Asymptotics of Integrals and Conditional Distributions
Lecturer’s Name
Introduction
Key motivation: Study behavior of integrals of the form
∫𝑓
𝐴(𝑥)𝑑𝑥, 𝐴 ⊂ ℝ𝑑
for sets 𝐴 far away from the origin.
Applications
• Statistics and probability theory
• Other areas of mathematics where such integrals arise
Challenges
• Difficult to compute numerically for high dimensions (10-50)
• Quadrature methods ineffective
• Monte-Carlo methods fail for small integral values (10−2, 10−3)
Examples in Probability and Statistics
Density Functions
Two example density functions on ℝ:
1. Weibull-like: 𝑤𝛼(𝑥)= 𝐾𝑤,𝛼exp(−|𝑥|𝛼/𝛼) 2. Student: 𝑠𝛼(𝑥)= 𝐾𝑠,𝛼(1+𝑥2/𝛼)−(𝛼+1)/2
Asymptotic Behavior of Integrals
For domain 𝐴𝑡= {𝑥 ∈ ℝ𝑑:⟨𝐶𝑥,𝑥⟩ > 𝑡}:
For 𝛼 ≠ 2:
∫ ∏𝑤𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(𝛼,𝑑,𝐶)𝑒−𝑡𝛼/2/𝑐2𝑡−(𝛼−2)𝑑/(4−𝛼)
For 𝛼 = 2:
∫ ∏𝑤2
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(2,𝑑,𝐶)𝑒−𝑡/𝜆𝑡(𝑘−1)/2
where 𝑘 is the dimension of the eigensubspace for the largest eigenvalue 𝜆 of 𝐶𝑇+𝐶.
For Student distribution:
∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝐾𝑠,𝛼𝛼(𝛼+1)/22𝑡−𝛼/2 ∑ 𝐶𝑖,𝑖
𝛼/2
𝑖:𝐶𝑖,𝑖>0
Applications to Random Matrices
For 𝐴𝑡= {𝑥 ∈ 𝑀(𝑛,ℝ):det𝑥 > 𝑡}:
Weibull-like distribution:
∫ ∏ 𝑤𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ {𝑐𝑒−𝑡𝛼/𝑛𝑡(𝛼(𝑛2−1)−2𝑛2)/2𝑛 if 𝛼 ≠ 2
𝑐𝑒−𝑛𝑡2/2𝑡(𝑛2−𝑛−2)/2 if 𝛼 = 2
Student distribution:
∫ ∏ 𝑠𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ 𝑐(log𝑡)𝑛−1
𝑡𝛼
Conditional Distributions and Gibbs Conditioning
Key question: Distribution of 𝑛 points in ℝ𝑛 given the volume of their parallelogram.
Examples of Conditional Distributions
Example 1: Standard Normal Variables
For 𝑋,𝑌 independent standard normal, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (1/2,1/2) in probability
Example 2: Exponential Variables
For 𝑋,𝑌 independent exponential, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → Uniform on {𝛼(1,0)+(1−𝛼)(0,1):𝛼 ∈ [0,1]}
Example 3: Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (𝑃𝑥+𝑃𝑦)/2
where 𝑃𝑥,𝑃𝑦 are Pareto distributions on x and y axes respectively.
Example 4: Product of Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
(Distribution to be analyzed)
Key insights:
• Limiting conditional distributions relate to tail behavior
• Geometry of level sets vs. conditioning set is crucial
• Naive interpretation of pictures can be misleading
Key motivation: Study behavior of integrals of the form
∫𝑓
𝐴(𝑥)𝑑𝑥, 𝐴 ⊂ ℝ𝑑
for sets 𝐴 far away from the origin.
Applications
• Statistics and probability theory
• Other areas of mathematics where such integrals arise
Challenges
• Difficult to compute numerically for high dimensions (10-50)
• Quadrature methods ineffective
• Monte-Carlo methods fail for small integral values (10−2, 10−3)
Examples in Probability and Statistics
Density Functions
Two example density functions on ℝ:
1. Weibull-like: 𝑤𝛼(𝑥)= 𝐾𝑤,𝛼exp(−|𝑥|𝛼/𝛼) 2. Student: 𝑠𝛼(𝑥)= 𝐾𝑠,𝛼(1+𝑥2/𝛼)−(𝛼+1)/2
Asymptotic Behavior of Integrals
For domain 𝐴𝑡= {𝑥 ∈ ℝ𝑑:⟨𝐶𝑥,𝑥⟩ > 𝑡}:
For 𝛼 ≠ 2:
∫ ∏𝑤𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(𝛼,𝑑,𝐶)𝑒−𝑡𝛼/2/𝑐2𝑡−(𝛼−2)𝑑/(4−𝛼)
For 𝛼 = 2:
∫ ∏𝑤2
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(2,𝑑,𝐶)𝑒−𝑡/𝜆𝑡(𝑘−1)/2
where 𝑘 is the dimension of the eigensubspace for the largest eigenvalue 𝜆 of 𝐶𝑇+𝐶.
For Student distribution:
∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝐾𝑠,𝛼𝛼(𝛼+1)/22𝑡−𝛼/2 ∑ 𝐶𝑖,𝑖
𝛼/2
𝑖:𝐶𝑖,𝑖>0
Applications to Random Matrices
For 𝐴𝑡= {𝑥 ∈ 𝑀(𝑛,ℝ):det𝑥 > 𝑡}:
Weibull-like distribution:
∫ ∏ 𝑤𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ {𝑐𝑒−𝑡𝛼/𝑛𝑡(𝛼(𝑛2−1)−2𝑛2)/2𝑛 if 𝛼 ≠ 2
𝑐𝑒−𝑛𝑡2/2𝑡(𝑛2−𝑛−2)/2 if 𝛼 = 2
Student distribution:
∫ ∏ 𝑠𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ 𝑐(log𝑡)𝑛−1
𝑡𝛼
Conditional Distributions and Gibbs Conditioning
Key question: Distribution of 𝑛 points in ℝ𝑛 given the volume of their parallelogram.
Examples of Conditional Distributions
Example 1: Standard Normal Variables
For 𝑋,𝑌 independent standard normal, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (1/2,1/2) in probability
Example 2: Exponential Variables
For 𝑋,𝑌 independent exponential, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → Uniform on {𝛼(1,0)+(1−𝛼)(0,1):𝛼 ∈ [0,1]}
Example 3: Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (𝑃𝑥+𝑃𝑦)/2
where 𝑃𝑥,𝑃𝑦 are Pareto distributions on x and y axes respectively.
Example 4: Product of Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
(Distribution to be analyzed)
Key insights:
• Limiting conditional distributions relate to tail behavior
• Geometry of level sets vs. conditioning set is crucial
• Naive interpretation of pictures can be misleading
Key motivation: Study behavior of integrals of the form
∫𝑓
𝐴(𝑥)𝑑𝑥, 𝐴 ⊂ ℝ𝑑
for sets 𝐴 far away from the origin.
Applications
• Statistics and probability theory
• Other areas of mathematics where such integrals arise
Challenges
• Difficult to compute numerically for high dimensions (10-50)
• Quadrature methods ineffective
• Monte-Carlo methods fail for small integral values (10−2, 10−3)
Examples in Probability and Statistics
Density Functions
Two example density functions on ℝ:
1. Weibull-like: 𝑤𝛼(𝑥)= 𝐾𝑤,𝛼exp(−|𝑥|𝛼/𝛼) 2. Student: 𝑠𝛼(𝑥)= 𝐾𝑠,𝛼(1+𝑥2/𝛼)−(𝛼+1)/2
Asymptotic Behavior of Integrals
For domain 𝐴𝑡= {𝑥 ∈ ℝ𝑑:⟨𝐶𝑥,𝑥⟩ > 𝑡}:
For 𝛼 ≠ 2:
∫ ∏𝑤𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(𝛼,𝑑,𝐶)𝑒−𝑡𝛼/2/𝑐2𝑡−(𝛼−2)𝑑/(4−𝛼)
For 𝛼 = 2:
∫ ∏𝑤2
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(2,𝑑,𝐶)𝑒−𝑡/𝜆𝑡(𝑘−1)/2
where 𝑘 is the dimension of the eigensubspace for the largest eigenvalue 𝜆 of 𝐶𝑇+𝐶.
For Student distribution:
∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝐾𝑠,𝛼𝛼(𝛼+1)/22𝑡−𝛼/2 ∑ 𝐶𝑖,𝑖
𝛼/2
𝑖:𝐶𝑖,𝑖>0
Applications to Random Matrices
For 𝐴𝑡= {𝑥 ∈ 𝑀(𝑛,ℝ):det𝑥 > 𝑡}:
Weibull-like distribution:
∫ ∏ 𝑤𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ {𝑐𝑒−𝑡𝛼/𝑛𝑡(𝛼(𝑛2−1)−2𝑛2)/2𝑛 if 𝛼 ≠ 2
𝑐𝑒−𝑛𝑡2/2𝑡(𝑛2−𝑛−2)/2 if 𝛼 = 2
Student distribution:
∫ ∏ 𝑠𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ 𝑐(log𝑡)𝑛−1
𝑡𝛼
Conditional Distributions and Gibbs Conditioning
Key question: Distribution of 𝑛 points in ℝ𝑛 given the volume of their parallelogram.
Examples of Conditional Distributions
Example 1: Standard Normal Variables
For 𝑋,𝑌 independent standard normal, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (1/2,1/2) in probability
Example 2: Exponential Variables
For 𝑋,𝑌 independent exponential, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → Uniform on {𝛼(1,0)+(1−𝛼)(0,1):𝛼 ∈ [0,1]}
Example 3: Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (𝑃𝑥+𝑃𝑦)/2
where 𝑃𝑥,𝑃𝑦 are Pareto distributions on x and y axes respectively.
Example 4: Product of Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
(Distribution to be analyzed)
Key insights:
• Limiting conditional distributions relate to tail behavior
• Geometry of level sets vs. conditioning set is crucial
• Naive interpretation of pictures can be misleading
Key motivation: Study behavior of integrals of the form
∫𝑓
𝐴(𝑥)𝑑𝑥, 𝐴 ⊂ ℝ𝑑
for sets 𝐴 far away from the origin.
Applications
• Statistics and probability theory
• Other areas of mathematics where such integrals arise
Challenges
• Difficult to compute numerically for high dimensions (10-50)
• Quadrature methods ineffective
• Monte-Carlo methods fail for small integral values (10−2, 10−3)
Examples in Probability and Statistics
Density Functions
Two example density functions on ℝ:
1. Weibull-like: 𝑤𝛼(𝑥)= 𝐾𝑤,𝛼exp(−|𝑥|𝛼/𝛼) 2. Student: 𝑠𝛼(𝑥)= 𝐾𝑠,𝛼(1+𝑥2/𝛼)−(𝛼+1)/2
Asymptotic Behavior of Integrals
For domain 𝐴𝑡= {𝑥 ∈ ℝ𝑑:⟨𝐶𝑥,𝑥⟩ > 𝑡}:
For 𝛼 ≠ 2:
∫ ∏𝑤𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(𝛼,𝑑,𝐶)𝑒−𝑡𝛼/2/𝑐2𝑡−(𝛼−2)𝑑/(4−𝛼)
For 𝛼 = 2:
∫ ∏𝑤2
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(2,𝑑,𝐶)𝑒−𝑡/𝜆𝑡(𝑘−1)/2
where 𝑘 is the dimension of the eigensubspace for the largest eigenvalue 𝜆 of 𝐶𝑇+𝐶.
For Student distribution:
∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝐾𝑠,𝛼𝛼(𝛼+1)/22𝑡−𝛼/2 ∑ 𝐶𝑖,𝑖
𝛼/2
𝑖:𝐶𝑖,𝑖>0
Applications to Random Matrices
For 𝐴𝑡= {𝑥 ∈ 𝑀(𝑛,ℝ):det𝑥 > 𝑡}:
Weibull-like distribution:
∫ ∏ 𝑤𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ {𝑐𝑒−𝑡𝛼/𝑛𝑡(𝛼(𝑛2−1)−2𝑛2)/2𝑛 if 𝛼 ≠ 2
𝑐𝑒−𝑛𝑡2/2𝑡(𝑛2−𝑛−2)/2 if 𝛼 = 2
Student distribution:
∫ ∏ 𝑠𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ 𝑐(log𝑡)𝑛−1
𝑡𝛼
Conditional Distributions and Gibbs Conditioning
Key question: Distribution of 𝑛 points in ℝ𝑛 given the volume of their parallelogram.
Examples of Conditional Distributions
Example 1: Standard Normal Variables
For 𝑋,𝑌 independent standard normal, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (1/2,1/2) in probability
Example 2: Exponential Variables
For 𝑋,𝑌 independent exponential, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → Uniform on {𝛼(1,0)+(1−𝛼)(0,1):𝛼 ∈ [0,1]}
Example 3: Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (𝑃𝑥+𝑃𝑦)/2
where 𝑃𝑥,𝑃𝑦 are Pareto distributions on x and y axes respectively.
Example 4: Product of Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
(Distribution to be analyzed)
Key insights:
• Limiting conditional distributions relate to tail behavior
• Geometry of level sets vs. conditioning set is crucial
• Naive interpretation of pictures can be misleading
Key motivation: Study behavior of integrals of the form
∫𝑓
𝐴(𝑥)𝑑𝑥, 𝐴 ⊂ ℝ𝑑
for sets 𝐴 far away from the origin.
Applications
• Statistics and probability theory
• Other areas of mathematics where such integrals arise
Challenges
• Difficult to compute numerically for high dimensions (10-50)
• Quadrature methods ineffective
• Monte-Carlo methods fail for small integral values (10−2, 10−3)
Examples in Probability and Statistics
Density Functions
Two example density functions on ℝ:
1. Weibull-like: 𝑤𝛼(𝑥)= 𝐾𝑤,𝛼exp(−|𝑥|𝛼/𝛼) 2. Student: 𝑠𝛼(𝑥)= 𝐾𝑠,𝛼(1+𝑥2/𝛼)−(𝛼+1)/2
Asymptotic Behavior of Integrals
For domain 𝐴𝑡= {𝑥 ∈ ℝ𝑑:⟨𝐶𝑥,𝑥⟩ > 𝑡}:
For 𝛼 ≠ 2:
∫ ∏𝑤𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(𝛼,𝑑,𝐶)𝑒−𝑡𝛼/2/𝑐2𝑡−(𝛼−2)𝑑/(4−𝛼)
For 𝛼 = 2:
∫ ∏𝑤2
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(2,𝑑,𝐶)𝑒−𝑡/𝜆𝑡(𝑘−1)/2
where 𝑘 is the dimension of the eigensubspace for the largest eigenvalue 𝜆 of 𝐶𝑇+𝐶.
For Student distribution:
∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝐾𝑠,𝛼𝛼(𝛼+1)/22𝑡−𝛼/2 ∑ 𝐶𝑖,𝑖
𝛼/2
𝑖:𝐶𝑖,𝑖>0
Applications to Random Matrices
For 𝐴𝑡= {𝑥 ∈ 𝑀(𝑛,ℝ):det𝑥 > 𝑡}:
Weibull-like distribution:
∫ ∏ 𝑤𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ {𝑐𝑒−𝑡𝛼/𝑛𝑡(𝛼(𝑛2−1)−2𝑛2)/2𝑛 if 𝛼 ≠ 2
𝑐𝑒−𝑛𝑡2/2𝑡(𝑛2−𝑛−2)/2 if 𝛼 = 2
Student distribution:
∫ ∏ 𝑠𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ 𝑐(log𝑡)𝑛−1
𝑡𝛼
Conditional Distributions and Gibbs Conditioning
Key question: Distribution of 𝑛 points in ℝ𝑛 given the volume of their parallelogram.
Examples of Conditional Distributions
Example 1: Standard Normal Variables
For 𝑋,𝑌 independent standard normal, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (1/2,1/2) in probability
Example 2: Exponential Variables
For 𝑋,𝑌 independent exponential, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → Uniform on {𝛼(1,0)+(1−𝛼)(0,1):𝛼 ∈ [0,1]}
Example 3: Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (𝑃𝑥+𝑃𝑦)/2
where 𝑃𝑥,𝑃𝑦 are Pareto distributions on x and y axes respectively.
Example 4: Product of Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
(Distribution to be analyzed)
Key insights:
• Limiting conditional distributions relate to tail behavior
• Geometry of level sets vs. conditioning set is crucial
• Naive interpretation of pictures can be misleading
Key motivation: Study behavior of integrals of the form
∫𝑓
𝐴(𝑥)𝑑𝑥, 𝐴 ⊂ ℝ𝑑
for sets 𝐴 far away from the origin.
Applications
• Statistics and probability theory
• Other areas of mathematics where such integrals arise
Challenges
• Difficult to compute numerically for high dimensions (10-50)
• Quadrature methods ineffective
• Monte-Carlo methods fail for small integral values (10−2, 10−3)
Examples in Probability and Statistics
Density Functions
Two example density functions on ℝ:
1. Weibull-like: 𝑤𝛼(𝑥)= 𝐾𝑤,𝛼exp(−|𝑥|𝛼/𝛼) 2. Student: 𝑠𝛼(𝑥)= 𝐾𝑠,𝛼(1+𝑥2/𝛼)−(𝛼+1)/2
Asymptotic Behavior of Integrals
For domain 𝐴𝑡= {𝑥 ∈ ℝ𝑑:⟨𝐶𝑥,𝑥⟩ > 𝑡}:
For 𝛼 ≠ 2:
∫ ∏𝑤𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(𝛼,𝑑,𝐶)𝑒−𝑡𝛼/2/𝑐2𝑡−(𝛼−2)𝑑/(4−𝛼)
For 𝛼 = 2:
∫ ∏𝑤2
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(2,𝑑,𝐶)𝑒−𝑡/𝜆𝑡(𝑘−1)/2
where 𝑘 is the dimension of the eigensubspace for the largest eigenvalue 𝜆 of 𝐶𝑇+𝐶.
For Student distribution:
∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝐾𝑠,𝛼𝛼(𝛼+1)/22𝑡−𝛼/2 ∑ 𝐶𝑖,𝑖
𝛼/2
𝑖:𝐶𝑖,𝑖>0
Applications to Random Matrices
For 𝐴𝑡= {𝑥 ∈ 𝑀(𝑛,ℝ):det𝑥 > 𝑡}:
Weibull-like distribution:
∫ ∏ 𝑤𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ {𝑐𝑒−𝑡𝛼/𝑛𝑡(𝛼(𝑛2−1)−2𝑛2)/2𝑛 if 𝛼 ≠ 2
𝑐𝑒−𝑛𝑡2/2𝑡(𝑛2−𝑛−2)/2 if 𝛼 = 2
Student distribution:
∫ ∏ 𝑠𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ 𝑐(log𝑡)𝑛−1
𝑡𝛼
Conditional Distributions and Gibbs Conditioning
Key question: Distribution of 𝑛 points in ℝ𝑛 given the volume of their parallelogram.
Examples of Conditional Distributions
Example 1: Standard Normal Variables
For 𝑋,𝑌 independent standard normal, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (1/2,1/2) in probability
Example 2: Exponential Variables
For 𝑋,𝑌 independent exponential, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → Uniform on {𝛼(1,0)+(1−𝛼)(0,1):𝛼 ∈ [0,1]}
Example 3: Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (𝑃𝑥+𝑃𝑦)/2
where 𝑃𝑥,𝑃𝑦 are Pareto distributions on x and y axes respectively.
Example 4: Product of Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
(Distribution to be analyzed)
Key insights:
• Limiting conditional distributions relate to tail behavior
• Geometry of level sets vs. conditioning set is crucial
• Naive interpretation of pictures can be misleading
Key motivation: Study behavior of integrals of the form
∫𝑓
𝐴(𝑥)𝑑𝑥, 𝐴 ⊂ ℝ𝑑
for sets 𝐴 far away from the origin.
Applications
• Statistics and probability theory
• Other areas of mathematics where such integrals arise
Challenges
• Difficult to compute numerically for high dimensions (10-50)
• Quadrature methods ineffective
• Monte-Carlo methods fail for small integral values (10−2, 10−3)
Examples in Probability and Statistics
Density Functions
Two example density functions on ℝ:
1. Weibull-like: 𝑤𝛼(𝑥)= 𝐾𝑤,𝛼exp(−|𝑥|𝛼/𝛼) 2. Student: 𝑠𝛼(𝑥)= 𝐾𝑠,𝛼(1+𝑥2/𝛼)−(𝛼+1)/2
Asymptotic Behavior of Integrals
For domain 𝐴𝑡= {𝑥 ∈ ℝ𝑑:⟨𝐶𝑥,𝑥⟩ > 𝑡}:
For 𝛼 ≠ 2:
∫ ∏𝑤𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(𝛼,𝑑,𝐶)𝑒−𝑡𝛼/2/𝑐2𝑡−(𝛼−2)𝑑/(4−𝛼)
For 𝛼 = 2:
∫ ∏𝑤2
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(2,𝑑,𝐶)𝑒−𝑡/𝜆𝑡(𝑘−1)/2
where 𝑘 is the dimension of the eigensubspace for the largest eigenvalue 𝜆 of 𝐶𝑇+𝐶.
