦 > 𝑡}
• Unique contact point between ∂𝐴𝑡 and maximal level set of density
• Decay of density crucial
• Shape of domain affects results
Asymptotic Behavior
𝑃{𝑋𝑌 > 𝑡} ∼ 2
𝜋2log𝑡
𝑡 as 𝑡 → ∞
For 𝛼 ∈ (0,1):
𝑃{𝑋 > 𝑡𝛼|𝑋𝑌 > 𝑡} ∼ 1−𝛼
2
Result: (𝑋,𝑌)/𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to 𝛿(0,0)
Logarithmic Scale Analysis
Writing 𝑋 = 𝜖1𝑒𝑈, 𝑌 = 𝜖2𝑒𝑉:
Key result: Distribution of (𝑈,𝑉)/log𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to uniform
distribution over {(𝑠,1−𝑠):𝑠 ∈ [0,1]}
Motivation for Systematic Methods
• Specific arguments for simple cases not generalizable
• Higher dimensions make visualization impossible
• Need for systematic procedures independent of intuition
Connection to Laplace’s Method
Theorem (Laplace’s Approximation): For ∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 with suitable conditions:
∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 ∼ (2𝜋)𝑑/2
√det𝐷2𝐼(𝑎∗)𝜆−𝑑/2𝑒−𝜆𝐼(𝑎∗) as 𝜆 → ∞
Relation to Our Problem
For 𝐴 = 𝑡𝐴1 and 𝐼 homogeneous of degree 𝛼:
∫ 𝑒−𝐼(𝑥)
𝑡𝐴1𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Key insight: Our method generalizes Laplace’s method with an infinite-dimensional
parameter
Challenges and Approach
• 𝐼 may not have a unique minimum on 𝐴
• Minimum may not be in the interior of 𝐴
• 𝐼 attains minimum on a set 𝐷𝐴 (potentially a k-dimensional manifold)
Approach:
• Apply Laplace’s approximation along fibers orthogonal to 𝐷𝐴
• Integrate approximations over ∂𝐴
• Careful change of variables and control of Jacobian
Scope of the Study
• Focus on sets with smooth boundary
• Assume 𝐼 is convex (can often transform problem to this case)
• Consider points where 𝐼(𝑥)−𝐼(𝐴) is bounded or not too large
Overview of Upcoming Chapters
• Chapters 2-5: Proof of main theorem for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Chapter 6: Special case of translated sets
• Chapter 7: Scaled sets and homogeneous 𝐼
• Chapter 8: Tail probabilities of quadratic forms
• Chapter 9: Random linear forms
• Chapter 10: Random matrices
• Chapter 11: Applications to time series analysis
• Chapter 12: Suprema of stochastic processes
Key Distribution Families
Symmetric Weibull-like Distribution
𝑤𝛼(𝑥)= 𝐾𝑤,𝛼𝑒−|𝑥|𝛼/𝛼,𝑥 ∈ ℝ
Student-like Distribution
Tail behavior:
𝑆𝛼(−𝑥)∼ 1−𝑆𝛼(𝑥)∼𝐾𝑠,𝛼𝛼(𝛼−1)/2
𝑥𝛼 as 𝑥 → ∞
Importance:
• Weibull-like: Includes Gaussian (𝛼 = 2), shows discontinuities at 𝛼 = 2
• Student-like: Representative of heavy-tailed distributions, includes symmetric
stable distributions
Product of Cauchy Variables
Consider 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
Key points:
• Set 𝐴𝑡= {(𝑥,𝑦):𝑥𝑦 > 𝑡}
• Unique contact point between ∂𝐴𝑡 and maximal level set of density
• Decay of density crucial
• Shape of domain affects results
Asymptotic Behavior
𝑃{𝑋𝑌 > 𝑡} ∼ 2
𝜋2log𝑡
𝑡 as 𝑡 → ∞
For 𝛼 ∈ (0,1):
𝑃{𝑋 > 𝑡𝛼|𝑋𝑌 > 𝑡} ∼ 1−𝛼
2
Result: (𝑋,𝑌)/𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to 𝛿(0,0)
Logarithmic Scale Analysis
Writing 𝑋 = 𝜖1𝑒𝑈, 𝑌 = 𝜖2𝑒𝑉:
Key result: Distribution of (𝑈,𝑉)/log𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to uniform
distribution over {(𝑠,1−𝑠):𝑠 ∈ [0,1]}
Motivation for Systematic Methods
• Specific arguments for simple cases not generalizable
• Higher dimensions make visualization impossible
• Need for systematic procedures independent of intuition
Connection to Laplace’s Method
Theorem (Laplace’s Approximation): For ∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 with suitable conditions:
∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 ∼ (2𝜋)𝑑/2
√det𝐷2𝐼(𝑎∗)𝜆−𝑑/2𝑒−𝜆𝐼(𝑎∗) as 𝜆 → ∞
Relation to Our Problem
For 𝐴 = 𝑡𝐴1 and 𝐼 homogeneous of degree 𝛼:
∫ 𝑒−𝐼(𝑥)
𝑡𝐴1𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Key insight: Our method generalizes Laplace’s method with an infinite-dimensional
parameter
Challenges and Approach