For Student distribution:
∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝐾𝑠,𝛼𝛼(𝛼+1)/22𝑡−𝛼/2 ∑ 𝐶𝑖,𝑖
𝛼/2
𝑖:𝐶𝑖,𝑖>0
Applications to Random Matrices
For 𝐴𝑡= {𝑥 ∈ 𝑀(𝑛,ℝ):det𝑥 > 𝑡}:
Weibull-like distribution:
∫ ∏ 𝑤𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ {𝑐𝑒−𝑡𝛼/𝑛𝑡(𝛼(𝑛2−1)−2𝑛2)/2𝑛 if 𝛼 ≠ 2
𝑐𝑒−𝑛𝑡2/2𝑡(𝑛2−𝑛−2)/2 if 𝛼 = 2
Student distribution:
∫ ∏ 𝑠𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ 𝑐(log𝑡)𝑛−1
𝑡𝛼
Conditional Distributions and Gibbs Conditioning
Key question: Distribution of 𝑛 points in ℝ𝑛 given the volume of their parallelogram.
Examples of Conditional Distributions
Example 1: Standard Normal Variables
For 𝑋,𝑌 independent standard normal, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (1/2,1/2) in probability
Example 2: Exponential Variables
For 𝑋,𝑌 independent exponential, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → Uniform on {𝛼(1,0)+(1−𝛼)(0,1):𝛼 ∈ [0,1]}
Example 3: Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (𝑃𝑥+𝑃𝑦)/2
where 𝑃𝑥,𝑃𝑦 are Pareto distributions on x and y axes respectively.
Example 4: Product of Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
(Distribution to be analyzed)
Key insights:
• Limiting conditional distributions relate to tail behavior
• Geometry of level sets vs. conditioning set is crucial
• Naive interpretation of pictures can be misleading
Key motivation: Study behavior of integrals of the form
∫𝑓
𝐴(𝑥)𝑑𝑥, 𝐴 ⊂ ℝ𝑑
for sets 𝐴 far away from the origin.
Applications
• Statistics and probability theory
• Other areas of mathematics where such integrals arise
Challenges
• Difficult to compute numerically for high dimensions (10-50)
• Quadrature methods ineffective
• Monte-Carlo methods fail for small integral values (10−2, 10−3)
Examples in Probability and Statistics
Density Functions
Two example density functions on ℝ:
1. Weibull-like: 𝑤𝛼(𝑥)= 𝐾𝑤,𝛼exp(−|𝑥|𝛼/𝛼) 2. Student: 𝑠𝛼(𝑥)= 𝐾𝑠,𝛼(1+𝑥2/𝛼)−(𝛼+1)/2
Asymptotic Behavior of Integrals
For domain 𝐴𝑡= {𝑥 ∈ ℝ𝑑:⟨𝐶𝑥,𝑥⟩ > 𝑡}:
For 𝛼 ≠ 2:
∫ ∏𝑤𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(𝛼,𝑑,𝐶)𝑒−𝑡𝛼/2/𝑐2𝑡−(𝛼−2)𝑑/(4−𝛼)
For 𝛼 = 2:
∫ ∏𝑤2
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(2,𝑑,𝐶)𝑒−𝑡/𝜆𝑡(𝑘−1)/2
where 𝑘 is the dimension of the eigensubspace for the largest eigenvalue 𝜆 of 𝐶𝑇+𝐶.
For Student distribution:
∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝐾𝑠,𝛼𝛼(𝛼+1)/22𝑡−𝛼/2 ∑ 𝐶𝑖,𝑖
𝛼/2
𝑖:𝐶𝑖,𝑖>0
Applications to Random Matrices
For 𝐴𝑡= {𝑥 ∈ 𝑀(𝑛,ℝ):det𝑥 > 𝑡}:
Weibull-like distribution:
∫ ∏ 𝑤𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ {𝑐𝑒−𝑡𝛼/𝑛𝑡(𝛼(𝑛2−1)−2𝑛2)/2𝑛 if 𝛼 ≠ 2
𝑐𝑒−𝑛𝑡2/2𝑡(𝑛2−𝑛−2)/2 if 𝛼 = 2
Student distribution:
∫ ∏ 𝑠𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ 𝑐(log𝑡)𝑛−1
𝑡𝛼
Conditional Distributions and Gibbs Conditioning
Key question: Distribution of 𝑛 points in ℝ𝑛 given the volume of their parallelogram.
Examples of Conditional Distributions
Example 1: Standard Normal Variables
For 𝑋,𝑌 independent standard normal, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (1/2,1/2) in probability
Example 2: Exponential Variables
For 𝑋,𝑌 independent exponential, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → Uniform on {𝛼(1,0)+(1−𝛼)(0,1):𝛼 ∈ [0,1]}
Example 3: Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (𝑃𝑥+𝑃𝑦)/2
where 𝑃𝑥,𝑃𝑦 are Pareto distributions on x and y axes respectively.
Example 4: Product of Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
(Distribution to be analyzed)
Key insights:
• Limiting conditional distributions relate to tail behavior
• Geometry of level sets vs. conditioning set is crucial
• Naive interpretation of pictures can be misleading
Key motivation: Study behavior of integrals of the form
∫𝑓
𝐴(𝑥)𝑑𝑥, 𝐴 ⊂ ℝ𝑑
for sets 𝐴 far away from the origin.
Applications
• Statistics and probability theory
• Other areas of mathematics where such integrals arise
Challenges
• Difficult to compute numerically for high dimensions (10-50)
• Quadrature methods ineffective
• Monte-Carlo methods fail for small integral values (10−2, 10−3)
Examples in Probability and Statistics
Density Functions
Two example density functions on ℝ:
1. Weibull-like: 𝑤𝛼(𝑥)= 𝐾𝑤,𝛼exp(−|𝑥|𝛼/𝛼) 2. Student: 𝑠𝛼(𝑥)= 𝐾𝑠,𝛼(1+𝑥2/𝛼)−(𝛼+1)/2
Asymptotic Behavior of Integrals
For domain 𝐴𝑡= {𝑥 ∈ ℝ𝑑:⟨𝐶𝑥,𝑥⟩ > 𝑡}:
For 𝛼 ≠ 2:
∫ ∏𝑤𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(𝛼,𝑑,𝐶)𝑒−𝑡𝛼/2/𝑐2𝑡−(𝛼−2)𝑑/(4−𝛼)
For 𝛼 = 2:
∫ ∏𝑤2
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(2,𝑑,𝐶)𝑒−𝑡/𝜆𝑡(𝑘−1)/2
where 𝑘 is the dimension of the eigensubspace for the largest eigenvalue 𝜆 of 𝐶𝑇+𝐶.
For Student distribution:
∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝐾𝑠,𝛼𝛼(𝛼+1)/22𝑡−𝛼/2 ∑ 𝐶𝑖,𝑖
𝛼/2
𝑖:𝐶𝑖,𝑖>0
Applications to Random Matrices
For 𝐴𝑡= {𝑥 ∈ 𝑀(𝑛,ℝ):det𝑥 > 𝑡}:
Weibull-like distribution:
∫ ∏ 𝑤𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ {𝑐𝑒−𝑡𝛼/𝑛𝑡(𝛼(𝑛2−1)−2𝑛2)/2𝑛 if 𝛼 ≠ 2
𝑐𝑒−𝑛𝑡2/2𝑡(𝑛2−𝑛−2)/2 if 𝛼 = 2
Student distribution:
∫ ∏ 𝑠𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ 𝑐(log𝑡)𝑛−1
𝑡𝛼
Conditional Distributions and Gibbs Conditioning
Key question: Distribution of 𝑛 points in ℝ𝑛 given the volume of their parallelogram.
Examples of Conditional Distributions
Example 1: Standard Normal Variables
For 𝑋,𝑌 independent standard normal, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (1/2,1/2) in probability
Example 2: Exponential Variables
For 𝑋,𝑌 independent exponential, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → Uniform on {𝛼(1,0)+(1−𝛼)(0,1):𝛼 ∈ [0,1]}
Example 3: Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (𝑃𝑥+𝑃𝑦)/2
where 𝑃𝑥,𝑃𝑦 are Pareto distributions on x and y axes respectively.
Example 4: Product of Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
(Distribution to be analyzed)
Key insights:
• Limiting conditional distributions relate to tail behavior
• Geometry of level sets vs. conditioning set is crucial
• Naive interpretation of pictures can be misleading
Key motivation: Study behavior of integrals of the form
∫𝑓
𝐴(𝑥)𝑑𝑥, 𝐴 ⊂ ℝ𝑑
for sets 𝐴 far away from the origin.
Applications
• Statistics and probability theory
• Other areas of mathematics where such integrals arise
Challenges
• Difficult to compute numerically for high dimensions (10-50)
• Quadrature methods ineffective
• Monte-Carlo methods fail for small integral values (10−2, 10−3)
Examples in Probability and Statistics
Density Functions
Two example density functions on ℝ:
1. Weibull-like: 𝑤𝛼(𝑥)= 𝐾𝑤,𝛼exp(−|𝑥|𝛼/𝛼) 2. Student: 𝑠𝛼(𝑥)= 𝐾𝑠,𝛼(1+𝑥2/𝛼)−(𝛼+1)/2
Asymptotic Behavior of Integrals
For domain 𝐴𝑡= {𝑥 ∈ ℝ𝑑:⟨𝐶𝑥,𝑥⟩ > 𝑡}:
For 𝛼 ≠ 2:
∫ ∏𝑤𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(𝛼,𝑑,𝐶)𝑒−𝑡𝛼/2/𝑐2𝑡−(𝛼−2)𝑑/(4−𝛼)
For 𝛼 = 2:
∫ ∏𝑤2
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(2,𝑑,𝐶)𝑒−𝑡/𝜆𝑡(𝑘−1)/2
where 𝑘 is the dimension of the eigensubspace for the largest eigenvalue 𝜆 of 𝐶𝑇+𝐶.
For Student distribution:
∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝐾𝑠,𝛼𝛼(𝛼+1)/22𝑡−𝛼/2 ∑ 𝐶𝑖,𝑖
𝛼/2
𝑖:𝐶𝑖,𝑖>0
Applications to Random Matrices
For 𝐴𝑡= {𝑥 ∈ 𝑀(𝑛,ℝ):det𝑥 > 𝑡}:
Weibull-like distribution:
∫ ∏ 𝑤𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ {𝑐𝑒−𝑡𝛼/𝑛𝑡(𝛼(𝑛2−1)−2𝑛2)/2𝑛 if 𝛼 ≠ 2
𝑐𝑒−𝑛𝑡2/2𝑡(𝑛2−𝑛−2)/2 if 𝛼 = 2
Student distribution:
∫ ∏ 𝑠𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ 𝑐(log𝑡)𝑛−1
𝑡𝛼
Conditional Distributions and Gibbs Conditioning
Key question: Distribution of 𝑛 points in ℝ𝑛 given the volume of their parallelogram.
Examples of Conditional Distributions
Example 1: Standard Normal Variables
For 𝑋,𝑌 independent standard normal, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (1/2,1/2) in probability
Example 2: Exponential Variables
For 𝑋,𝑌 independent exponential, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → Uniform on {𝛼(1,0)+(1−𝛼)(0,1):𝛼 ∈ [0,1]}
Example 3: Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (𝑃𝑥+𝑃𝑦)/2
where 𝑃𝑥,𝑃𝑦 are Pareto distributions on x and y axes respectively.
Example 4: Product of Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
(Distribution to be analyzed)
Key insights:
• Limiting conditional distributions relate to tail behavior
• Geometry of level sets vs. conditioning set is crucial
• Naive interpretation of pictures can be misleading
Key motivation: Study behavior of integrals of the form
∫𝑓
𝐴(𝑥)𝑑𝑥, 𝐴 ⊂ ℝ𝑑
for sets 𝐴 far away from the origin.
Applications
• Statistics and probability theory
• Other areas of mathematics where such integrals arise
Challenges
• Difficult to compute numerically for high dimensions (10-50)
• Quadrature methods ineffective
• Monte-Carlo methods fail for small integral values (10−2, 10−3)
Examples in Probability and Statistics
Density Functions
Two example density functions on ℝ:
1. Weibull-like: 𝑤𝛼(𝑥)= 𝐾𝑤,𝛼exp(−|𝑥|𝛼/𝛼) 2. Student: 𝑠𝛼(𝑥)= 𝐾𝑠,𝛼(1+𝑥2/𝛼)−(𝛼+1)/2
Asymptotic Behavior of Integrals
For domain 𝐴𝑡= {𝑥 ∈ ℝ𝑑:⟨𝐶𝑥,𝑥⟩ > 𝑡}:
For 𝛼 ≠ 2:
∫ ∏𝑤𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(𝛼,𝑑,𝐶)𝑒−𝑡𝛼/2/𝑐2𝑡−(𝛼−2)𝑑/(4−𝛼)
For 𝛼 = 2:
∫ ∏𝑤2
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(2,𝑑,𝐶)𝑒−𝑡/𝜆𝑡(𝑘−1)/2
where 𝑘 is the dimension of the eigensubspace for the largest eigenvalue 𝜆 of 𝐶𝑇+𝐶.
For Student distribution:
∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝐾𝑠,𝛼𝛼(𝛼+1)/22𝑡−𝛼/2 ∑ 𝐶𝑖,𝑖
𝛼/2
𝑖:𝐶𝑖,𝑖>0
Applications to Random Matrices
For 𝐴𝑡= {𝑥 ∈ 𝑀(𝑛,ℝ):det𝑥 > 𝑡}:
Weibull-like distribution:
∫ ∏ 𝑤𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ {𝑐𝑒−𝑡𝛼/𝑛𝑡(𝛼(𝑛2−1)−2𝑛2)/2𝑛 if 𝛼 ≠ 2
𝑐𝑒−𝑛𝑡2/2𝑡(𝑛2−𝑛−2)/2 if 𝛼 = 2
Student distribution:
∫ ∏ 𝑠𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ 𝑐(log𝑡)𝑛−1
𝑡𝛼
Conditional Distributions and Gibbs Conditioning
Key question: Distribution of 𝑛 points in ℝ𝑛 given the volume of their parallelogram.
Examples of Conditional Distributions
Example 1: Standard Normal Variables
For 𝑋,𝑌 independent standard normal, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (1/2,1/2) in probability
Example 2: Exponential Variables
For 𝑋,𝑌 independent exponential, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → Uniform on {𝛼(1,0)+(1−𝛼)(0,1):𝛼 ∈ [0,1]}
Example 3: Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (𝑃𝑥+𝑃𝑦)/2
where 𝑃𝑥,𝑃𝑦 are Pareto distributions on x and y axes respectively.
Example 4: Product of Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
(Distribution to be analyzed)
Key insights:
• Limiting conditional distributions relate to tail behavior
• Geometry of level sets vs. conditioning set is crucial
• Naive interpretation of pictures can be misleading
Key motivation: Study behavior of integrals of the form
∫𝑓
𝐴(𝑥)𝑑𝑥, 𝐴 ⊂ ℝ𝑑
for sets 𝐴 far away from the origin.
Applications
• Statistics and probability theory
• Other areas of mathematics where such integrals arise
Challenges
• Difficult to compute numerically for high dimensions (10-50)
• Quadrature methods ineffective
• Monte-Carlo methods fail for small integral values (10−2, 10−3)
Examples in Probability and Statistics
Density Functions
Two example density functions on ℝ:
1. Weibull-like: 𝑤𝛼(𝑥)= 𝐾𝑤,𝛼exp(−|𝑥|𝛼/𝛼) 2. Student: 𝑠𝛼(𝑥)= 𝐾𝑠,𝛼(1+𝑥2/𝛼)−(𝛼+1)/2
Asymptotic Behavior of Integrals
For domain 𝐴𝑡= {𝑥 ∈ ℝ𝑑:⟨𝐶𝑥,𝑥⟩ > 𝑡}:
For 𝛼 ≠ 2:
∫ ∏𝑤𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(𝛼,𝑑,𝐶)𝑒−𝑡𝛼/2/𝑐2𝑡−(𝛼−2)𝑑/(4−𝛼)
For 𝛼 = 2:
∫ ∏𝑤2
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(2,𝑑,𝐶)𝑒−𝑡/𝜆𝑡(𝑘−1)/2
where 𝑘 is the dimension of the eigensubspace for the largest eigenvalue 𝜆 of 𝐶𝑇+𝐶.
For Student distribution:
∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝐾𝑠,𝛼𝛼(𝛼+1)/22𝑡−𝛼/2 ∑ 𝐶𝑖,𝑖
𝛼/2
𝑖:𝐶𝑖,𝑖>0
Applications to Random Matrices
For 𝐴𝑡= {𝑥 ∈ 𝑀(𝑛,ℝ):det𝑥 > 𝑡}:
Weibull-like distribution:
∫ ∏ 𝑤𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ {𝑐𝑒−𝑡𝛼/𝑛𝑡(𝛼(𝑛2−1)−2𝑛2)/2𝑛 if 𝛼 ≠ 2
𝑐𝑒−𝑛𝑡2/2𝑡(𝑛2−𝑛−2)/2 if 𝛼 = 2
Student distribution:
∫ ∏ 𝑠𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ 𝑐(log𝑡)𝑛−1
𝑡𝛼
Conditional Distributions and Gibbs Conditioning
Key question: Distribution of 𝑛 points in ℝ𝑛 given the volume of their parallelogram.
Examples of Conditional Distributions
Example 1: Standard Normal Variables
For 𝑋,𝑌 independent standard normal, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (1/2,1/2) in probability
Example 2: Exponential Variables
For 𝑋,𝑌 independent exponential, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → Uniform on {𝛼(1,0)+(1−𝛼)(0,1):𝛼 ∈ [0,1]}
Example 3: Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (𝑃𝑥+𝑃𝑦)/2
where 𝑃𝑥,𝑃𝑦 are Pareto distributions on x and y axes respectively.
Example 4: Product of Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
(Distribution to be analyzed)
Key insights:
• Limiting conditional distributions relate to tail behavior
• Geometry of level sets vs. conditioning set is crucial
• Naive interpretation of pictures can be misleading
Key motivation: Study behavior of integrals of the form
∫𝑓
𝐴(𝑥)𝑑𝑥, 𝐴 ⊂ ℝ𝑑
for sets 𝐴 far away from the origin.
Applications
• Statistics and probability theory
• Other areas of mathematics where such integrals arise
Challenges
• Difficult to compute numerically for high dimensions (10-50)
• Quadrature methods ineffective
• Monte-Carlo methods fail for small integral values (10−2, 10−3)
Examples in Probability and Statistics
Density Functions
Two example density functions on ℝ:
1. Weibull-like: 𝑤𝛼(𝑥)= 𝐾𝑤,𝛼exp(−|𝑥|𝛼/𝛼) 2. Student: 𝑠𝛼(𝑥)= 𝐾𝑠,𝛼(1+𝑥2/𝛼)−(𝛼+1)/2
Asymptotic Behavior of Integrals
For domain 𝐴𝑡= {𝑥 ∈ ℝ𝑑:⟨𝐶𝑥,𝑥⟩ > 𝑡}:
For 𝛼 ≠ 2:
∫ ∏𝑤𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(𝛼,𝑑,𝐶)𝑒−𝑡𝛼/2/𝑐2𝑡−(𝛼−2)𝑑/(4−𝛼)
For 𝛼 = 2:
∫ ∏𝑤2
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(2,𝑑,𝐶)𝑒−𝑡/𝜆𝑡(𝑘−1)/2
where 𝑘 is the dimension of the eigensubspace for the largest eigenvalue 𝜆 of 𝐶𝑇+𝐶.
For Student distribution:
∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝐾𝑠,𝛼𝛼(𝛼+1)/22𝑡−𝛼/2 ∑ 𝐶𝑖,𝑖
𝛼/2
𝑖:𝐶𝑖,𝑖>0
Applications to Random Matrices
For 𝐴𝑡= {𝑥 ∈ 𝑀(𝑛,ℝ):det𝑥 > 𝑡}:
Weibull-like distribution:
∫ ∏ 𝑤𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ {𝑐𝑒−𝑡𝛼/𝑛𝑡(𝛼(𝑛2−1)−2𝑛2)/2𝑛 if 𝛼 ≠ 2
𝑐𝑒−𝑛𝑡2/2𝑡(𝑛2−𝑛−2)/2 if 𝛼 = 2
Student distribution:
∫ ∏ 𝑠𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ 𝑐(log𝑡)𝑛−1
𝑡𝛼
Conditional Distributions and Gibbs Conditioning
Key question: Distribution of 𝑛 points in ℝ𝑛 given the volume of their parallelogram.
Examples of Conditional Distributions
Example 1: Standard Normal Variables
For 𝑋,𝑌 independent standard normal, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (1/2,1/2) in probability
Example 2: Exponential Variables
For 𝑋,𝑌 independent exponential, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → Uniform on {𝛼(1,0)+(1−𝛼)(0,1):𝛼 ∈ [0,1]}
Example 3: Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (𝑃𝑥+𝑃𝑦)/2
where 𝑃𝑥,𝑃𝑦 are Pareto distributions on x and y axes respectively.
Example 4: Product of Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
(Distribution to be analyzed)
Key insights:
• Limiting conditional distributions relate to tail behavior
• Geometry of level sets vs. conditioning set is crucial
• Naive interpretation of pictures can be misleading
Key motivation: Study behavior of integrals of the form
∫𝑓
𝐴(𝑥)𝑑𝑥, 𝐴 ⊂ ℝ𝑑
for sets 𝐴 far away from the origin.
Applications
• Statistics and probability theory
• Other areas of mathematics where such integrals arise
Challenges
• Difficult to compute numerically for high dimensions (10-50)
• Quadrature methods ineffective
• Monte-Carlo methods fail for small integral values (10−2, 10−3)
Examples in Probability and Statistics
Density Functions
Two example density functions on ℝ:
1. Weibull-like: 𝑤𝛼(𝑥)= 𝐾𝑤,𝛼exp(−|𝑥|𝛼/𝛼) 2. Student: 𝑠𝛼(𝑥)= 𝐾𝑠,𝛼(1+𝑥2/𝛼)−(𝛼+1)/2
Asymptotic Behavior of Integrals
For domain 𝐴𝑡= {𝑥 ∈ ℝ𝑑:⟨𝐶𝑥,𝑥⟩ > 𝑡}:
For 𝛼 ≠ 2:
∫ ∏𝑤𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(𝛼,𝑑,𝐶)𝑒−𝑡𝛼/2/𝑐2𝑡−(𝛼−2)𝑑/(4−𝛼)
For 𝛼 = 2:
∫ ∏𝑤2
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(2,𝑑,𝐶)𝑒−𝑡/𝜆𝑡(𝑘−1)/2
where 𝑘 is the dimension of the eigensubspace for the largest eigenvalue 𝜆 of 𝐶𝑇+𝐶.
For Student distribution:
∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝐾𝑠,𝛼𝛼(𝛼+1)/22𝑡−𝛼/2 ∑ 𝐶𝑖,𝑖
𝛼/2
𝑖:𝐶𝑖,𝑖>0
Applications to Random Matrices
For 𝐴𝑡= {𝑥 ∈ 𝑀(𝑛,ℝ):det𝑥 > 𝑡}:
Weibull-like distribution:
∫ ∏ 𝑤𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ {𝑐𝑒−𝑡𝛼/𝑛𝑡(𝛼(𝑛2−1)−2𝑛2)/2𝑛 if 𝛼 ≠ 2
𝑐𝑒−𝑛𝑡2/2𝑡(𝑛2−𝑛−2)/2 if 𝛼 = 2
Student distribution:
∫ ∏ 𝑠𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ 𝑐(log𝑡)𝑛−1
𝑡𝛼
Conditional Distributions and Gibbs Conditioning
Key question: Distribution of 𝑛 points in ℝ𝑛 given the volume of their parallelogram.
Examples of Conditional Distributions
Example 1: Standard Normal Variables
For 𝑋,𝑌 independent standard normal, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (1/2,1/2) in probability
Example 2: Exponential Variables
For 𝑋,𝑌 independent exponential, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → Uniform on {𝛼(1,0)+(1−𝛼)(0,1):𝛼 ∈ [0,1]}
Example 3: Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (𝑃𝑥+𝑃𝑦)/2
where 𝑃𝑥,𝑃𝑦 are Pareto distributions on x and y axes respectively.
Example 4: Product of Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
(Distribution to be analyzed)
Key insights:
• Limiting conditional distributions relate to tail behavior
• Geometry of level sets vs. conditioning set is crucial
• Naive interpretation of pictures can be misleading
Key motivation: Study behavior of integrals of the form
∫𝑓
𝐴(𝑥)𝑑𝑥, 𝐴 ⊂ ℝ𝑑
for sets 𝐴 far away from the origin.
Applications
• Statistics and probability theory
• Other areas of mathematics where such integrals arise
Challenges
• Difficult to compute numerically for high dimensions (10-50)
• Quadrature methods ineffective
• Monte-Carlo methods fail for small integral values (10−2, 10−3)
Examples in Probability and Statistics
Density Functions
Two example density functions on ℝ:
1. Weibull-like: 𝑤𝛼(𝑥)= 𝐾𝑤,𝛼exp(−|𝑥|𝛼/𝛼) 2. Student: 𝑠𝛼(𝑥)= 𝐾𝑠,𝛼(1+𝑥2/𝛼)−(𝛼+1)/2
Asymptotic Behavior of Integrals
For domain 𝐴𝑡= {𝑥 ∈ ℝ𝑑:⟨𝐶𝑥,𝑥⟩ > 𝑡}:
For 𝛼 ≠ 2:
∫ ∏𝑤𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(𝛼,𝑑,𝐶)𝑒−𝑡𝛼/2/𝑐2𝑡−(𝛼−2)𝑑/(4−𝛼)
For 𝛼 = 2:
∫ ∏𝑤2
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(2,𝑑,𝐶)𝑒−𝑡/𝜆𝑡(𝑘−1)/2
where 𝑘 is the dimension of the eigensubspace for the largest eigenvalue 𝜆 of 𝐶𝑇+𝐶.
For Student distribution:
∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝐾𝑠,𝛼𝛼(𝛼+1)/22𝑡−𝛼/2 ∑ 𝐶𝑖,𝑖
𝛼/2
𝑖:𝐶𝑖,𝑖>0
Applications to Random Matrices
For 𝐴𝑡= {𝑥 ∈ 𝑀(𝑛,ℝ):det𝑥 > 𝑡}:
Weibull-like distribution:
∫ ∏ 𝑤𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ {𝑐𝑒−𝑡𝛼/𝑛𝑡(𝛼(𝑛2−1)−2𝑛2)/2𝑛 if 𝛼 ≠ 2
𝑐𝑒−𝑛𝑡2/2𝑡(𝑛2−𝑛−2)/2 if 𝛼 = 2
Student distribution:
∫ ∏ 𝑠𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ 𝑐(log𝑡)𝑛−1
𝑡𝛼
Conditional Distributions and Gibbs Conditioning
Key question: Distribution of 𝑛 points in ℝ𝑛 given the volume of their parallelogram.
Examples of Conditional Distributions
Example 1: Standard Normal Variables
For 𝑋,𝑌 independent standard normal, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (1/2,1/2) in probability
Example 2: Exponential Variables
For 𝑋,𝑌 independent exponential, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → Uniform on {𝛼(1,0)+(1−𝛼)(0,1):𝛼 ∈ [0,1]}
Example 3: Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (𝑃𝑥+𝑃𝑦)/2
where 𝑃𝑥,𝑃𝑦 are Pareto distributions on x and y axes respectively.
Example 4: Product of Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
(Distribution to be analyzed)
Key insights:
• Limiting conditional distributions relate to tail behavior
• Geometry of level sets vs. conditioning set is crucial
• Naive interpretation of pictures can be misleading
Key motivation: Study behavior of integrals of the form
∫𝑓
𝐴(𝑥)𝑑𝑥, 𝐴 ⊂ ℝ𝑑
for sets 𝐴 far away from the origin.
Applications
• Statistics and probability theory
• Other areas of mathematics where such integrals arise
Challenges
• Difficult to compute numerically for high dimensions (10-50)
• Quadrature methods ineffective
• Monte-Carlo methods fail for small integral values (10−2, 10−3)
Examples in Probability and Statistics
Density Functions
Two example density functions on ℝ:
1. Weibull-like: 𝑤𝛼(𝑥)= 𝐾𝑤,𝛼exp(−|𝑥|𝛼/𝛼) 2. Student: 𝑠𝛼(𝑥)= 𝐾𝑠,𝛼(1+𝑥2/𝛼)−(𝛼+1)/2
Asymptotic Behavior of Integrals
For domain 𝐴𝑡= {𝑥 ∈ ℝ𝑑:⟨𝐶𝑥,𝑥⟩ > 𝑡}:
For 𝛼 ≠ 2:
∫ ∏𝑤𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(𝛼,𝑑,𝐶)𝑒−𝑡𝛼/2/𝑐2𝑡−(𝛼−2)𝑑/(4−𝛼)
For 𝛼 = 2:
∫ ∏𝑤2
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(2,𝑑,𝐶)𝑒−𝑡/𝜆𝑡(𝑘−1)/2
where 𝑘 is the dimension of the eigensubspace for the largest eigenvalue 𝜆 of 𝐶𝑇+𝐶.
For Student distribution:
∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝐾𝑠,𝛼𝛼(𝛼+1)/22𝑡−𝛼/2 ∑ 𝐶𝑖,𝑖
𝛼/2
𝑖:𝐶𝑖,𝑖>0
Applications to Random Matrices
For 𝐴𝑡= {𝑥 ∈ 𝑀(𝑛,ℝ):det𝑥 > 𝑡}:
Weibull-like distribution:
∫ ∏ 𝑤𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ {𝑐𝑒−𝑡𝛼/𝑛𝑡(𝛼(𝑛2−1)−2𝑛2)/2𝑛 if 𝛼 ≠ 2
𝑐𝑒−𝑛𝑡2/2𝑡(𝑛2−𝑛−2)/2 if 𝛼 = 2
Student distribution:
∫ ∏ 𝑠𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ 𝑐(log𝑡)𝑛−1
𝑡𝛼
Conditional Distributions and Gibbs Conditioning
Key question: Distribution of 𝑛 points in ℝ𝑛 given the volume of their parallelogram.
Examples of Conditional Distributions
Example 1: Standard Normal Variables
For 𝑋,𝑌 independent standard normal, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (1/2,1/2) in probability
Example 2: Exponential Variables
For 𝑋,𝑌 independent exponential, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → Uniform on {𝛼(1,0)+(1−𝛼)(0,1):𝛼 ∈ [0,1]}
Example 3: Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (𝑃𝑥+𝑃𝑦)/2
where 𝑃𝑥,𝑃𝑦 are Pareto distributions on x and y axes respectively.
Example 4: Product of Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
(Distribution to be analyzed)
Key insights:
• Limiting conditional distributions relate to tail behavior
• Geometry of level sets vs. conditioning set is crucial
• Naive interpretation of pictures can be misleading
Key motivation: Study behavior of integrals of the form
∫𝑓
𝐴(𝑥)𝑑𝑥, 𝐴 ⊂ ℝ𝑑
for sets 𝐴 far away from the origin.
Applications
• Statistics and probability theory
• Other areas of mathematics where such integrals arise
Challenges
• Difficult to compute numerically for high dimensions (10-50)
• Quadrature methods ineffective
• Monte-Carlo methods fail for small integral values (10−2, 10−3)
Examples in Probability and Statistics
Density Functions
Two example density functions on ℝ:
1. Weibull-like: 𝑤𝛼(𝑥)= 𝐾𝑤,𝛼exp(−|𝑥|𝛼/𝛼) 2. Student: 𝑠𝛼(𝑥)= 𝐾𝑠,𝛼(1+𝑥2/𝛼)−(𝛼+1)/2
Asymptotic Behavior of Integrals
For domain 𝐴𝑡= {𝑥 ∈ ℝ𝑑:⟨𝐶𝑥,𝑥⟩ > 𝑡}:
For 𝛼 ≠ 2:
∫ ∏𝑤𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(𝛼,𝑑,𝐶)𝑒−𝑡𝛼/2/𝑐2𝑡−(𝛼−2)𝑑/(4−𝛼)
For 𝛼 = 2:
∫ ∏𝑤2
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(2,𝑑,𝐶)𝑒−𝑡/𝜆𝑡(𝑘−1)/2
where 𝑘 is the dimension of the eigensubspace for the largest eigenvalue 𝜆 of 𝐶𝑇+𝐶.
For Student distribution:
∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝐾𝑠,𝛼𝛼(𝛼+1)/22𝑡−𝛼/2 ∑ 𝐶𝑖,𝑖
𝛼/2
𝑖:𝐶𝑖,𝑖>0
Applications to Random Matrices
For 𝐴𝑡= {𝑥 ∈ 𝑀(𝑛,ℝ):det𝑥 > 𝑡}:
Weibull-like distribution:
∫ ∏ 𝑤𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ {𝑐𝑒−𝑡𝛼/𝑛𝑡(𝛼(𝑛2−1)−2𝑛2)/2𝑛 if 𝛼 ≠ 2
𝑐𝑒−𝑛𝑡2/2𝑡(𝑛2−𝑛−2)/2 if 𝛼 = 2
Student distribution:
∫ ∏ 𝑠𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ 𝑐(log𝑡)𝑛−1
𝑡𝛼
Conditional Distributions and Gibbs Conditioning
Key question: Distribution of 𝑛 points in ℝ𝑛 given the volume of their parallelogram.
Examples of Conditional Distributions
Example 1: Standard Normal Variables
For 𝑋,𝑌 independent standard normal, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (1/2,1/2) in probability
Example 2: Exponential Variables
For 𝑋,𝑌 independent exponential, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → Uniform on {𝛼(1,0)+(1−𝛼)(0,1):𝛼 ∈ [0,1]}
Example 3: Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (𝑃𝑥+𝑃𝑦)/2
where 𝑃𝑥,𝑃𝑦 are Pareto distributions on x and y axes respectively.
Example 4: Product of Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
(Distribution to be analyzed)
Key insights:
• Limiting conditional distributions relate to tail behavior
• Geometry of level sets vs. conditioning set is crucial
• Naive interpretation of pictures can be misleading
Key motivation: Study behavior of integrals of the form
∫𝑓
𝐴(𝑥)𝑑𝑥, 𝐴 ⊂ ℝ𝑑
for sets 𝐴 far away from the origin.
Applications
• Statistics and probability theory
• Other areas of mathematics where such integrals arise
Challenges
• Difficult to compute numerically for high dimensions (10-50)
• Quadrature methods ineffective
• Monte-Carlo methods fail for small integral values (10−2, 10−3)
Examples in Probability and Statistics
Density Functions
Two example density functions on ℝ:
1. Weibull-like: 𝑤𝛼(𝑥)= 𝐾𝑤,𝛼exp(−|𝑥|𝛼/𝛼) 2. Student: 𝑠𝛼(𝑥)= 𝐾𝑠,𝛼(1+𝑥2/𝛼)−(𝛼+1)/2
Asymptotic Behavior of Integrals
For domain 𝐴𝑡= {𝑥 ∈ ℝ𝑑:⟨𝐶𝑥,𝑥⟩ > 𝑡}:
For 𝛼 ≠ 2:
∫ ∏𝑤𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(𝛼,𝑑,𝐶)𝑒−𝑡𝛼/2/𝑐2𝑡−(𝛼−2)𝑑/(4−𝛼)
For 𝛼 = 2:
∫ ∏𝑤2
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(2,𝑑,𝐶)𝑒−𝑡/𝜆𝑡(𝑘−1)/2
where 𝑘 is the dimension of the eigensubspace for the largest eigenvalue 𝜆 of 𝐶𝑇+𝐶.
For Student distribution:
∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝐾𝑠,𝛼𝛼(𝛼+1)/22𝑡−𝛼/2 ∑ 𝐶𝑖,𝑖
𝛼/2
𝑖:𝐶𝑖,𝑖>0
Applications to Random Matrices
For 𝐴𝑡= {𝑥 ∈ 𝑀(𝑛,ℝ):det𝑥 > 𝑡}:
Weibull-like distribution:
∫ ∏ 𝑤𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ {𝑐𝑒−𝑡𝛼/𝑛𝑡(𝛼(𝑛2−1)−2𝑛2)/2𝑛 if 𝛼 ≠ 2
𝑐𝑒−𝑛𝑡2/2𝑡(𝑛2−𝑛−2)/2 if 𝛼 = 2
Student distribution:
∫ ∏ 𝑠𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ 𝑐(log𝑡)𝑛−1
𝑡𝛼
Conditional Distributions and Gibbs Conditioning
Key question: Distribution of 𝑛 points in ℝ𝑛 given the volume of their parallelogram.
Examples of Conditional Distributions
Example 1: Standard Normal Variables
For 𝑋,𝑌 independent standard normal, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (1/2,1/2) in probability
Example 2: Exponential Variables
For 𝑋,𝑌 independent exponential, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → Uniform on {𝛼(1,0)+(1−𝛼)(0,1):𝛼 ∈ [0,1]}
Example 3: Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (𝑃𝑥+𝑃𝑦)/2
where 𝑃𝑥,𝑃𝑦 are Pareto distributions on x and y axes respectively.
Example 4: Product of Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
(Distribution to be analyzed)
Key insights:
• Limiting conditional distributions relate to tail behavior
• Geometry of level sets vs. conditioning set is crucial
• Naive interpretation of pictures can be misleading
Key motivation: Study behavior of integrals of the form
∫𝑓
𝐴(𝑥)𝑑𝑥, 𝐴 ⊂ ℝ𝑑
for sets 𝐴 far away from the origin.
Applications
• Statistics and probability theory
• Other areas of mathematics where such integrals arise
Challenges
• Difficult to compute numerically for high dimensions (10-50)
• Quadrature methods ineffective
• Monte-Carlo methods fail for small integral values (10−2, 10−3)
Examples in Probability and Statistics
Density Functions
Two example density functions on ℝ:
1. Weibull-like: 𝑤𝛼(𝑥)= 𝐾𝑤,𝛼exp(−|𝑥|𝛼/𝛼) 2. Student: 𝑠𝛼(𝑥)= 𝐾𝑠,𝛼(1+𝑥2/𝛼)−(𝛼+1)/2
Asymptotic Behavior of Integrals
For domain 𝐴𝑡= {𝑥 ∈ ℝ𝑑:⟨𝐶𝑥,𝑥⟩ > 𝑡}:
For 𝛼 ≠ 2:
∫ ∏𝑤𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(𝛼,𝑑,𝐶)𝑒−𝑡𝛼/2/𝑐2𝑡−(𝛼−2)𝑑/(4−𝛼)
For 𝛼 = 2:
∫ ∏𝑤2
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(2,𝑑,𝐶)𝑒−𝑡/𝜆𝑡(𝑘−1)/2
where 𝑘 is the dimension of the eigensubspace for the largest eigenvalue 𝜆 of 𝐶𝑇+𝐶.
For Student distribution:
∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝐾𝑠,𝛼𝛼(𝛼+1)/22𝑡−𝛼/2 ∑ 𝐶𝑖,𝑖
𝛼/2
𝑖:𝐶𝑖,𝑖>0
Applications to Random Matrices
For 𝐴𝑡= {𝑥 ∈ 𝑀(𝑛,ℝ):det𝑥 > 𝑡}:
Weibull-like distribution:
∫ ∏ 𝑤𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ {𝑐𝑒−𝑡𝛼/𝑛𝑡(𝛼(𝑛2−1)−2𝑛2)/2𝑛 if 𝛼 ≠ 2
𝑐𝑒−𝑛𝑡2/2𝑡(𝑛2−𝑛−2)/2 if 𝛼 = 2
Student distribution:
∫ ∏ 𝑠𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ 𝑐(log𝑡)𝑛−1
𝑡𝛼
Conditional Distributions and Gibbs Conditioning
Key question: Distribution of 𝑛 points in ℝ𝑛 given the volume of their parallelogram.
Examples of Conditional Distributions
Example 1: Standard Normal Variables
For 𝑋,𝑌 independent standard normal, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (1/2,1/2) in probability
Example 2: Exponential Variables
For 𝑋,𝑌 independent exponential, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → Uniform on {𝛼(1,0)+(1−𝛼)(0,1):𝛼 ∈ [0,1]}
Example 3: Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (𝑃𝑥+𝑃𝑦)/2
where 𝑃𝑥,𝑃𝑦 are Pareto distributions on x and y axes respectively.
Example 4: Product of Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
(Distribution to be analyzed)
Key insights:
• Limiting conditional distributions relate to tail behavior
• Geometry of level sets vs. conditioning set is crucial
• Naive interpretation of pictures can be misleading
Key motivation: Study behavior of integrals of the form
∫𝑓
𝐴(𝑥)𝑑𝑥, 𝐴 ⊂ ℝ𝑑
for sets 𝐴 far away from the origin.
Applications
• Statistics and probability theory
• Other areas of mathematics where such integrals arise
Challenges
• Difficult to compute numerically for high dimensions (10-50)
• Quadrature methods ineffective
• Monte-Carlo methods fail for small integral values (10−2, 10−3)
Examples in Probability and Statistics
Density Functions
Two example density functions on ℝ:
1. Weibull-like: 𝑤𝛼(𝑥)= 𝐾𝑤,𝛼exp(−|𝑥|𝛼/𝛼) 2. Student: 𝑠𝛼(𝑥)= 𝐾𝑠,𝛼(1+𝑥2/𝛼)−(𝛼+1)/2
Asymptotic Behavior of Integrals
For domain 𝐴𝑡= {𝑥 ∈ ℝ𝑑:⟨𝐶𝑥,𝑥⟩ > 𝑡}:
For 𝛼 ≠ 2:
∫ ∏𝑤𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(𝛼,𝑑,𝐶)𝑒−𝑡𝛼/2/𝑐2𝑡−(𝛼−2)𝑑/(4−𝛼)
For 𝛼 = 2:
∫ ∏𝑤2
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(2,𝑑,𝐶)𝑒−𝑡/𝜆𝑡(𝑘−1)/2
where 𝑘 is the dimension of the eigensubspace for the largest eigenvalue 𝜆 of 𝐶𝑇+𝐶.