• 𝐼 may not have a unique minimum on 𝐴
• Minimum may not be in the interior of 𝐴
• 𝐼 attains minimum on a set 𝐷𝐴 (potentially a k-dimensional manifold)
Approach:
• Apply Laplace’s approximation along fibers orthogonal to 𝐷𝐴
• Integrate approximations over ∂𝐴
• Careful change of variables and control of Jacobian
Scope of the Study
• Focus on sets with smooth boundary
• Assume 𝐼 is convex (can often transform problem to this case)
• Consider points where 𝐼(𝑥)−𝐼(𝐴) is bounded or not too large
Overview of Upcoming Chapters
• Chapters 2-5: Proof of main theorem for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Chapter 6: Special case of translated sets
• Chapter 7: Scaled sets and homogeneous 𝐼
• Chapter 8: Tail probabilities of quadratic forms
• Chapter 9: Random linear forms
• Chapter 10: Random matrices
• Chapter 11: Applications to time series analysis
• Chapter 12: Suprema of stochastic processes
Key Distribution Families
Symmetric Weibull-like Distribution
𝑤𝛼(𝑥)= 𝐾𝑤,𝛼𝑒−|𝑥|𝛼/𝛼,𝑥 ∈ ℝ
Student-like Distribution
Tail behavior:
𝑆𝛼(−𝑥)∼ 1−𝑆𝛼(𝑥)∼𝐾𝑠,𝛼𝛼(𝛼−1)/2
𝑥𝛼 as 𝑥 → ∞
Importance:
• Weibull-like: Includes Gaussian (𝛼 = 2), shows discontinuities at 𝛼 = 2
• Student-like: Representative of heavy-tailed distributions, includes symmetric
stable distributions
Product of Cauchy Variables
Consider 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
Key points:
• Set 𝐴𝑡= {(𝑥,𝑦):𝑥𝑦 > 𝑡}
• Unique contact point between ∂𝐴𝑡 and maximal level set of density
• Decay of density crucial
• Shape of domain affects results
Asymptotic Behavior
𝑃{𝑋𝑌 > 𝑡} ∼ 2
𝜋2log𝑡
𝑡 as 𝑡 → ∞
For 𝛼 ∈ (0,1):
𝑃{𝑋 > 𝑡𝛼|𝑋𝑌 > 𝑡} ∼ 1−𝛼
2
Result: (𝑋,𝑌)/𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to 𝛿(0,0)
Logarithmic Scale Analysis
Writing 𝑋 = 𝜖1𝑒𝑈, 𝑌 = 𝜖2𝑒𝑉:
Key result: Distribution of (𝑈,𝑉)/log𝑡 given 𝑋𝑌 > 𝑡 converges weakly* to uniform
distribution over {(𝑠,1−𝑠):𝑠 ∈ [0,1]}
Motivation for Systematic Methods
• Specific arguments for simple cases not generalizable
• Higher dimensions make visualization impossible
• Need for systematic procedures independent of intuition
Connection to Laplace’s Method
Theorem (Laplace’s Approximation): For ∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 with suitable conditions:
∫𝑒−𝜆𝐼(𝑥)
𝐴𝑑𝑥 ∼ (2𝜋)𝑑/2
√det𝐷2𝐼(𝑎∗)𝜆−𝑑/2𝑒−𝜆𝐼(𝑎∗) as 𝜆 → ∞
Relation to Our Problem
For 𝐴 = 𝑡𝐴1 and 𝐼 homogeneous of degree 𝛼:
∫ 𝑒−𝐼(𝑥)
𝑡𝐴1𝑑𝑥 = 𝑡𝑑∫ 𝑒−𝑡𝛼𝐼(𝑦)
𝐴1𝑑𝑦
Key insight: Our method generalizes Laplace’s method with an infinite-dimensional
parameter
Challenges and Approach
• 𝐼 may not have a unique minimum on 𝐴
• Minimum may not be in the interior of 𝐴
• 𝐼 attains minimum on a set 𝐷𝐴 (potentially a k-dimensional manifold)
Approach:
• Apply Laplace’s approximation along fibers orthogonal to 𝐷𝐴
• Integrate approximations over ∂𝐴
• Careful change of variables and control of Jacobian
Scope of the Study
• Focus on sets with smooth boundary
• Assume 𝐼 is convex (can often transform problem to this case)
• Consider points where 𝐼(𝑥)−𝐼(𝐴) is bounded or not too large
Overview of Upcoming Chapters
• Chapters 2-5: Proof of main theorem for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Chapter 6: Special case of translated sets
• Chapter 7: Scaled sets and homogeneous 𝐼
• Chapter 8: Tail probabilities of quadratic forms
• Chapter 9: Random linear forms
• Chapter 10: Random matrices
• Chapter 11: Applications to time series analysis
• Chapter 12: Suprema of stochastic processes
Key Distribution Families
Symmetric Weibull-like Distribution
𝑤𝛼(𝑥)= 𝐾𝑤,𝛼𝑒−|𝑥|𝛼/𝛼,𝑥 ∈ ℝ
Student-like Distribution
Tail behavior:
𝑆𝛼(−𝑥)∼ 1−𝑆𝛼(𝑥)∼𝐾𝑠,𝛼𝛼(𝛼−1)/2
𝑥𝛼 as 𝑥 → ∞
Importance:
• Weibull-like: Includes Gaussian (𝛼 = 2), shows discontinuities at 𝛼 = 2
• Student-like: Representative of heavy-tailed distributions, includes symmetric
stable distributions