For Student distribution:
∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝐾𝑠,𝛼𝛼(𝛼+1)/22𝑡−𝛼/2 ∑ 𝐶𝑖,𝑖
𝛼/2
𝑖:𝐶𝑖,𝑖>0
Applications to Random Matrices
For 𝐴𝑡= {𝑥 ∈ 𝑀(𝑛,ℝ):det𝑥 > 𝑡}:
Weibull-like distribution:
∫ ∏ 𝑤𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ {𝑐𝑒−𝑡𝛼/𝑛𝑡(𝛼(𝑛2−1)−2𝑛2)/2𝑛 if 𝛼 ≠ 2
𝑐𝑒−𝑛𝑡2/2𝑡(𝑛2−𝑛−2)/2 if 𝛼 = 2
Student distribution:
∫ ∏ 𝑠𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ 𝑐(log𝑡)𝑛−1
𝑡𝛼
Conditional Distributions and Gibbs Conditioning
Key question: Distribution of 𝑛 points in ℝ𝑛 given the volume of their parallelogram.
Examples of Conditional Distributions
Example 1: Standard Normal Variables
For 𝑋,𝑌 independent standard normal, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (1/2,1/2) in probability
Example 2: Exponential Variables
For 𝑋,𝑌 independent exponential, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → Uniform on {𝛼(1,0)+(1−𝛼)(0,1):𝛼 ∈ [0,1]}
Example 3: Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (𝑃𝑥+𝑃𝑦)/2
where 𝑃𝑥,𝑃𝑦 are Pareto distributions on x and y axes respectively.
Example 4: Product of Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
(Distribution to be analyzed)
Key insights:
• Limiting conditional distributions relate to tail behavior
• Geometry of level sets vs. conditioning set is crucial
• Naive interpretation of pictures can be misleading
Key motivation: Study behavior of integrals of the form
∫𝑓
𝐴(𝑥)𝑑𝑥, 𝐴 ⊂ ℝ𝑑
for sets 𝐴 far away from the origin.
Applications
• Statistics and probability theory
• Other areas of mathematics where such integrals arise
Challenges
• Difficult to compute numerically for high dimensions (10-50)
• Quadrature methods ineffective
• Monte-Carlo methods fail for small integral values (10−2, 10−3)
Examples in Probability and Statistics
Density Functions
Two example density functions on ℝ:
1. Weibull-like: 𝑤𝛼(𝑥)= 𝐾𝑤,𝛼exp(−|𝑥|𝛼/𝛼) 2. Student: 𝑠𝛼(𝑥)= 𝐾𝑠,𝛼(1+𝑥2/𝛼)−(𝛼+1)/2
Asymptotic Behavior of Integrals
For domain 𝐴𝑡= {𝑥 ∈ ℝ𝑑:⟨𝐶𝑥,𝑥⟩ > 𝑡}:
For 𝛼 ≠ 2:
∫ ∏𝑤𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(𝛼,𝑑,𝐶)𝑒−𝑡𝛼/2/𝑐2𝑡−(𝛼−2)𝑑/(4−𝛼)
For 𝛼 = 2:
∫ ∏𝑤2
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(2,𝑑,𝐶)𝑒−𝑡/𝜆𝑡(𝑘−1)/2
where 𝑘 is the dimension of the eigensubspace for the largest eigenvalue 𝜆 of 𝐶𝑇+𝐶.
For Student distribution:
∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝐾𝑠,𝛼𝛼(𝛼+1)/22𝑡−𝛼/2 ∑ 𝐶𝑖,𝑖
𝛼/2
𝑖:𝐶𝑖,𝑖>0
Applications to Random Matrices
For 𝐴𝑡= {𝑥 ∈ 𝑀(𝑛,ℝ):det𝑥 > 𝑡}:
Weibull-like distribution:
∫ ∏ 𝑤𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ {𝑐𝑒−𝑡𝛼/𝑛𝑡(𝛼(𝑛2−1)−2𝑛2)/2𝑛 if 𝛼 ≠ 2
𝑐𝑒−𝑛𝑡2/2𝑡(𝑛2−𝑛−2)/2 if 𝛼 = 2
Student distribution:
∫ ∏ 𝑠𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ 𝑐(log𝑡)𝑛−1
𝑡𝛼
Conditional Distributions and Gibbs Conditioning
Key question: Distribution of 𝑛 points in ℝ𝑛 given the volume of their parallelogram.
Examples of Conditional Distributions
Example 1: Standard Normal Variables
For 𝑋,𝑌 independent standard normal, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (1/2,1/2) in probability
Example 2: Exponential Variables
For 𝑋,𝑌 independent exponential, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → Uniform on {𝛼(1,0)+(1−𝛼)(0,1):𝛼 ∈ [0,1]}
Example 3: Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (𝑃𝑥+𝑃𝑦)/2
where 𝑃𝑥,𝑃𝑦 are Pareto distributions on x and y axes respectively.
Example 4: Product of Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
(Distribution to be analyzed)
Key insights:
• Limiting conditional distributions relate to tail behavior
• Geometry of level sets vs. conditioning set is crucial
• Naive interpretation of pictures can be misleading
Key motivation: Study behavior of integrals of the form
∫𝑓
𝐴(𝑥)𝑑𝑥, 𝐴 ⊂ ℝ𝑑
for sets 𝐴 far away from the origin.
Applications
• Statistics and probability theory
• Other areas of mathematics where such integrals arise
Challenges
• Difficult to compute numerically for high dimensions (10-50)
• Quadrature methods ineffective
• Monte-Carlo methods fail for small integral values (10−2, 10−3)
Examples in Probability and Statistics
Density Functions
Two example density functions on ℝ:
1. Weibull-like: 𝑤𝛼(𝑥)= 𝐾𝑤,𝛼exp(−|𝑥|𝛼/𝛼) 2. Student: 𝑠𝛼(𝑥)= 𝐾𝑠,𝛼(1+𝑥2/𝛼)−(𝛼+1)/2
Asymptotic Behavior of Integrals
For domain 𝐴𝑡= {𝑥 ∈ ℝ𝑑:⟨𝐶𝑥,𝑥⟩ > 𝑡}:
For 𝛼 ≠ 2:
∫ ∏𝑤𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(𝛼,𝑑,𝐶)𝑒−𝑡𝛼/2/𝑐2𝑡−(𝛼−2)𝑑/(4−𝛼)
For 𝛼 = 2:
∫ ∏𝑤2
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(2,𝑑,𝐶)𝑒−𝑡/𝜆𝑡(𝑘−1)/2
where 𝑘 is the dimension of the eigensubspace for the largest eigenvalue 𝜆 of 𝐶𝑇+𝐶.
For Student distribution:
∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝐾𝑠,𝛼𝛼(𝛼+1)/22𝑡−𝛼/2 ∑ 𝐶𝑖,𝑖
𝛼/2
𝑖:𝐶𝑖,𝑖>0
Applications to Random Matrices
For 𝐴𝑡= {𝑥 ∈ 𝑀(𝑛,ℝ):det𝑥 > 𝑡}:
Weibull-like distribution:
∫ ∏ 𝑤𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ {𝑐𝑒−𝑡𝛼/𝑛𝑡(𝛼(𝑛2−1)−2𝑛2)/2𝑛 if 𝛼 ≠ 2
𝑐𝑒−𝑛𝑡2/2𝑡(𝑛2−𝑛−2)/2 if 𝛼 = 2
Student distribution:
∫ ∏ 𝑠𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ 𝑐(log𝑡)𝑛−1
𝑡𝛼
Conditional Distributions and Gibbs Conditioning
Key question: Distribution of 𝑛 points in ℝ𝑛 given the volume of their parallelogram.
Examples of Conditional Distributions
Example 1: Standard Normal Variables
For 𝑋,𝑌 independent standard normal, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (1/2,1/2) in probability
Example 2: Exponential Variables
For 𝑋,𝑌 independent exponential, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → Uniform on {𝛼(1,0)+(1−𝛼)(0,1):𝛼 ∈ [0,1]}
Example 3: Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (𝑃𝑥+𝑃𝑦)/2
where 𝑃𝑥,𝑃𝑦 are Pareto distributions on x and y axes respectively.
Example 4: Product of Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
(Distribution to be analyzed)
Key insights:
• Limiting conditional distributions relate to tail behavior
• Geometry of level sets vs. conditioning set is crucial
• Naive interpretation of pictures can be misleading
Key motivation: Study behavior of integrals of the form
∫𝑓
𝐴(𝑥)𝑑𝑥, 𝐴 ⊂ ℝ𝑑
for sets 𝐴 far away from the origin.
Applications
• Statistics and probability theory
• Other areas of mathematics where such integrals arise
Challenges
• Difficult to compute numerically for high dimensions (10-50)
• Quadrature methods ineffective
• Monte-Carlo methods fail for small integral values (10−2, 10−3)
Examples in Probability and Statistics
Density Functions
Two example density functions on ℝ:
1. Weibull-like: 𝑤𝛼(𝑥)= 𝐾𝑤,𝛼exp(−|𝑥|𝛼/𝛼) 2. Student: 𝑠𝛼(𝑥)= 𝐾𝑠,𝛼(1+𝑥2/𝛼)−(𝛼+1)/2
Asymptotic Behavior of Integrals
For domain 𝐴𝑡= {𝑥 ∈ ℝ𝑑:⟨𝐶𝑥,𝑥⟩ > 𝑡}:
For 𝛼 ≠ 2:
∫ ∏𝑤𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(𝛼,𝑑,𝐶)𝑒−𝑡𝛼/2/𝑐2𝑡−(𝛼−2)𝑑/(4−𝛼)
For 𝛼 = 2:
∫ ∏𝑤2
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(2,𝑑,𝐶)𝑒−𝑡/𝜆𝑡(𝑘−1)/2
where 𝑘 is the dimension of the eigensubspace for the largest eigenvalue 𝜆 of 𝐶𝑇+𝐶.
For Student distribution:
∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝐾𝑠,𝛼𝛼(𝛼+1)/22𝑡−𝛼/2 ∑ 𝐶𝑖,𝑖
𝛼/2
𝑖:𝐶𝑖,𝑖>0
Applications to Random Matrices
For 𝐴𝑡= {𝑥 ∈ 𝑀(𝑛,ℝ):det𝑥 > 𝑡}:
Weibull-like distribution:
∫ ∏ 𝑤𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ {𝑐𝑒−𝑡𝛼/𝑛𝑡(𝛼(𝑛2−1)−2𝑛2)/2𝑛 if 𝛼 ≠ 2
𝑐𝑒−𝑛𝑡2/2𝑡(𝑛2−𝑛−2)/2 if 𝛼 = 2
Student distribution:
∫ ∏ 𝑠𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ 𝑐(log𝑡)𝑛−1
𝑡𝛼
Conditional Distributions and Gibbs Conditioning
Key question: Distribution of 𝑛 points in ℝ𝑛 given the volume of their parallelogram.
Examples of Conditional Distributions
Example 1: Standard Normal Variables
For 𝑋,𝑌 independent standard normal, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (1/2,1/2) in probability
Example 2: Exponential Variables
For 𝑋,𝑌 independent exponential, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → Uniform on {𝛼(1,0)+(1−𝛼)(0,1):𝛼 ∈ [0,1]}
Example 3: Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (𝑃𝑥+𝑃𝑦)/2
where 𝑃𝑥,𝑃𝑦 are Pareto distributions on x and y axes respectively.
Example 4: Product of Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
(Distribution to be analyzed)
Key insights:
• Limiting conditional distributions relate to tail behavior
• Geometry of level sets vs. conditioning set is crucial
• Naive interpretation of pictures can be misleading
Key motivation: Study behavior of integrals of the form
∫𝑓
𝐴(𝑥)𝑑𝑥, 𝐴 ⊂ ℝ𝑑
for sets 𝐴 far away from the origin.
Applications
• Statistics and probability theory
• Other areas of mathematics where such integrals arise
Challenges
• Difficult to compute numerically for high dimensions (10-50)
• Quadrature methods ineffective
• Monte-Carlo methods fail for small integral values (10−2, 10−3)
Examples in Probability and Statistics
Density Functions
Two example density functions on ℝ:
1. Weibull-like: 𝑤𝛼(𝑥)= 𝐾𝑤,𝛼exp(−|𝑥|𝛼/𝛼) 2. Student: 𝑠𝛼(𝑥)= 𝐾𝑠,𝛼(1+𝑥2/𝛼)−(𝛼+1)/2
Asymptotic Behavior of Integrals
For domain 𝐴𝑡= {𝑥 ∈ ℝ𝑑:⟨𝐶𝑥,𝑥⟩ > 𝑡}:
For 𝛼 ≠ 2:
∫ ∏𝑤𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(𝛼,𝑑,𝐶)𝑒−𝑡𝛼/2/𝑐2𝑡−(𝛼−2)𝑑/(4−𝛼)
For 𝛼 = 2:
∫ ∏𝑤2
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(2,𝑑,𝐶)𝑒−𝑡/𝜆𝑡(𝑘−1)/2
where 𝑘 is the dimension of the eigensubspace for the largest eigenvalue 𝜆 of 𝐶𝑇+𝐶.
For Student distribution:
∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝐾𝑠,𝛼𝛼(𝛼+1)/22𝑡−𝛼/2 ∑ 𝐶𝑖,𝑖
𝛼/2
𝑖:𝐶𝑖,𝑖>0
Applications to Random Matrices
For 𝐴𝑡= {𝑥 ∈ 𝑀(𝑛,ℝ):det𝑥 > 𝑡}:
Weibull-like distribution:
∫ ∏ 𝑤𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ {𝑐𝑒−𝑡𝛼/𝑛𝑡(𝛼(𝑛2−1)−2𝑛2)/2𝑛 if 𝛼 ≠ 2
𝑐𝑒−𝑛𝑡2/2𝑡(𝑛2−𝑛−2)/2 if 𝛼 = 2
Student distribution:
∫ ∏ 𝑠𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ 𝑐(log𝑡)𝑛−1
𝑡𝛼
Conditional Distributions and Gibbs Conditioning
Key question: Distribution of 𝑛 points in ℝ𝑛 given the volume of their parallelogram.
Examples of Conditional Distributions
Example 1: Standard Normal Variables
For 𝑋,𝑌 independent standard normal, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (1/2,1/2) in probability
Example 2: Exponential Variables
For 𝑋,𝑌 independent exponential, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → Uniform on {𝛼(1,0)+(1−𝛼)(0,1):𝛼 ∈ [0,1]}
Example 3: Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (𝑃𝑥+𝑃𝑦)/2
where 𝑃𝑥,𝑃𝑦 are Pareto distributions on x and y axes respectively.
Example 4: Product of Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
(Distribution to be analyzed)
Key insights:
• Limiting conditional distributions relate to tail behavior
• Geometry of level sets vs. conditioning set is crucial
• Naive interpretation of pictures can be misleading
Key motivation: Study behavior of integrals of the form
∫𝑓
𝐴(𝑥)𝑑𝑥, 𝐴 ⊂ ℝ𝑑
for sets 𝐴 far away from the origin.
Applications
• Statistics and probability theory
• Other areas of mathematics where such integrals arise
Challenges
• Difficult to compute numerically for high dimensions (10-50)
• Quadrature methods ineffective
• Monte-Carlo methods fail for small integral values (10−2, 10−3)
Examples in Probability and Statistics
Density Functions
Two example density functions on ℝ:
1. Weibull-like: 𝑤𝛼(𝑥)= 𝐾𝑤,𝛼exp(−|𝑥|𝛼/𝛼) 2. Student: 𝑠𝛼(𝑥)= 𝐾𝑠,𝛼(1+𝑥2/𝛼)−(𝛼+1)/2
Asymptotic Behavior of Integrals
For domain 𝐴𝑡= {𝑥 ∈ ℝ𝑑:⟨𝐶𝑥,𝑥⟩ > 𝑡}:
For 𝛼 ≠ 2:
∫ ∏𝑤𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(𝛼,𝑑,𝐶)𝑒−𝑡𝛼/2/𝑐2𝑡−(𝛼−2)𝑑/(4−𝛼)
For 𝛼 = 2:
∫ ∏𝑤2
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(2,𝑑,𝐶)𝑒−𝑡/𝜆𝑡(𝑘−1)/2
where 𝑘 is the dimension of the eigensubspace for the largest eigenvalue 𝜆 of 𝐶𝑇+𝐶.
For Student distribution:
∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝐾𝑠,𝛼𝛼(𝛼+1)/22𝑡−𝛼/2 ∑ 𝐶𝑖,𝑖
𝛼/2
𝑖:𝐶𝑖,𝑖>0
Applications to Random Matrices
For 𝐴𝑡= {𝑥 ∈ 𝑀(𝑛,ℝ):det𝑥 > 𝑡}:
Weibull-like distribution:
∫ ∏ 𝑤𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ {𝑐𝑒−𝑡𝛼/𝑛𝑡(𝛼(𝑛2−1)−2𝑛2)/2𝑛 if 𝛼 ≠ 2
𝑐𝑒−𝑛𝑡2/2𝑡(𝑛2−𝑛−2)/2 if 𝛼 = 2
Student distribution:
∫ ∏ 𝑠𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ 𝑐(log𝑡)𝑛−1
𝑡𝛼
Conditional Distributions and Gibbs Conditioning
Key question: Distribution of 𝑛 points in ℝ𝑛 given the volume of their parallelogram.
Examples of Conditional Distributions
Example 1: Standard Normal Variables
For 𝑋,𝑌 independent standard normal, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (1/2,1/2) in probability
Example 2: Exponential Variables
For 𝑋,𝑌 independent exponential, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → Uniform on {𝛼(1,0)+(1−𝛼)(0,1):𝛼 ∈ [0,1]}
Example 3: Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (𝑃𝑥+𝑃𝑦)/2
where 𝑃𝑥,𝑃𝑦 are Pareto distributions on x and y axes respectively.
Example 4: Product of Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
(Distribution to be analyzed)
Key insights:
• Limiting conditional distributions relate to tail behavior
• Geometry of level sets vs. conditioning set is crucial
• Naive interpretation of pictures can be misleading
Key motivation: Study behavior of integrals of the form
∫𝑓
𝐴(𝑥)𝑑𝑥, 𝐴 ⊂ ℝ𝑑
for sets 𝐴 far away from the origin.
Applications
• Statistics and probability theory
• Other areas of mathematics where such integrals arise
Challenges
• Difficult to compute numerically for high dimensions (10-50)
• Quadrature methods ineffective
• Monte-Carlo methods fail for small integral values (10−2, 10−3)
Examples in Probability and Statistics
Density Functions
Two example density functions on ℝ:
1. Weibull-like: 𝑤𝛼(𝑥)= 𝐾𝑤,𝛼exp(−|𝑥|𝛼/𝛼) 2. Student: 𝑠𝛼(𝑥)= 𝐾𝑠,𝛼(1+𝑥2/𝛼)−(𝛼+1)/2
Asymptotic Behavior of Integrals
For domain 𝐴𝑡= {𝑥 ∈ ℝ𝑑:⟨𝐶𝑥,𝑥⟩ > 𝑡}:
For 𝛼 ≠ 2:
∫ ∏𝑤𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(𝛼,𝑑,𝐶)𝑒−𝑡𝛼/2/𝑐2𝑡−(𝛼−2)𝑑/(4−𝛼)
For 𝛼 = 2:
∫ ∏𝑤2
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(2,𝑑,𝐶)𝑒−𝑡/𝜆𝑡(𝑘−1)/2
where 𝑘 is the dimension of the eigensubspace for the largest eigenvalue 𝜆 of 𝐶𝑇+𝐶.
For Student distribution:
∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝐾𝑠,𝛼𝛼(𝛼+1)/22𝑡−𝛼/2 ∑ 𝐶𝑖,𝑖
𝛼/2
𝑖:𝐶𝑖,𝑖>0
Applications to Random Matrices
For 𝐴𝑡= {𝑥 ∈ 𝑀(𝑛,ℝ):det𝑥 > 𝑡}:
Weibull-like distribution:
∫ ∏ 𝑤𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ {𝑐𝑒−𝑡𝛼/𝑛𝑡(𝛼(𝑛2−1)−2𝑛2)/2𝑛 if 𝛼 ≠ 2
𝑐𝑒−𝑛𝑡2/2𝑡(𝑛2−𝑛−2)/2 if 𝛼 = 2
Student distribution:
∫ ∏ 𝑠𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ 𝑐(log𝑡)𝑛−1
𝑡𝛼
Conditional Distributions and Gibbs Conditioning
Key question: Distribution of 𝑛 points in ℝ𝑛 given the volume of their parallelogram.
Examples of Conditional Distributions
Example 1: Standard Normal Variables
For 𝑋,𝑌 independent standard normal, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (1/2,1/2) in probability
Example 2: Exponential Variables
For 𝑋,𝑌 independent exponential, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → Uniform on {𝛼(1,0)+(1−𝛼)(0,1):𝛼 ∈ [0,1]}
Example 3: Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (𝑃𝑥+𝑃𝑦)/2
where 𝑃𝑥,𝑃𝑦 are Pareto distributions on x and y axes respectively.
Example 4: Product of Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
(Distribution to be analyzed)
Key insights:
• Limiting conditional distributions relate to tail behavior
• Geometry of level sets vs. conditioning set is crucial
• Naive interpretation of pictures can be misleading
Key motivation: Study behavior of integrals of the form
∫𝑓
𝐴(𝑥)𝑑𝑥, 𝐴 ⊂ ℝ𝑑
for sets 𝐴 far away from the origin.
Applications
• Statistics and probability theory
• Other areas of mathematics where such integrals arise
Challenges
• Difficult to compute numerically for high dimensions (10-50)
• Quadrature methods ineffective
• Monte-Carlo methods fail for small integral values (10−2, 10−3)
Examples in Probability and Statistics
Density Functions
Two example density functions on ℝ:
1. Weibull-like: 𝑤𝛼(𝑥)= 𝐾𝑤,𝛼exp(−|𝑥|𝛼/𝛼) 2. Student: 𝑠𝛼(𝑥)= 𝐾𝑠,𝛼(1+𝑥2/𝛼)−(𝛼+1)/2
Asymptotic Behavior of Integrals
For domain 𝐴𝑡= {𝑥 ∈ ℝ𝑑:⟨𝐶𝑥,𝑥⟩ > 𝑡}:
For 𝛼 ≠ 2:
∫ ∏𝑤𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(𝛼,𝑑,𝐶)𝑒−𝑡𝛼/2/𝑐2𝑡−(𝛼−2)𝑑/(4−𝛼)
For 𝛼 = 2:
∫ ∏𝑤2
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(2,𝑑,𝐶)𝑒−𝑡/𝜆𝑡(𝑘−1)/2
where 𝑘 is the dimension of the eigensubspace for the largest eigenvalue 𝜆 of 𝐶𝑇+𝐶.
For Student distribution:
∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝐾𝑠,𝛼𝛼(𝛼+1)/22𝑡−𝛼/2 ∑ 𝐶𝑖,𝑖
𝛼/2
𝑖:𝐶𝑖,𝑖>0
Applications to Random Matrices
For 𝐴𝑡= {𝑥 ∈ 𝑀(𝑛,ℝ):det𝑥 > 𝑡}:
Weibull-like distribution:
∫ ∏ 𝑤𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ {𝑐𝑒−𝑡𝛼/𝑛𝑡(𝛼(𝑛2−1)−2𝑛2)/2𝑛 if 𝛼 ≠ 2
𝑐𝑒−𝑛𝑡2/2𝑡(𝑛2−𝑛−2)/2 if 𝛼 = 2
Student distribution:
∫ ∏ 𝑠𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ 𝑐(log𝑡)𝑛−1
𝑡𝛼
Conditional Distributions and Gibbs Conditioning
Key question: Distribution of 𝑛 points in ℝ𝑛 given the volume of their parallelogram.
Examples of Conditional Distributions
Example 1: Standard Normal Variables
For 𝑋,𝑌 independent standard normal, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (1/2,1/2) in probability
Example 2: Exponential Variables
For 𝑋,𝑌 independent exponential, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → Uniform on {𝛼(1,0)+(1−𝛼)(0,1):𝛼 ∈ [0,1]}
Example 3: Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (𝑃𝑥+𝑃𝑦)/2
where 𝑃𝑥,𝑃𝑦 are Pareto distributions on x and y axes respectively.
Example 4: Product of Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
(Distribution to be analyzed)
Key insights:
• Limiting conditional distributions relate to tail behavior
• Geometry of level sets vs. conditioning set is crucial
• Naive interpretation of pictures can be misleading
Key motivation: Study behavior of integrals of the form
∫𝑓
𝐴(𝑥)𝑑𝑥, 𝐴 ⊂ ℝ𝑑
for sets 𝐴 far away from the origin.
Applications
• Statistics and probability theory
• Other areas of mathematics where such integrals arise
Challenges
• Difficult to compute numerically for high dimensions (10-50)
• Quadrature methods ineffective
• Monte-Carlo methods fail for small integral values (10−2, 10−3)
Examples in Probability and Statistics
Density Functions
Two example density functions on ℝ:
1. Weibull-like: 𝑤𝛼(𝑥)= 𝐾𝑤,𝛼exp(−|𝑥|𝛼/𝛼) 2. Student: 𝑠𝛼(𝑥)= 𝐾𝑠,𝛼(1+𝑥2/𝛼)−(𝛼+1)/2
Asymptotic Behavior of Integrals
For domain 𝐴𝑡= {𝑥 ∈ ℝ𝑑:⟨𝐶𝑥,𝑥⟩ > 𝑡}:
For 𝛼 ≠ 2:
∫ ∏𝑤𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(𝛼,𝑑,𝐶)𝑒−𝑡𝛼/2/𝑐2𝑡−(𝛼−2)𝑑/(4−𝛼)
For 𝛼 = 2:
∫ ∏𝑤2
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(2,𝑑,𝐶)𝑒−𝑡/𝜆𝑡(𝑘−1)/2
where 𝑘 is the dimension of the eigensubspace for the largest eigenvalue 𝜆 of 𝐶𝑇+𝐶.
For Student distribution:
∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝐾𝑠,𝛼𝛼(𝛼+1)/22𝑡−𝛼/2 ∑ 𝐶𝑖,𝑖
𝛼/2
𝑖:𝐶𝑖,𝑖>0
Applications to Random Matrices
For 𝐴𝑡= {𝑥 ∈ 𝑀(𝑛,ℝ):det𝑥 > 𝑡}:
Weibull-like distribution:
∫ ∏ 𝑤𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ {𝑐𝑒−𝑡𝛼/𝑛𝑡(𝛼(𝑛2−1)−2𝑛2)/2𝑛 if 𝛼 ≠ 2
𝑐𝑒−𝑛𝑡2/2𝑡(𝑛2−𝑛−2)/2 if 𝛼 = 2
Student distribution:
∫ ∏ 𝑠𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ 𝑐(log𝑡)𝑛−1
𝑡𝛼
Conditional Distributions and Gibbs Conditioning
Key question: Distribution of 𝑛 points in ℝ𝑛 given the volume of their parallelogram.
Examples of Conditional Distributions
Example 1: Standard Normal Variables
For 𝑋,𝑌 independent standard normal, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (1/2,1/2) in probability
Example 2: Exponential Variables
For 𝑋,𝑌 independent exponential, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → Uniform on {𝛼(1,0)+(1−𝛼)(0,1):𝛼 ∈ [0,1]}
Example 3: Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (𝑃𝑥+𝑃𝑦)/2
where 𝑃𝑥,𝑃𝑦 are Pareto distributions on x and y axes respectively.
Example 4: Product of Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
(Distribution to be analyzed)
Key insights:
• Limiting conditional distributions relate to tail behavior
• Geometry of level sets vs. conditioning set is crucial
• Naive interpretation of pictures can be misleading
Key motivation: Study behavior of integrals of the form
∫𝑓
𝐴(𝑥)𝑑𝑥, 𝐴 ⊂ ℝ𝑑
for sets 𝐴 far away from the origin.
Applications
• Statistics and probability theory
• Other areas of mathematics where such integrals arise
Challenges
• Difficult to compute numerically for high dimensions (10-50)
• Quadrature methods ineffective
• Monte-Carlo methods fail for small integral values (10−2, 10−3)
Examples in Probability and Statistics
Density Functions
Two example density functions on ℝ:
1. Weibull-like: 𝑤𝛼(𝑥)= 𝐾𝑤,𝛼exp(−|𝑥|𝛼/𝛼) 2. Student: 𝑠𝛼(𝑥)= 𝐾𝑠,𝛼(1+𝑥2/𝛼)−(𝛼+1)/2
Asymptotic Behavior of Integrals
For domain 𝐴𝑡= {𝑥 ∈ ℝ𝑑:⟨𝐶𝑥,𝑥⟩ > 𝑡}:
For 𝛼 ≠ 2:
∫ ∏𝑤𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(𝛼,𝑑,𝐶)𝑒−𝑡𝛼/2/𝑐2𝑡−(𝛼−2)𝑑/(4−𝛼)
For 𝛼 = 2:
∫ ∏𝑤2
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(2,𝑑,𝐶)𝑒−𝑡/𝜆𝑡(𝑘−1)/2
where 𝑘 is the dimension of the eigensubspace for the largest eigenvalue 𝜆 of 𝐶𝑇+𝐶.
For Student distribution:
∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝐾𝑠,𝛼𝛼(𝛼+1)/22𝑡−𝛼/2 ∑ 𝐶𝑖,𝑖
𝛼/2
𝑖:𝐶𝑖,𝑖>0
Applications to Random Matrices
For 𝐴𝑡= {𝑥 ∈ 𝑀(𝑛,ℝ):det𝑥 > 𝑡}:
Weibull-like distribution:
∫ ∏ 𝑤𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ {𝑐𝑒−𝑡𝛼/𝑛𝑡(𝛼(𝑛2−1)−2𝑛2)/2𝑛 if 𝛼 ≠ 2
𝑐𝑒−𝑛𝑡2/2𝑡(𝑛2−𝑛−2)/2 if 𝛼 = 2
Student distribution:
∫ ∏ 𝑠𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ 𝑐(log𝑡)𝑛−1
𝑡𝛼
Conditional Distributions and Gibbs Conditioning
Key question: Distribution of 𝑛 points in ℝ𝑛 given the volume of their parallelogram.
Examples of Conditional Distributions
Example 1: Standard Normal Variables
For 𝑋,𝑌 independent standard normal, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (1/2,1/2) in probability
Example 2: Exponential Variables
For 𝑋,𝑌 independent exponential, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → Uniform on {𝛼(1,0)+(1−𝛼)(0,1):𝛼 ∈ [0,1]}
Example 3: Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (𝑃𝑥+𝑃𝑦)/2
where 𝑃𝑥,𝑃𝑦 are Pareto distributions on x and y axes respectively.
Example 4: Product of Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
(Distribution to be analyzed)
Key insights:
• Limiting conditional distributions relate to tail behavior
• Geometry of level sets vs. conditioning set is crucial
• Naive interpretation of pictures can be misleading
Key motivation: Study behavior of integrals of the form
∫𝑓
𝐴(𝑥)𝑑𝑥, 𝐴 ⊂ ℝ𝑑
for sets 𝐴 far away from the origin.
Applications
• Statistics and probability theory
• Other areas of mathematics where such integrals arise
Challenges
• Difficult to compute numerically for high dimensions (10-50)
• Quadrature methods ineffective
• Monte-Carlo methods fail for small integral values (10−2, 10−3)
Examples in Probability and Statistics
Density Functions
Two example density functions on ℝ:
1. Weibull-like: 𝑤𝛼(𝑥)= 𝐾𝑤,𝛼exp(−|𝑥|𝛼/𝛼) 2. Student: 𝑠𝛼(𝑥)= 𝐾𝑠,𝛼(1+𝑥2/𝛼)−(𝛼+1)/2
Asymptotic Behavior of Integrals
For domain 𝐴𝑡= {𝑥 ∈ ℝ𝑑:⟨𝐶𝑥,𝑥⟩ > 𝑡}:
For 𝛼 ≠ 2:
∫ ∏𝑤𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(𝛼,𝑑,𝐶)𝑒−𝑡𝛼/2/𝑐2𝑡−(𝛼−2)𝑑/(4−𝛼)
For 𝛼 = 2:
∫ ∏𝑤2
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(2,𝑑,𝐶)𝑒−𝑡/𝜆𝑡(𝑘−1)/2
where 𝑘 is the dimension of the eigensubspace for the largest eigenvalue 𝜆 of 𝐶𝑇+𝐶.
For Student distribution:
∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝐾𝑠,𝛼𝛼(𝛼+1)/22𝑡−𝛼/2 ∑ 𝐶𝑖,𝑖
𝛼/2
𝑖:𝐶𝑖,𝑖>0
Applications to Random Matrices
For 𝐴𝑡= {𝑥 ∈ 𝑀(𝑛,ℝ):det𝑥 > 𝑡}:
Weibull-like distribution:
∫ ∏ 𝑤𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ {𝑐𝑒−𝑡𝛼/𝑛𝑡(𝛼(𝑛2−1)−2𝑛2)/2𝑛 if 𝛼 ≠ 2
𝑐𝑒−𝑛𝑡2/2𝑡(𝑛2−𝑛−2)/2 if 𝛼 = 2
Student distribution:
∫ ∏ 𝑠𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ 𝑐(log𝑡)𝑛−1
𝑡𝛼
Conditional Distributions and Gibbs Conditioning
Key question: Distribution of 𝑛 points in ℝ𝑛 given the volume of their parallelogram.
Examples of Conditional Distributions
Example 1: Standard Normal Variables
For 𝑋,𝑌 independent standard normal, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (1/2,1/2) in probability
Example 2: Exponential Variables
For 𝑋,𝑌 independent exponential, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → Uniform on {𝛼(1,0)+(1−𝛼)(0,1):𝛼 ∈ [0,1]}
Example 3: Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (𝑃𝑥+𝑃𝑦)/2
where 𝑃𝑥,𝑃𝑦 are Pareto distributions on x and y axes respectively.
Example 4: Product of Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
(Distribution to be analyzed)
Key insights:
• Limiting conditional distributions relate to tail behavior
• Geometry of level sets vs. conditioning set is crucial
• Naive interpretation of pictures can be misleading
Key motivation: Study behavior of integrals of the form
∫𝑓
𝐴(𝑥)𝑑𝑥, 𝐴 ⊂ ℝ𝑑
for sets 𝐴 far away from the origin.
Applications
• Statistics and probability theory
• Other areas of mathematics where such integrals arise
Challenges
• Difficult to compute numerically for high dimensions (10-50)
• Quadrature methods ineffective
• Monte-Carlo methods fail for small integral values (10−2, 10−3)
Examples in Probability and Statistics
Density Functions
Two example density functions on ℝ:
1. Weibull-like: 𝑤𝛼(𝑥)= 𝐾𝑤,𝛼exp(−|𝑥|𝛼/𝛼) 2. Student: 𝑠𝛼(𝑥)= 𝐾𝑠,𝛼(1+𝑥2/𝛼)−(𝛼+1)/2
Asymptotic Behavior of Integrals
For domain 𝐴𝑡= {𝑥 ∈ ℝ𝑑:⟨𝐶𝑥,𝑥⟩ > 𝑡}:
For 𝛼 ≠ 2:
∫ ∏𝑤𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(𝛼,𝑑,𝐶)𝑒−𝑡𝛼/2/𝑐2𝑡−(𝛼−2)𝑑/(4−𝛼)
For 𝛼 = 2:
∫ ∏𝑤2
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(2,𝑑,𝐶)𝑒−𝑡/𝜆𝑡(𝑘−1)/2
where 𝑘 is the dimension of the eigensubspace for the largest eigenvalue 𝜆 of 𝐶𝑇+𝐶.
For Student distribution:
∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝐾𝑠,𝛼𝛼(𝛼+1)/22𝑡−𝛼/2 ∑ 𝐶𝑖,𝑖
𝛼/2
𝑖:𝐶𝑖,𝑖>0
Applications to Random Matrices
For 𝐴𝑡= {𝑥 ∈ 𝑀(𝑛,ℝ):det𝑥 > 𝑡}:
Weibull-like distribution:
∫ ∏ 𝑤𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ {𝑐𝑒−𝑡𝛼/𝑛𝑡(𝛼(𝑛2−1)−2𝑛2)/2𝑛 if 𝛼 ≠ 2
𝑐𝑒−𝑛𝑡2/2𝑡(𝑛2−𝑛−2)/2 if 𝛼 = 2
Student distribution:
∫ ∏ 𝑠𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ 𝑐(log𝑡)𝑛−1
𝑡𝛼
Conditional Distributions and Gibbs Conditioning
Key question: Distribution of 𝑛 points in ℝ𝑛 given the volume of their parallelogram.
Examples of Conditional Distributions
Example 1: Standard Normal Variables
For 𝑋,𝑌 independent standard normal, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (1/2,1/2) in probability
Example 2: Exponential Variables
For 𝑋,𝑌 independent exponential, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → Uniform on {𝛼(1,0)+(1−𝛼)(0,1):𝛼 ∈ [0,1]}
Example 3: Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (𝑃𝑥+𝑃𝑦)/2
where 𝑃𝑥,𝑃𝑦 are Pareto distributions on x and y axes respectively.
Example 4: Product of Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
(Distribution to be analyzed)
Key insights:
• Limiting conditional distributions relate to tail behavior
• Geometry of level sets vs. conditioning set is crucial
• Naive interpretation of pictures can be misleading
Key motivation: Study behavior of integrals of the form
∫𝑓
𝐴(𝑥)𝑑𝑥, 𝐴 ⊂ ℝ𝑑
for sets 𝐴 far away from the origin.
Applications
• Statistics and probability theory
• Other areas of mathematics where such integrals arise
Challenges
• Difficult to compute numerically for high dimensions (10-50)
• Quadrature methods ineffective
• Monte-Carlo methods fail for small integral values (10−2, 10−3)
Examples in Probability and Statistics
Density Functions
Two example density functions on ℝ:
1. Weibull-like: 𝑤𝛼(𝑥)= 𝐾𝑤,𝛼exp(−|𝑥|𝛼/𝛼) 2. Student: 𝑠𝛼(𝑥)= 𝐾𝑠,𝛼(1+𝑥2/𝛼)−(𝛼+1)/2
Asymptotic Behavior of Integrals
For domain 𝐴𝑡= {𝑥 ∈ ℝ𝑑:⟨𝐶𝑥,𝑥⟩ > 𝑡}:
For 𝛼 ≠ 2:
∫ ∏𝑤𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(𝛼,𝑑,𝐶)𝑒−𝑡𝛼/2/𝑐2𝑡−(𝛼−2)𝑑/(4−𝛼)
For 𝛼 = 2:
∫ ∏𝑤2
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(2,𝑑,𝐶)𝑒−𝑡/𝜆𝑡(𝑘−1)/2
where 𝑘 is the dimension of the eigensubspace for the largest eigenvalue 𝜆 of 𝐶𝑇+𝐶.
For Student distribution:
∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝐾𝑠,𝛼𝛼(𝛼+1)/22𝑡−𝛼/2 ∑ 𝐶𝑖,𝑖
𝛼/2
𝑖:𝐶𝑖,𝑖>0
Applications to Random Matrices
For 𝐴𝑡= {𝑥 ∈ 𝑀(𝑛,ℝ):det𝑥 > 𝑡}:
Weibull-like distribution:
∫ ∏ 𝑤𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ {𝑐𝑒−𝑡𝛼/𝑛𝑡(𝛼(𝑛2−1)−2𝑛2)/2𝑛 if 𝛼 ≠ 2
𝑐𝑒−𝑛𝑡2/2𝑡(𝑛2−𝑛−2)/2 if 𝛼 = 2
Student distribution:
∫ ∏ 𝑠𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ 𝑐(log𝑡)𝑛−1
𝑡𝛼
Conditional Distributions and Gibbs Conditioning
Key question: Distribution of 𝑛 points in ℝ𝑛 given the volume of their parallelogram.
Examples of Conditional Distributions
Example 1: Standard Normal Variables
For 𝑋,𝑌 independent standard normal, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (1/2,1/2) in probability
Example 2: Exponential Variables
For 𝑋,𝑌 independent exponential, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → Uniform on {𝛼(1,0)+(1−𝛼)(0,1):𝛼 ∈ [0,1]}
Example 3: Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (𝑃𝑥+𝑃𝑦)/2
where 𝑃𝑥,𝑃𝑦 are Pareto distributions on x and y axes respectively.
Example 4: Product of Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
(Distribution to be analyzed)
Key insights:
• Limiting conditional distributions relate to tail behavior
• Geometry of level sets vs. conditioning set is crucial
• Naive interpretation of pictures can be misleading
Key motivation: Study behavior of integrals of the form
∫𝑓
𝐴(𝑥)𝑑𝑥, 𝐴 ⊂ ℝ𝑑
for sets 𝐴 far away from the origin.
Applications
• Statistics and probability theory
• Other areas of mathematics where such integrals arise
Challenges
• Difficult to compute numerically for high dimensions (10-50)
• Quadrature methods ineffective
• Monte-Carlo methods fail for small integral values (10−2, 10−3)
Examples in Probability and Statistics
Density Functions
Two example density functions on ℝ:
1. Weibull-like: 𝑤𝛼(𝑥)= 𝐾𝑤,𝛼exp(−|𝑥|𝛼/𝛼) 2. Student: 𝑠𝛼(𝑥)= 𝐾𝑠,𝛼(1+𝑥2/𝛼)−(𝛼+1)/2
Asymptotic Behavior of Integrals
For domain 𝐴𝑡= {𝑥 ∈ ℝ𝑑:⟨𝐶𝑥,𝑥⟩ > 𝑡}:
For 𝛼 ≠ 2:
∫ ∏𝑤𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(𝛼,𝑑,𝐶)𝑒−𝑡𝛼/2/𝑐2𝑡−(𝛼−2)𝑑/(4−𝛼)
For 𝛼 = 2:
∫ ∏𝑤2
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(2,𝑑,𝐶)𝑒−𝑡/𝜆𝑡(𝑘−1)/2
where 𝑘 is the dimension of the eigensubspace for the largest eigenvalue 𝜆 of 𝐶𝑇+𝐶.
For Student distribution:
∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝐾𝑠,𝛼𝛼(𝛼+1)/22𝑡−𝛼/2 ∑ 𝐶𝑖,𝑖
𝛼/2
𝑖:𝐶𝑖,𝑖>0
Applications to Random Matrices
For 𝐴𝑡= {𝑥 ∈ 𝑀(𝑛,ℝ):det𝑥 > 𝑡}:
Weibull-like distribution:
∫ ∏ 𝑤𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ {𝑐𝑒−𝑡𝛼/𝑛𝑡(𝛼(𝑛2−1)−2𝑛2)/2𝑛 if 𝛼 ≠ 2
𝑐𝑒−𝑛𝑡2/2𝑡(𝑛2−𝑛−2)/2 if 𝛼 = 2
Student distribution:
∫ ∏ 𝑠𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ 𝑐(log𝑡)𝑛−1
𝑡𝛼
Conditional Distributions and Gibbs Conditioning
Key question: Distribution of 𝑛 points in ℝ𝑛 given the volume of their parallelogram.
Examples of Conditional Distributions
Example 1: Standard Normal Variables
For 𝑋,𝑌 independent standard normal, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (1/2,1/2) in probability
Example 2: Exponential Variables
For 𝑋,𝑌 independent exponential, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → Uniform on {𝛼(1,0)+(1−𝛼)(0,1):𝛼 ∈ [0,1]}
Example 3: Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (𝑃𝑥+𝑃𝑦)/2
where 𝑃𝑥,𝑃𝑦 are Pareto distributions on x and y axes respectively.
Example 4: Product of Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
(Distribution to be analyzed)
Key insights:
• Limiting conditional distributions relate to tail behavior
• Geometry of level sets vs. conditioning set is crucial
• Naive interpretation of pictures can be misleading
Key motivation: Study behavior of integrals of the form
∫𝑓
𝐴(𝑥)𝑑𝑥, 𝐴 ⊂ ℝ𝑑
for sets 𝐴 far away from the origin.
Applications
• Statistics and probability theory
• Other areas of mathematics where such integrals arise
Challenges
• Difficult to compute numerically for high dimensions (10-50)
• Quadrature methods ineffective
• Monte-Carlo methods fail for small integral values (10−2, 10−3)
Examples in Probability and Statistics
Density Functions
Two example density functions on ℝ:
1. Weibull-like: 𝑤𝛼(𝑥)= 𝐾𝑤,𝛼exp(−|𝑥|𝛼/𝛼) 2. Student: 𝑠𝛼(𝑥)= 𝐾𝑠,𝛼(1+𝑥2/𝛼)−(𝛼+1)/2
Asymptotic Behavior of Integrals
For domain 𝐴𝑡= {𝑥 ∈ ℝ𝑑:⟨𝐶𝑥,𝑥⟩ > 𝑡}:
For 𝛼 ≠ 2:
∫ ∏𝑤𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(𝛼,𝑑,𝐶)𝑒−𝑡𝛼/2/𝑐2𝑡−(𝛼−2)𝑑/(4−𝛼)
For 𝛼 = 2:
∫ ∏𝑤2
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(2,𝑑,𝐶)𝑒−𝑡/𝜆𝑡(𝑘−1)/2
where 𝑘 is the dimension of the eigensubspace for the largest eigenvalue 𝜆 of 𝐶𝑇+𝐶.
For Student distribution:
∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝐾𝑠,𝛼𝛼(𝛼+1)/22𝑡−𝛼/2 ∑ 𝐶𝑖,𝑖
𝛼/2
𝑖:𝐶𝑖,𝑖>0
Applications to Random Matrices
For 𝐴𝑡= {𝑥 ∈ 𝑀(𝑛,ℝ):det𝑥 > 𝑡}:
Weibull-like distribution:
∫ ∏ 𝑤𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ {𝑐𝑒−𝑡𝛼/𝑛𝑡(𝛼(𝑛2−1)−2𝑛2)/2𝑛 if 𝛼 ≠ 2
𝑐𝑒−𝑛𝑡2/2𝑡(𝑛2−𝑛−2)/2 if 𝛼 = 2
Student distribution:
∫ ∏ 𝑠𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ 𝑐(log𝑡)𝑛−1
𝑡𝛼
Conditional Distributions and Gibbs Conditioning
Key question: Distribution of 𝑛 points in ℝ𝑛 given the volume of their parallelogram.
Examples of Conditional Distributions
Example 1: Standard Normal Variables
For 𝑋,𝑌 independent standard normal, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (1/2,1/2) in probability
Example 2: Exponential Variables
For 𝑋,𝑌 independent exponential, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → Uniform on {𝛼(1,0)+(1−𝛼)(0,1):𝛼 ∈ [0,1]}
Example 3: Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (𝑃𝑥+𝑃𝑦)/2
where 𝑃𝑥,𝑃𝑦 are Pareto distributions on x and y axes respectively.
Example 4: Product of Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
(Distribution to be analyzed)
Key insights:
• Limiting conditional distributions relate to tail behavior
• Geometry of level sets vs. conditioning set is crucial
• Naive interpretation of pictures can be misleading
Key motivation: Study behavior of integrals of the form
∫𝑓
𝐴(𝑥)𝑑𝑥, 𝐴 ⊂ ℝ𝑑
for sets 𝐴 far away from the origin.
Applications
• Statistics and probability theory
• Other areas of mathematics where such integrals arise
Challenges
• Difficult to compute numerically for high dimensions (10-50)
• Quadrature methods ineffective
• Monte-Carlo methods fail for small integral values (10−2, 10−3)
Examples in Probability and Statistics
Density Functions
Two example density functions on ℝ:
1. Weibull-like: 𝑤𝛼(𝑥)= 𝐾𝑤,𝛼exp(−|𝑥|𝛼/𝛼) 2. Student: 𝑠𝛼(𝑥)= 𝐾𝑠,𝛼(1+𝑥2/𝛼)−(𝛼+1)/2
Asymptotic Behavior of Integrals
For domain 𝐴𝑡= {𝑥 ∈ ℝ𝑑:⟨𝐶𝑥,𝑥⟩ > 𝑡}:
For 𝛼 ≠ 2:
∫ ∏𝑤𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(𝛼,𝑑,𝐶)𝑒−𝑡𝛼/2/𝑐2𝑡−(𝛼−2)𝑑/(4−𝛼)
For 𝛼 = 2:
∫ ∏𝑤2
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(2,𝑑,𝐶)𝑒−𝑡/𝜆𝑡(𝑘−1)/2
where 𝑘 is the dimension of the eigensubspace for the largest eigenvalue 𝜆 of 𝐶𝑇+𝐶.
For Student distribution:
∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝐾𝑠,𝛼𝛼(𝛼+1)/22𝑡−𝛼/2 ∑ 𝐶𝑖,𝑖
𝛼/2
𝑖:𝐶𝑖,𝑖>0
Applications to Random Matrices
For 𝐴𝑡= {𝑥 ∈ 𝑀(𝑛,ℝ):det𝑥 > 𝑡}:
Weibull-like distribution:
∫ ∏ 𝑤𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ {𝑐𝑒−𝑡𝛼/𝑛𝑡(𝛼(𝑛2−1)−2𝑛2)/2𝑛 if 𝛼 ≠ 2
𝑐𝑒−𝑛𝑡2/2𝑡(𝑛2−𝑛−2)/2 if 𝛼 = 2
Student distribution:
∫ ∏ 𝑠𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ 𝑐(log𝑡)𝑛−1
𝑡𝛼
Conditional Distributions and Gibbs Conditioning
Key question: Distribution of 𝑛 points in ℝ𝑛 given the volume of their parallelogram.
Examples of Conditional Distributions
Example 1: Standard Normal Variables
For 𝑋,𝑌 independent standard normal, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (1/2,1/2) in probability
Example 2: Exponential Variables
For 𝑋,𝑌 independent exponential, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → Uniform on {𝛼(1,0)+(1−𝛼)(0,1):𝛼 ∈ [0,1]}
Example 3: Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (𝑃𝑥+𝑃𝑦)/2
where 𝑃𝑥,𝑃𝑦 are Pareto distributions on x and y axes respectively.
Example 4: Product of Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
(Distribution to be analyzed)
Key insights:
• Limiting conditional distributions relate to tail behavior
• Geometry of level sets vs. conditioning set is crucial
• Naive interpretation of pictures can be misleading
Key motivation: Study behavior of integrals of the form
∫𝑓
𝐴(𝑥)𝑑𝑥, 𝐴 ⊂ ℝ𝑑
for sets 𝐴 far away from the origin.
Applications
• Statistics and probability theory
• Other areas of mathematics where such integrals arise
Challenges
• Difficult to compute numerically for high dimensions (10-50)
• Quadrature methods ineffective
• Monte-Carlo methods fail for small integral values (10−2, 10−3)
Examples in Probability and Statistics
Density Functions
Two example density functions on ℝ:
1. Weibull-like: 𝑤𝛼(𝑥)= 𝐾𝑤,𝛼exp(−|𝑥|𝛼/𝛼) 2. Student: 𝑠𝛼(𝑥)= 𝐾𝑠,𝛼(1+𝑥2/𝛼)−(𝛼+1)/2
Asymptotic Behavior of Integrals
For domain 𝐴𝑡= {𝑥 ∈ ℝ𝑑:⟨𝐶𝑥,𝑥⟩ > 𝑡}:
For 𝛼 ≠ 2:
∫ ∏𝑤𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(𝛼,𝑑,𝐶)𝑒−𝑡𝛼/2/𝑐2𝑡−(𝛼−2)𝑑/(4−𝛼)
For 𝛼 = 2:
∫ ∏𝑤2
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(2,𝑑,𝐶)𝑒−𝑡/𝜆𝑡(𝑘−1)/2
where 𝑘 is the dimension of the eigensubspace for the largest eigenvalue 𝜆 of 𝐶𝑇+𝐶.
For Student distribution:
∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝐾𝑠,𝛼𝛼(𝛼+1)/22𝑡−𝛼/2 ∑ 𝐶𝑖,𝑖
𝛼/2
𝑖:𝐶𝑖,𝑖>0
Applications to Random Matrices
For 𝐴𝑡= {𝑥 ∈ 𝑀(𝑛,ℝ):det𝑥 > 𝑡}:
Weibull-like distribution:
∫ ∏ 𝑤𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ {𝑐𝑒−𝑡𝛼/𝑛𝑡(𝛼(𝑛2−1)−2𝑛2)/2𝑛 if 𝛼 ≠ 2
𝑐𝑒−𝑛𝑡2/2𝑡(𝑛2−𝑛−2)/2 if 𝛼 = 2
Student distribution:
∫ ∏ 𝑠𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ 𝑐(log𝑡)𝑛−1
𝑡𝛼
Conditional Distributions and Gibbs Conditioning
Key question: Distribution of 𝑛 points in ℝ𝑛 given the volume of their parallelogram.
Examples of Conditional Distributions
Example 1: Standard Normal Variables
For 𝑋,𝑌 independent standard normal, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (1/2,1/2) in probability
Example 2: Exponential Variables
For 𝑋,𝑌 independent exponential, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → Uniform on {𝛼(1,0)+(1−𝛼)(0,1):𝛼 ∈ [0,1]}
Example 3: Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (𝑃𝑥+𝑃𝑦)/2
where 𝑃𝑥,𝑃𝑦 are Pareto distributions on x and y axes respectively.
Example 4: Product of Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
(Distribution to be analyzed)
Key insights:
• Limiting conditional distributions relate to tail behavior
• Geometry of level sets vs. conditioning set is crucial
• Naive interpretation of pictures can be misleading
Key motivation: Study behavior of integrals of the form
∫𝑓
𝐴(𝑥)𝑑𝑥, 𝐴 ⊂ ℝ𝑑
for sets 𝐴 far away from the origin.
Applications
• Statistics and probability theory
• Other areas of mathematics where such integrals arise
Challenges
• Difficult to compute numerically for high dimensions (10-50)
• Quadrature methods ineffective
• Monte-Carlo methods fail for small integral values (10−2, 10−3)
Examples in Probability and Statistics
Density Functions
Two example density functions on ℝ:
1. Weibull-like: 𝑤𝛼(𝑥)= 𝐾𝑤,𝛼exp(−|𝑥|𝛼/𝛼) 2. Student: 𝑠𝛼(𝑥)= 𝐾𝑠,𝛼(1+𝑥2/𝛼)−(𝛼+1)/2
Asymptotic Behavior of Integrals
For domain 𝐴𝑡= {𝑥 ∈ ℝ𝑑:⟨𝐶𝑥,𝑥⟩ > 𝑡}:
For 𝛼 ≠ 2:
∫ ∏𝑤𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(𝛼,𝑑,𝐶)𝑒−𝑡𝛼/2/𝑐2𝑡−(𝛼−2)𝑑/(4−𝛼)
For 𝛼 = 2:
∫ ∏𝑤2
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(2,𝑑,𝐶)𝑒−𝑡/𝜆𝑡(𝑘−1)/2
where 𝑘 is the dimension of the eigensubspace for the largest eigenvalue 𝜆 of 𝐶𝑇+𝐶.
For Student distribution:
∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝐾𝑠,𝛼𝛼(𝛼+1)/22𝑡−𝛼/2 ∑ 𝐶𝑖,𝑖
𝛼/2
𝑖:𝐶𝑖,𝑖>0
Applications to Random Matrices
For 𝐴𝑡= {𝑥 ∈ 𝑀(𝑛,ℝ):det𝑥 > 𝑡}:
Weibull-like distribution:
∫ ∏ 𝑤𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ {𝑐𝑒−𝑡𝛼/𝑛𝑡(𝛼(𝑛2−1)−2𝑛2)/2𝑛 if 𝛼 ≠ 2
𝑐𝑒−𝑛𝑡2/2𝑡(𝑛2−𝑛−2)/2 if 𝛼 = 2
Student distribution:
∫ ∏ 𝑠𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ 𝑐(log𝑡)𝑛−1
𝑡𝛼
Conditional Distributions and Gibbs Conditioning
Key question: Distribution of 𝑛 points in ℝ𝑛 given the volume of their parallelogram.
Examples of Conditional Distributions
Example 1: Standard Normal Variables
For 𝑋,𝑌 independent standard normal, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (1/2,1/2) in probability
Example 2: Exponential Variables
For 𝑋,𝑌 independent exponential, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → Uniform on {𝛼(1,0)+(1−𝛼)(0,1):𝛼 ∈ [0,1]}
Example 3: Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (𝑃𝑥+𝑃𝑦)/2
where 𝑃𝑥,𝑃𝑦 are Pareto distributions on x and y axes respectively.
Example 4: Product of Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
(Distribution to be analyzed)
Key insights:
• Limiting conditional distributions relate to tail behavior
• Geometry of level sets vs. conditioning set is crucial
• Naive interpretation of pictures can be misleading
Key motivation: Study behavior of integrals of the form
∫𝑓
𝐴(𝑥)𝑑𝑥, 𝐴 ⊂ ℝ𝑑
for sets 𝐴 far away from the origin.
Applications
• Statistics and probability theory
• Other areas of mathematics where such integrals arise
Challenges
• Difficult to compute numerically for high dimensions (10-50)
• Quadrature methods ineffective
• Monte-Carlo methods fail for small integral values (10−2, 10−3)
Examples in Probability and Statistics
Density Functions
Two example density functions on ℝ:
1. Weibull-like: 𝑤𝛼(𝑥)= 𝐾𝑤,𝛼exp(−|𝑥|𝛼/𝛼) 2. Student: 𝑠𝛼(𝑥)= 𝐾𝑠,𝛼(1+𝑥2/𝛼)−(𝛼+1)/2
Asymptotic Behavior of Integrals
For domain 𝐴𝑡= {𝑥 ∈ ℝ𝑑:⟨𝐶𝑥,𝑥⟩ > 𝑡}:
For 𝛼 ≠ 2:
∫ ∏𝑤𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(𝛼,𝑑,𝐶)𝑒−𝑡𝛼/2/𝑐2𝑡−(𝛼−2)𝑑/(4−𝛼)
For 𝛼 = 2:
∫ ∏𝑤2
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(2,𝑑,𝐶)𝑒−𝑡/𝜆𝑡(𝑘−1)/2
where 𝑘 is the dimension of the eigensubspace for the largest eigenvalue 𝜆 of 𝐶𝑇+𝐶.
For Student distribution:
∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝐾𝑠,𝛼𝛼(𝛼+1)/22𝑡−𝛼/2 ∑ 𝐶𝑖,𝑖
𝛼/2
𝑖:𝐶𝑖,𝑖>0
Applications to Random Matrices
For 𝐴𝑡= {𝑥 ∈ 𝑀(𝑛,ℝ):det𝑥 > 𝑡}:
Weibull-like distribution:
∫ ∏ 𝑤𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ {𝑐𝑒−𝑡𝛼/𝑛𝑡(𝛼(𝑛2−1)−2𝑛2)/2𝑛 if 𝛼 ≠ 2
𝑐𝑒−𝑛𝑡2/2𝑡(𝑛2−𝑛−2)/2 if 𝛼 = 2
Student distribution:
∫ ∏ 𝑠𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ 𝑐(log𝑡)𝑛−1
𝑡𝛼
Conditional Distributions and Gibbs Conditioning
Key question: Distribution of 𝑛 points in ℝ𝑛 given the volume of their parallelogram.
Examples of Conditional Distributions
Example 1: Standard Normal Variables
For 𝑋,𝑌 independent standard normal, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (1/2,1/2) in probability
Example 2: Exponential Variables
For 𝑋,𝑌 independent exponential, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → Uniform on {𝛼(1,0)+(1−𝛼)(0,1):𝛼 ∈ [0,1]}
Example 3: Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (𝑃𝑥+𝑃𝑦)/2
where 𝑃𝑥,𝑃𝑦 are Pareto distributions on x and y axes respectively.
Example 4: Product of Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
(Distribution to be analyzed)
Key insights:
• Limiting conditional distributions relate to tail behavior
• Geometry of level sets vs. conditioning set is crucial
• Naive interpretation of pictures can be misleading
Key motivation: Study behavior of integrals of the form
∫𝑓
𝐴(𝑥)𝑑𝑥, 𝐴 ⊂ ℝ𝑑
for sets 𝐴 far away from the origin.
Applications
• Statistics and probability theory
• Other areas of mathematics where such integrals arise
Challenges
• Difficult to compute numerically for high dimensions (10-50)
• Quadrature methods ineffective
• Monte-Carlo methods fail for small integral values (10−2, 10−3)
Examples in Probability and Statistics
Density Functions
Two example density functions on ℝ:
1. Weibull-like: 𝑤𝛼(𝑥)= 𝐾𝑤,𝛼exp(−|𝑥|𝛼/𝛼) 2. Student: 𝑠𝛼(𝑥)= 𝐾𝑠,𝛼(1+𝑥2/𝛼)−(𝛼+1)/2
Asymptotic Behavior of Integrals
For domain 𝐴𝑡= {𝑥 ∈ ℝ𝑑:⟨𝐶𝑥,𝑥⟩ > 𝑡}:
For 𝛼 ≠ 2:
∫ ∏𝑤𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(𝛼,𝑑,𝐶)𝑒−𝑡𝛼/2/𝑐2𝑡−(𝛼−2)𝑑/(4−𝛼)
For 𝛼 = 2:
∫ ∏𝑤2
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(2,𝑑,𝐶)𝑒−𝑡/𝜆𝑡(𝑘−1)/2
where 𝑘 is the dimension of the eigensubspace for the largest eigenvalue 𝜆 of 𝐶𝑇+𝐶.
For Student distribution:
∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝐾𝑠,𝛼𝛼(𝛼+1)/22𝑡−𝛼/2 ∑ 𝐶𝑖,𝑖
𝛼/2
𝑖:𝐶𝑖,𝑖>0
Applications to Random Matrices
For 𝐴𝑡= {𝑥 ∈ 𝑀(𝑛,ℝ):det𝑥 > 𝑡}:
Weibull-like distribution:
∫ ∏ 𝑤𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ {𝑐𝑒−𝑡𝛼/𝑛𝑡(𝛼(𝑛2−1)−2𝑛2)/2𝑛 if 𝛼 ≠ 2
𝑐𝑒−𝑛𝑡2/2𝑡(𝑛2−𝑛−2)/2 if 𝛼 = 2
Student distribution:
∫ ∏ 𝑠𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ 𝑐(log𝑡)𝑛−1
𝑡𝛼
Conditional Distributions and Gibbs Conditioning
Key question: Distribution of 𝑛 points in ℝ𝑛 given the volume of their parallelogram.
Examples of Conditional Distributions
Example 1: Standard Normal Variables
For 𝑋,𝑌 independent standard normal, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (1/2,1/2) in probability
Example 2: Exponential Variables
For 𝑋,𝑌 independent exponential, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → Uniform on {𝛼(1,0)+(1−𝛼)(0,1):𝛼 ∈ [0,1]}
Example 3: Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (𝑃𝑥+𝑃𝑦)/2
where 𝑃𝑥,𝑃𝑦 are Pareto distributions on x and y axes respectively.
Example 4: Product of Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
(Distribution to be analyzed)
Key insights:
• Limiting conditional distributions relate to tail behavior
• Geometry of level sets vs. conditioning set is crucial
• Naive interpretation of pictures can be misleading
Key motivation: Study behavior of integrals of the form
∫𝑓
𝐴(𝑥)𝑑𝑥, 𝐴 ⊂ ℝ𝑑
for sets 𝐴 far away from the origin.
Applications
• Statistics and probability theory
• Other areas of mathematics where such integrals arise
Challenges
• Difficult to compute numerically for high dimensions (10-50)
• Quadrature methods ineffective
• Monte-Carlo methods fail for small integral values (10−2, 10−3)
Examples in Probability and Statistics
Density Functions
Two example density functions on ℝ:
1. Weibull-like: 𝑤𝛼(𝑥)= 𝐾𝑤,𝛼exp(−|𝑥|𝛼/𝛼) 2. Student: 𝑠𝛼(𝑥)= 𝐾𝑠,𝛼(1+𝑥2/𝛼)−(𝛼+1)/2
Asymptotic Behavior of Integrals
For domain 𝐴𝑡= {𝑥 ∈ ℝ𝑑:⟨𝐶𝑥,𝑥⟩ > 𝑡}:
For 𝛼 ≠ 2:
∫ ∏𝑤𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(𝛼,𝑑,𝐶)𝑒−𝑡𝛼/2/𝑐2𝑡−(𝛼−2)𝑑/(4−𝛼)
For 𝛼 = 2:
∫ ∏𝑤2
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(2,𝑑,𝐶)𝑒−𝑡/𝜆𝑡(𝑘−1)/2
where 𝑘 is the dimension of the eigensubspace for the largest eigenvalue 𝜆 of 𝐶𝑇+𝐶.
For Student distribution:
∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝐾𝑠,𝛼𝛼(𝛼+1)/22𝑡−𝛼/2 ∑ 𝐶𝑖,𝑖
𝛼/2
𝑖:𝐶𝑖,𝑖>0
Applications to Random Matrices
For 𝐴𝑡= {𝑥 ∈ 𝑀(𝑛,ℝ):det𝑥 > 𝑡}:
Weibull-like distribution:
∫ ∏ 𝑤𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ {𝑐𝑒−𝑡𝛼/𝑛𝑡(𝛼(𝑛2−1)−2𝑛2)/2𝑛 if 𝛼 ≠ 2
𝑐𝑒−𝑛𝑡2/2𝑡(𝑛2−𝑛−2)/2 if 𝛼 = 2
Student distribution:
∫ ∏ 𝑠𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ 𝑐(log𝑡)𝑛−1
𝑡𝛼
Conditional Distributions and Gibbs Conditioning
Key question: Distribution of 𝑛 points in ℝ𝑛 given the volume of their parallelogram.
Examples of Conditional Distributions
Example 1: Standard Normal Variables
For 𝑋,𝑌 independent standard normal, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (1/2,1/2) in probability
Example 2: Exponential Variables
For 𝑋,𝑌 independent exponential, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → Uniform on {𝛼(1,0)+(1−𝛼)(0,1):𝛼 ∈ [0,1]}
Example 3: Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (𝑃𝑥+𝑃𝑦)/2
where 𝑃𝑥,𝑃𝑦 are Pareto distributions on x and y axes respectively.
Example 4: Product of Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
(Distribution to be analyzed)
Key insights:
• Limiting conditional distributions relate to tail behavior
• Geometry of level sets vs. conditioning set is crucial
• Naive interpretation of pictures can be misleading
Key motivation: Study behavior of integrals of the form
∫𝑓
𝐴(𝑥)𝑑𝑥, 𝐴 ⊂ ℝ𝑑
for sets 𝐴 far away from the origin.
Applications
• Statistics and probability theory
• Other areas of mathematics where such integrals arise
Challenges
• Difficult to compute numerically for high dimensions (10-50)
• Quadrature methods ineffective
• Monte-Carlo methods fail for small integral values (10−2, 10−3)
Examples in Probability and Statistics
Density Functions
Two example density functions on ℝ:
1. Weibull-like: 𝑤𝛼(𝑥)= 𝐾𝑤,𝛼exp(−|𝑥|𝛼/𝛼) 2. Student: 𝑠𝛼(𝑥)= 𝐾𝑠,𝛼(1+𝑥2/𝛼)−(𝛼+1)/2
Asymptotic Behavior of Integrals
For domain 𝐴𝑡= {𝑥 ∈ ℝ𝑑:⟨𝐶𝑥,𝑥⟩ > 𝑡}:
For 𝛼 ≠ 2:
∫ ∏𝑤𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(𝛼,𝑑,𝐶)𝑒−𝑡𝛼/2/𝑐2𝑡−(𝛼−2)𝑑/(4−𝛼)
For 𝛼 = 2:
∫ ∏𝑤2
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(2,𝑑,𝐶)𝑒−𝑡/𝜆𝑡(𝑘−1)/2
where 𝑘 is the dimension of the eigensubspace for the largest eigenvalue 𝜆 of 𝐶𝑇+𝐶.
For Student distribution:
∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝐾𝑠,𝛼𝛼(𝛼+1)/22𝑡−𝛼/2 ∑ 𝐶𝑖,𝑖
𝛼/2
𝑖:𝐶𝑖,𝑖>0
Applications to Random Matrices
For 𝐴𝑡= {𝑥 ∈ 𝑀(𝑛,ℝ):det𝑥 > 𝑡}:
Weibull-like distribution:
∫ ∏ 𝑤𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ {𝑐𝑒−𝑡𝛼/𝑛𝑡(𝛼(𝑛2−1)−2𝑛2)/2𝑛 if 𝛼 ≠ 2
𝑐𝑒−𝑛𝑡2/2𝑡(𝑛2−𝑛−2)/2 if 𝛼 = 2
Student distribution:
∫ ∏ 𝑠𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ 𝑐(log𝑡)𝑛−1
𝑡𝛼
Conditional Distributions and Gibbs Conditioning
Key question: Distribution of 𝑛 points in ℝ𝑛 given the volume of their parallelogram.
Examples of Conditional Distributions
Example 1: Standard Normal Variables
For 𝑋,𝑌 independent standard normal, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (1/2,1/2) in probability
Example 2: Exponential Variables
For 𝑋,𝑌 independent exponential, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → Uniform on {𝛼(1,0)+(1−𝛼)(0,1):𝛼 ∈ [0,1]}
Example 3: Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (𝑃𝑥+𝑃𝑦)/2
where 𝑃𝑥,𝑃𝑦 are Pareto distributions on x and y axes respectively.
Example 4: Product of Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
(Distribution to be analyzed)
Key insights:
• Limiting conditional distributions relate to tail behavior
• Geometry of level sets vs. conditioning set is crucial
• Naive interpretation of pictures can be misleading
Key motivation: Study behavior of integrals of the form
∫𝑓
𝐴(𝑥)𝑑𝑥, 𝐴 ⊂ ℝ𝑑
for sets 𝐴 far away from the origin.
Applications
• Statistics and probability theory
• Other areas of mathematics where such integrals arise
Challenges
• Difficult to compute numerically for high dimensions (10-50)
• Quadrature methods ineffective
• Monte-Carlo methods fail for small integral values (10−2, 10−3)
Examples in Probability and Statistics
Density Functions
Two example density functions on ℝ:
1. Weibull-like: 𝑤𝛼(𝑥)= 𝐾𝑤,𝛼exp(−|𝑥|𝛼/𝛼) 2. Student: 𝑠𝛼(𝑥)= 𝐾𝑠,𝛼(1+𝑥2/𝛼)−(𝛼+1)/2
Asymptotic Behavior of Integrals
For domain 𝐴𝑡= {𝑥 ∈ ℝ𝑑:⟨𝐶𝑥,𝑥⟩ > 𝑡}:
For 𝛼 ≠ 2:
∫ ∏𝑤𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(𝛼,𝑑,𝐶)𝑒−𝑡𝛼/2/𝑐2𝑡−(𝛼−2)𝑑/(4−𝛼)
For 𝛼 = 2:
∫ ∏𝑤2
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(2,𝑑,𝐶)𝑒−𝑡/𝜆𝑡(𝑘−1)/2
where 𝑘 is the dimension of the eigensubspace for the largest eigenvalue 𝜆 of 𝐶𝑇+𝐶.
For Student distribution:
∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝐾𝑠,𝛼𝛼(𝛼+1)/22𝑡−𝛼/2 ∑ 𝐶𝑖,𝑖
𝛼/2
𝑖:𝐶𝑖,𝑖>0
Applications to Random Matrices
For 𝐴𝑡= {𝑥 ∈ 𝑀(𝑛,ℝ):det𝑥 > 𝑡}:
Weibull-like distribution:
∫ ∏ 𝑤𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ {𝑐𝑒−𝑡𝛼/𝑛𝑡(𝛼(𝑛2−1)−2𝑛2)/2𝑛 if 𝛼 ≠ 2
𝑐𝑒−𝑛𝑡2/2𝑡(𝑛2−𝑛−2)/2 if 𝛼 = 2
Student distribution:
∫ ∏ 𝑠𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ 𝑐(log𝑡)𝑛−1
𝑡𝛼
Conditional Distributions and Gibbs Conditioning
Key question: Distribution of 𝑛 points in ℝ𝑛 given the volume of their parallelogram.
Examples of Conditional Distributions
Example 1: Standard Normal Variables
For 𝑋,𝑌 independent standard normal, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (1/2,1/2) in probability
Example 2: Exponential Variables
For 𝑋,𝑌 independent exponential, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → Uniform on {𝛼(1,0)+(1−𝛼)(0,1):𝛼 ∈ [0,1]}
Example 3: Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (𝑃𝑥+𝑃𝑦)/2
where 𝑃𝑥,𝑃𝑦 are Pareto distributions on x and y axes respectively.
Example 4: Product of Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
(Distribution to be analyzed)
Key insights:
• Limiting conditional distributions relate to tail behavior
• Geometry of level sets vs. conditioning set is crucial
• Naive interpretation of pictures can be misleading
Key motivation: Study behavior of integrals of the form
∫𝑓
𝐴(𝑥)𝑑𝑥, 𝐴 ⊂ ℝ𝑑
for sets 𝐴 far away from the origin.
Applications
• Statistics and probability theory
• Other areas of mathematics where such integrals arise
Challenges
• Difficult to compute numerically for high dimensions (10-50)
• Quadrature methods ineffective
• Monte-Carlo methods fail for small integral values (10−2, 10−3)
Examples in Probability and Statistics
Density Functions
Two example density functions on ℝ:
1. Weibull-like: 𝑤𝛼(𝑥)= 𝐾𝑤,𝛼exp(−|𝑥|𝛼/𝛼) 2. Student: 𝑠𝛼(𝑥)= 𝐾𝑠,𝛼(1+𝑥2/𝛼)−(𝛼+1)/2
Asymptotic Behavior of Integrals
For domain 𝐴𝑡= {𝑥 ∈ ℝ𝑑:⟨𝐶𝑥,𝑥⟩ > 𝑡}:
For 𝛼 ≠ 2:
∫ ∏𝑤𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(𝛼,𝑑,𝐶)𝑒−𝑡𝛼/2/𝑐2𝑡−(𝛼−2)𝑑/(4−𝛼)
For 𝛼 = 2:
∫ ∏𝑤2
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(2,𝑑,𝐶)𝑒−𝑡/𝜆𝑡(𝑘−1)/2
where 𝑘 is the dimension of the eigensubspace for the largest eigenvalue 𝜆 of 𝐶𝑇+𝐶.
For Student distribution:
∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝐾𝑠,𝛼𝛼(𝛼+1)/22𝑡−𝛼/2 ∑ 𝐶𝑖,𝑖
𝛼/2
𝑖:𝐶𝑖,𝑖>0
Applications to Random Matrices
For 𝐴𝑡= {𝑥 ∈ 𝑀(𝑛,ℝ):det𝑥 > 𝑡}:
Weibull-like distribution:
∫ ∏ 𝑤𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ {𝑐𝑒−𝑡𝛼/𝑛𝑡(𝛼(𝑛2−1)−2𝑛2)/2𝑛 if 𝛼 ≠ 2
𝑐𝑒−𝑛𝑡2/2𝑡(𝑛2−𝑛−2)/2 if 𝛼 = 2
Student distribution:
∫ ∏ 𝑠𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ 𝑐(log𝑡)𝑛−1
𝑡𝛼
Conditional Distributions and Gibbs Conditioning
Key question: Distribution of 𝑛 points in ℝ𝑛 given the volume of their parallelogram.
Examples of Conditional Distributions
Example 1: Standard Normal Variables
For 𝑋,𝑌 independent standard normal, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (1/2,1/2) in probability
Example 2: Exponential Variables
For 𝑋,𝑌 independent exponential, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → Uniform on {𝛼(1,0)+(1−𝛼)(0,1):𝛼 ∈ [0,1]}
Example 3: Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (𝑃𝑥+𝑃𝑦)/2
where 𝑃𝑥,𝑃𝑦 are Pareto distributions on x and y axes respectively.
Example 4: Product of Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
(Distribution to be analyzed)
Key insights:
• Limiting conditional distributions relate to tail behavior
• Geometry of level sets vs. conditioning set is crucial
• Naive interpretation of pictures can be misleading
Key motivation: Study behavior of integrals of the form
∫𝑓
𝐴(𝑥)𝑑𝑥, 𝐴 ⊂ ℝ𝑑
for sets 𝐴 far away from the origin.
Applications
• Statistics and probability theory
• Other areas of mathematics where such integrals arise
Challenges
• Difficult to compute numerically for high dimensions (10-50)
• Quadrature methods ineffective
• Monte-Carlo methods fail for small integral values (10−2, 10−3)
Examples in Probability and Statistics
Density Functions
Two example density functions on ℝ:
1. Weibull-like: 𝑤𝛼(𝑥)= 𝐾𝑤,𝛼exp(−|𝑥|𝛼/𝛼) 2. Student: 𝑠𝛼(𝑥)= 𝐾𝑠,𝛼(1+𝑥2/𝛼)−(𝛼+1)/2
Asymptotic Behavior of Integrals
For domain 𝐴𝑡= {𝑥 ∈ ℝ𝑑:⟨𝐶𝑥,𝑥⟩ > 𝑡}:
For 𝛼 ≠ 2:
∫ ∏𝑤𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(𝛼,𝑑,𝐶)𝑒−𝑡𝛼/2/𝑐2𝑡−(𝛼−2)𝑑/(4−𝛼)
For 𝛼 = 2:
∫ ∏𝑤2
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(2,𝑑,𝐶)𝑒−𝑡/𝜆𝑡(𝑘−1)/2
where 𝑘 is the dimension of the eigensubspace for the largest eigenvalue 𝜆 of 𝐶𝑇+𝐶.
For Student distribution:
∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝐾𝑠,𝛼𝛼(𝛼+1)/22𝑡−𝛼/2 ∑ 𝐶𝑖,𝑖
𝛼/2
𝑖:𝐶𝑖,𝑖>0
Applications to Random Matrices
For 𝐴𝑡= {𝑥 ∈ 𝑀(𝑛,ℝ):det𝑥 > 𝑡}:
Weibull-like distribution:
∫ ∏ 𝑤𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ {𝑐𝑒−𝑡𝛼/𝑛𝑡(𝛼(𝑛2−1)−2𝑛2)/2𝑛 if 𝛼 ≠ 2
𝑐𝑒−𝑛𝑡2/2𝑡(𝑛2−𝑛−2)/2 if 𝛼 = 2
Student distribution:
∫ ∏ 𝑠𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ 𝑐(log𝑡)𝑛−1
𝑡𝛼
Conditional Distributions and Gibbs Conditioning
Key question: Distribution of 𝑛 points in ℝ𝑛 given the volume of their parallelogram.
Examples of Conditional Distributions
Example 1: Standard Normal Variables
For 𝑋,𝑌 independent standard normal, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (1/2,1/2) in probability
Example 2: Exponential Variables
For 𝑋,𝑌 independent exponential, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → Uniform on {𝛼(1,0)+(1−𝛼)(0,1):𝛼 ∈ [0,1]}
Example 3: Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (𝑃𝑥+𝑃𝑦)/2
where 𝑃𝑥,𝑃𝑦 are Pareto distributions on x and y axes respectively.
Example 4: Product of Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
(Distribution to be analyzed)
Key insights:
• Limiting conditional distributions relate to tail behavior
• Geometry of level sets vs. conditioning set is crucial
• Naive interpretation of pictures can be misleading
Key motivation: Study behavior of integrals of the form
∫𝑓
𝐴(𝑥)𝑑𝑥, 𝐴 ⊂ ℝ𝑑
for sets 𝐴 far away from the origin.
Applications
• Statistics and probability theory
• Other areas of mathematics where such integrals arise
Challenges
• Difficult to compute numerically for high dimensions (10-50)
• Quadrature methods ineffective
• Monte-Carlo methods fail for small integral values (10−2, 10−3)
Examples in Probability and Statistics
Density Functions
Two example density functions on ℝ:
1. Weibull-like: 𝑤𝛼(𝑥)= 𝐾𝑤,𝛼exp(−|𝑥|𝛼/𝛼) 2. Student: 𝑠𝛼(𝑥)= 𝐾𝑠,𝛼(1+𝑥2/𝛼)−(𝛼+1)/2
Asymptotic Behavior of Integrals
For domain 𝐴𝑡= {𝑥 ∈ ℝ𝑑:⟨𝐶𝑥,𝑥⟩ > 𝑡}:
For 𝛼 ≠ 2:
∫ ∏𝑤𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(𝛼,𝑑,𝐶)𝑒−𝑡𝛼/2/𝑐2𝑡−(𝛼−2)𝑑/(4−𝛼)
For 𝛼 = 2:
∫ ∏𝑤2
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(2,𝑑,𝐶)𝑒−𝑡/𝜆𝑡(𝑘−1)/2
where 𝑘 is the dimension of the eigensubspace for the largest eigenvalue 𝜆 of 𝐶𝑇+𝐶.
For Student distribution:
∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝐾𝑠,𝛼𝛼(𝛼+1)/22𝑡−𝛼/2 ∑ 𝐶𝑖,𝑖
𝛼/2
𝑖:𝐶𝑖,𝑖>0
Applications to Random Matrices
For 𝐴𝑡= {𝑥 ∈ 𝑀(𝑛,ℝ):det𝑥 > 𝑡}:
Weibull-like distribution:
∫ ∏ 𝑤𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ {𝑐𝑒−𝑡𝛼/𝑛𝑡(𝛼(𝑛2−1)−2𝑛2)/2𝑛 if 𝛼 ≠ 2
𝑐𝑒−𝑛𝑡2/2𝑡(𝑛2−𝑛−2)/2 if 𝛼 = 2
Student distribution:
∫ ∏ 𝑠𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ 𝑐(log𝑡)𝑛−1
𝑡𝛼
Conditional Distributions and Gibbs Conditioning
Key question: Distribution of 𝑛 points in ℝ𝑛 given the volume of their parallelogram.
Examples of Conditional Distributions
Example 1: Standard Normal Variables
For 𝑋,𝑌 independent standard normal, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (1/2,1/2) in probability
Example 2: Exponential Variables
For 𝑋,𝑌 independent exponential, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → Uniform on {𝛼(1,0)+(1−𝛼)(0,1):𝛼 ∈ [0,1]}
Example 3: Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (𝑃𝑥+𝑃𝑦)/2
where 𝑃𝑥,𝑃𝑦 are Pareto distributions on x and y axes respectively.
Example 4: Product of Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
(Distribution to be analyzed)
Key insights:
• Limiting conditional distributions relate to tail behavior
• Geometry of level sets vs. conditioning set is crucial
• Naive interpretation of pictures can be misleading
Key motivation: Study behavior of integrals of the form
∫𝑓
𝐴(𝑥)𝑑𝑥, 𝐴 ⊂ ℝ𝑑
for sets 𝐴 far away from the origin.
Applications
• Statistics and probability theory
• Other areas of mathematics where such integrals arise
Challenges
• Difficult to compute numerically for high dimensions (10-50)
• Quadrature methods ineffective
• Monte-Carlo methods fail for small integral values (10−2, 10−3)
Examples in Probability and Statistics
Density Functions
Two example density functions on ℝ:
1. Weibull-like: 𝑤𝛼(𝑥)= 𝐾𝑤,𝛼exp(−|𝑥|𝛼/𝛼) 2. Student: 𝑠𝛼(𝑥)= 𝐾𝑠,𝛼(1+𝑥2/𝛼)−(𝛼+1)/2
Asymptotic Behavior of Integrals
For domain 𝐴𝑡= {𝑥 ∈ ℝ𝑑:⟨𝐶𝑥,𝑥⟩ > 𝑡}:
For 𝛼 ≠ 2:
∫ ∏𝑤𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(𝛼,𝑑,𝐶)𝑒−𝑡𝛼/2/𝑐2𝑡−(𝛼−2)𝑑/(4−𝛼)
For 𝛼 = 2:
∫ ∏𝑤2
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(2,𝑑,𝐶)𝑒−𝑡/𝜆𝑡(𝑘−1)/2
where 𝑘 is the dimension of the eigensubspace for the largest eigenvalue 𝜆 of 𝐶𝑇+𝐶.
For Student distribution:
∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝐾𝑠,𝛼𝛼(𝛼+1)/22𝑡−𝛼/2 ∑ 𝐶𝑖,𝑖
𝛼/2
𝑖:𝐶𝑖,𝑖>0
Applications to Random Matrices
For 𝐴𝑡= {𝑥 ∈ 𝑀(𝑛,ℝ):det𝑥 > 𝑡}:
Weibull-like distribution:
∫ ∏ 𝑤𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ {𝑐𝑒−𝑡𝛼/𝑛𝑡(𝛼(𝑛2−1)−2𝑛2)/2𝑛 if 𝛼 ≠ 2
𝑐𝑒−𝑛𝑡2/2𝑡(𝑛2−𝑛−2)/2 if 𝛼 = 2
Student distribution:
∫ ∏ 𝑠𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ 𝑐(log𝑡)𝑛−1
𝑡𝛼
Conditional Distributions and Gibbs Conditioning
Key question: Distribution of 𝑛 points in ℝ𝑛 given the volume of their parallelogram.
Examples of Conditional Distributions
Example 1: Standard Normal Variables
For 𝑋,𝑌 independent standard normal, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (1/2,1/2) in probability
Example 2: Exponential Variables
For 𝑋,𝑌 independent exponential, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → Uniform on {𝛼(1,0)+(1−𝛼)(0,1):𝛼 ∈ [0,1]}
Example 3: Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (𝑃𝑥+𝑃𝑦)/2
where 𝑃𝑥,𝑃𝑦 are Pareto distributions on x and y axes respectively.
Example 4: Product of Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
(Distribution to be analyzed)
Key insights:
• Limiting conditional distributions relate to tail behavior
• Geometry of level sets vs. conditioning set is crucial
• Naive interpretation of pictures can be misleading
Key motivation: Study behavior of integrals of the form
∫𝑓
𝐴(𝑥)𝑑𝑥, 𝐴 ⊂ ℝ𝑑
for sets 𝐴 far away from the origin.
Applications
• Statistics and probability theory
• Other areas of mathematics where such integrals arise
Challenges
• Difficult to compute numerically for high dimensions (10-50)
• Quadrature methods ineffective
• Monte-Carlo methods fail for small integral values (10−2, 10−3)
Examples in Probability and Statistics
Density Functions
Two example density functions on ℝ:
1. Weibull-like: 𝑤𝛼(𝑥)= 𝐾𝑤,𝛼exp(−|𝑥|𝛼/𝛼) 2. Student: 𝑠𝛼(𝑥)= 𝐾𝑠,𝛼(1+𝑥2/𝛼)−(𝛼+1)/2
Asymptotic Behavior of Integrals
For domain 𝐴𝑡= {𝑥 ∈ ℝ𝑑:⟨𝐶𝑥,𝑥⟩ > 𝑡}:
For 𝛼 ≠ 2:
∫ ∏𝑤𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(𝛼,𝑑,𝐶)𝑒−𝑡𝛼/2/𝑐2𝑡−(𝛼−2)𝑑/(4−𝛼)
For 𝛼 = 2:
∫ ∏𝑤2
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(2,𝑑,𝐶)𝑒−𝑡/𝜆𝑡(𝑘−1)/2
where 𝑘 is the dimension of the eigensubspace for the largest eigenvalue 𝜆 of 𝐶𝑇+𝐶.
For Student distribution:
∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝐾𝑠,𝛼𝛼(𝛼+1)/22𝑡−𝛼/2 ∑ 𝐶𝑖,𝑖
𝛼/2
𝑖:𝐶𝑖,𝑖>0
Applications to Random Matrices
For 𝐴𝑡= {𝑥 ∈ 𝑀(𝑛,ℝ):det𝑥 > 𝑡}:
Weibull-like distribution:
∫ ∏ 𝑤𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ {𝑐𝑒−𝑡𝛼/𝑛𝑡(𝛼(𝑛2−1)−2𝑛2)/2𝑛 if 𝛼 ≠ 2
𝑐𝑒−𝑛𝑡2/2𝑡(𝑛2−𝑛−2)/2 if 𝛼 = 2
Student distribution:
∫ ∏ 𝑠𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ 𝑐(log𝑡)𝑛−1
𝑡𝛼
Conditional Distributions and Gibbs Conditioning
Key question: Distribution of 𝑛 points in ℝ𝑛 given the volume of their parallelogram.
Examples of Conditional Distributions
Example 1: Standard Normal Variables
For 𝑋,𝑌 independent standard normal, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (1/2,1/2) in probability
Example 2: Exponential Variables
For 𝑋,𝑌 independent exponential, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → Uniform on {𝛼(1,0)+(1−𝛼)(0,1):𝛼 ∈ [0,1]}
Example 3: Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (𝑃𝑥+𝑃𝑦)/2
where 𝑃𝑥,𝑃𝑦 are Pareto distributions on x and y axes respectively.
Example 4: Product of Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
(Distribution to be analyzed)
Key insights:
• Limiting conditional distributions relate to tail behavior
• Geometry of level sets vs. conditioning set is crucial
• Naive interpretation of pictures can be misleading
Key motivation: Study behavior of integrals of the form
∫𝑓
𝐴(𝑥)𝑑𝑥, 𝐴 ⊂ ℝ𝑑
for sets 𝐴 far away from the origin.
Applications
• Statistics and probability theory
• Other areas of mathematics where such integrals arise
Challenges
• Difficult to compute numerically for high dimensions (10-50)
• Quadrature methods ineffective
• Monte-Carlo methods fail for small integral values (10−2, 10−3)
Examples in Probability and Statistics
Density Functions
Two example density functions on ℝ:
1. Weibull-like: 𝑤𝛼(𝑥)= 𝐾𝑤,𝛼exp(−|𝑥|𝛼/𝛼) 2. Student: 𝑠𝛼(𝑥)= 𝐾𝑠,𝛼(1+𝑥2/𝛼)−(𝛼+1)/2
Asymptotic Behavior of Integrals
For domain 𝐴𝑡= {𝑥 ∈ ℝ𝑑:⟨𝐶𝑥,𝑥⟩ > 𝑡}:
For 𝛼 ≠ 2:
∫ ∏𝑤𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(𝛼,𝑑,𝐶)𝑒−𝑡𝛼/2/𝑐2𝑡−(𝛼−2)𝑑/(4−𝛼)
For 𝛼 = 2:
∫ ∏𝑤2
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(2,𝑑,𝐶)𝑒−𝑡/𝜆𝑡(𝑘−1)/2
where 𝑘 is the dimension of the eigensubspace for the largest eigenvalue 𝜆 of 𝐶𝑇+𝐶.
For Student distribution:
∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝐾𝑠,𝛼𝛼(𝛼+1)/22𝑡−𝛼/2 ∑ 𝐶𝑖,𝑖
𝛼/2
𝑖:𝐶𝑖,𝑖>0
Applications to Random Matrices
For 𝐴𝑡= {𝑥 ∈ 𝑀(𝑛,ℝ):det𝑥 > 𝑡}:
Weibull-like distribution:
∫ ∏ 𝑤𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ {𝑐𝑒−𝑡𝛼/𝑛𝑡(𝛼(𝑛2−1)−2𝑛2)/2𝑛 if 𝛼 ≠ 2
𝑐𝑒−𝑛𝑡2/2𝑡(𝑛2−𝑛−2)/2 if 𝛼 = 2
Student distribution:
∫ ∏ 𝑠𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ 𝑐(log𝑡)𝑛−1
𝑡𝛼
Conditional Distributions and Gibbs Conditioning
Key question: Distribution of 𝑛 points in ℝ𝑛 given the volume of their parallelogram.
Examples of Conditional Distributions
Example 1: Standard Normal Variables
For 𝑋,𝑌 independent standard normal, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (1/2,1/2) in probability
Example 2: Exponential Variables
For 𝑋,𝑌 independent exponential, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → Uniform on {𝛼(1,0)+(1−𝛼)(0,1):𝛼 ∈ [0,1]}
Example 3: Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (𝑃𝑥+𝑃𝑦)/2
where 𝑃𝑥,𝑃𝑦 are Pareto distributions on x and y axes respectively.
Example 4: Product of Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
(Distribution to be analyzed)
Key insights:
• Limiting conditional distributions relate to tail behavior
• Geometry of level sets vs. conditioning set is crucial
• Naive interpretation of pictures can be misleading
Key motivation: Study behavior of integrals of the form
∫𝑓
𝐴(𝑥)𝑑𝑥, 𝐴 ⊂ ℝ𝑑
for sets 𝐴 far away from the origin.
Applications
• Statistics and probability theory
• Other areas of mathematics where such integrals arise
Challenges
• Difficult to compute numerically for high dimensions (10-50)
• Quadrature methods ineffective
• Monte-Carlo methods fail for small integral values (10−2, 10−3)
Examples in Probability and Statistics
Density Functions
Two example density functions on ℝ:
1. Weibull-like: 𝑤𝛼(𝑥)= 𝐾𝑤,𝛼exp(−|𝑥|𝛼/𝛼) 2. Student: 𝑠𝛼(𝑥)= 𝐾𝑠,𝛼(1+𝑥2/𝛼)−(𝛼+1)/2
Asymptotic Behavior of Integrals
For domain 𝐴𝑡= {𝑥 ∈ ℝ𝑑:⟨𝐶𝑥,𝑥⟩ > 𝑡}:
For 𝛼 ≠ 2:
∫ ∏𝑤𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(𝛼,𝑑,𝐶)𝑒−𝑡𝛼/2/𝑐2𝑡−(𝛼−2)𝑑/(4−𝛼)
For 𝛼 = 2:
∫ ∏𝑤2
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(2,𝑑,𝐶)𝑒−𝑡/𝜆𝑡(𝑘−1)/2
where 𝑘 is the dimension of the eigensubspace for the largest eigenvalue 𝜆 of 𝐶𝑇+𝐶.
For Student distribution:
∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝐾𝑠,𝛼𝛼(𝛼+1)/22𝑡−𝛼/2 ∑ 𝐶𝑖,𝑖
𝛼/2
𝑖:𝐶𝑖,𝑖>0
Applications to Random Matrices
For 𝐴𝑡= {𝑥 ∈ 𝑀(𝑛,ℝ):det𝑥 > 𝑡}:
Weibull-like distribution:
∫ ∏ 𝑤𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ {𝑐𝑒−𝑡𝛼/𝑛𝑡(𝛼(𝑛2−1)−2𝑛2)/2𝑛 if 𝛼 ≠ 2
𝑐𝑒−𝑛𝑡2/2𝑡(𝑛2−𝑛−2)/2 if 𝛼 = 2
Student distribution:
∫ ∏ 𝑠𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ 𝑐(log𝑡)𝑛−1
𝑡𝛼
Conditional Distributions and Gibbs Conditioning
Key question: Distribution of 𝑛 points in ℝ𝑛 given the volume of their parallelogram.
Examples of Conditional Distributions
Example 1: Standard Normal Variables
For 𝑋,𝑌 independent standard normal, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (1/2,1/2) in probability
Example 2: Exponential Variables
For 𝑋,𝑌 independent exponential, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → Uniform on {𝛼(1,0)+(1−𝛼)(0,1):𝛼 ∈ [0,1]}
Example 3: Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (𝑃𝑥+𝑃𝑦)/2
where 𝑃𝑥,𝑃𝑦 are Pareto distributions on x and y axes respectively.
Example 4: Product of Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
(Distribution to be analyzed)
Key insights:
• Limiting conditional distributions relate to tail behavior
• Geometry of level sets vs. conditioning set is crucial
• Naive interpretation of pictures can be misleading
Key motivation: Study behavior of integrals of the form
∫𝑓
𝐴(𝑥)𝑑𝑥, 𝐴 ⊂ ℝ𝑑
for sets 𝐴 far away from the origin.
Applications
• Statistics and probability theory
• Other areas of mathematics where such integrals arise
Challenges
• Difficult to compute numerically for high dimensions (10-50)
• Quadrature methods ineffective
• Monte-Carlo methods fail for small integral values (10−2, 10−3)
Examples in Probability and Statistics
Density Functions
Two example density functions on ℝ:
1. Weibull-like: 𝑤𝛼(𝑥)= 𝐾𝑤,𝛼exp(−|𝑥|𝛼/𝛼) 2. Student: 𝑠𝛼(𝑥)= 𝐾𝑠,𝛼(1+𝑥2/𝛼)−(𝛼+1)/2
Asymptotic Behavior of Integrals
For domain 𝐴𝑡= {𝑥 ∈ ℝ𝑑:⟨𝐶𝑥,𝑥⟩ > 𝑡}:
For 𝛼 ≠ 2:
∫ ∏𝑤𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(𝛼,𝑑,𝐶)𝑒−𝑡𝛼/2/𝑐2𝑡−(𝛼−2)𝑑/(4−𝛼)
For 𝛼 = 2:
∫ ∏𝑤2
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(2,𝑑,𝐶)𝑒−𝑡/𝜆𝑡(𝑘−1)/2
where 𝑘 is the dimension of the eigensubspace for the largest eigenvalue 𝜆 of 𝐶𝑇+𝐶.
For Student distribution:
∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝐾𝑠,𝛼𝛼(𝛼+1)/22𝑡−𝛼/2 ∑ 𝐶𝑖,𝑖
𝛼/2
𝑖:𝐶𝑖,𝑖>0
Applications to Random Matrices
For 𝐴𝑡= {𝑥 ∈ 𝑀(𝑛,ℝ):det𝑥 > 𝑡}:
Weibull-like distribution:
∫ ∏ 𝑤𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ {𝑐𝑒−𝑡𝛼/𝑛𝑡(𝛼(𝑛2−1)−2𝑛2)/2𝑛 if 𝛼 ≠ 2
𝑐𝑒−𝑛𝑡2/2𝑡(𝑛2−𝑛−2)/2 if 𝛼 = 2
Student distribution:
∫ ∏ 𝑠𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ 𝑐(log𝑡)𝑛−1
𝑡𝛼
Conditional Distributions and Gibbs Conditioning
Key question: Distribution of 𝑛 points in ℝ𝑛 given the volume of their parallelogram.
Examples of Conditional Distributions
Example 1: Standard Normal Variables
For 𝑋,𝑌 independent standard normal, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (1/2,1/2) in probability
Example 2: Exponential Variables
For 𝑋,𝑌 independent exponential, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → Uniform on {𝛼(1,0)+(1−𝛼)(0,1):𝛼 ∈ [0,1]}
Example 3: Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (𝑃𝑥+𝑃𝑦)/2
where 𝑃𝑥,𝑃𝑦 are Pareto distributions on x and y axes respectively.
Example 4: Product of Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
(Distribution to be analyzed)
Key insights:
• Limiting conditional distributions relate to tail behavior
• Geometry of level sets vs. conditioning set is crucial
• Naive interpretation of pictures can be misleading
Key motivation: Study behavior of integrals of the form
∫𝑓
𝐴(𝑥)𝑑𝑥, 𝐴 ⊂ ℝ𝑑
for sets 𝐴 far away from the origin.
Applications
• Statistics and probability theory
• Other areas of mathematics where such integrals arise
Challenges
• Difficult to compute numerically for high dimensions (10-50)
• Quadrature methods ineffective
• Monte-Carlo methods fail for small integral values (10−2, 10−3)
Examples in Probability and Statistics
Density Functions
Two example density functions on ℝ:
1. Weibull-like: 𝑤𝛼(𝑥)= 𝐾𝑤,𝛼exp(−|𝑥|𝛼/𝛼) 2. Student: 𝑠𝛼(𝑥)= 𝐾𝑠,𝛼(1+𝑥2/𝛼)−(𝛼+1)/2
Asymptotic Behavior of Integrals
For domain 𝐴𝑡= {𝑥 ∈ ℝ𝑑:⟨𝐶𝑥,𝑥⟩ > 𝑡}:
For 𝛼 ≠ 2:
∫ ∏𝑤𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(𝛼,𝑑,𝐶)𝑒−𝑡𝛼/2/𝑐2𝑡−(𝛼−2)𝑑/(4−𝛼)
For 𝛼 = 2:
∫ ∏𝑤2
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(2,𝑑,𝐶)𝑒−𝑡/𝜆𝑡(𝑘−1)/2
where 𝑘 is the dimension of the eigensubspace for the largest eigenvalue 𝜆 of 𝐶𝑇+𝐶.
For Student distribution:
∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝐾𝑠,𝛼𝛼(𝛼+1)/22𝑡−𝛼/2 ∑ 𝐶𝑖,𝑖
𝛼/2
𝑖:𝐶𝑖,𝑖>0
Applications to Random Matrices
For 𝐴𝑡= {𝑥 ∈ 𝑀(𝑛,ℝ):det𝑥 > 𝑡}:
Weibull-like distribution:
∫ ∏ 𝑤𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ {𝑐𝑒−𝑡𝛼/𝑛𝑡(𝛼(𝑛2−1)−2𝑛2)/2𝑛 if 𝛼 ≠ 2
𝑐𝑒−𝑛𝑡2/2𝑡(𝑛2−𝑛−2)/2 if 𝛼 = 2
Student distribution:
∫ ∏ 𝑠𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ 𝑐(log𝑡)𝑛−1
𝑡𝛼
Conditional Distributions and Gibbs Conditioning
Key question: Distribution of 𝑛 points in ℝ𝑛 given the volume of their parallelogram.
Examples of Conditional Distributions
Example 1: Standard Normal Variables
For 𝑋,𝑌 independent standard normal, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (1/2,1/2) in probability
Example 2: Exponential Variables
For 𝑋,𝑌 independent exponential, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → Uniform on {𝛼(1,0)+(1−𝛼)(0,1):𝛼 ∈ [0,1]}
Example 3: Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (𝑃𝑥+𝑃𝑦)/2
where 𝑃𝑥,𝑃𝑦 are Pareto distributions on x and y axes respectively.
Example 4: Product of Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
(Distribution to be analyzed)
Key insights:
• Limiting conditional distributions relate to tail behavior
• Geometry of level sets vs. conditioning set is crucial
• Naive interpretation of pictures can be misleading
Key motivation: Study behavior of integrals of the form
∫𝑓
𝐴(𝑥)𝑑𝑥, 𝐴 ⊂ ℝ𝑑
for sets 𝐴 far away from the origin.
Applications
• Statistics and probability theory
• Other areas of mathematics where such integrals arise
Challenges
• Difficult to compute numerically for high dimensions (10-50)
• Quadrature methods ineffective
• Monte-Carlo methods fail for small integral values (10−2, 10−3)
Examples in Probability and Statistics
Density Functions
Two example density functions on ℝ:
1. Weibull-like: 𝑤𝛼(𝑥)= 𝐾𝑤,𝛼exp(−|𝑥|𝛼/𝛼) 2. Student: 𝑠𝛼(𝑥)= 𝐾𝑠,𝛼(1+𝑥2/𝛼)−(𝛼+1)/2
Asymptotic Behavior of Integrals
For domain 𝐴𝑡= {𝑥 ∈ ℝ𝑑:⟨𝐶𝑥,𝑥⟩ > 𝑡}:
For 𝛼 ≠ 2:
∫ ∏𝑤𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(𝛼,𝑑,𝐶)𝑒−𝑡𝛼/2/𝑐2𝑡−(𝛼−2)𝑑/(4−𝛼)
For 𝛼 = 2:
∫ ∏𝑤2
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(2,𝑑,𝐶)𝑒−𝑡/𝜆𝑡(𝑘−1)/2
where 𝑘 is the dimension of the eigensubspace for the largest eigenvalue 𝜆 of 𝐶𝑇+𝐶.
For Student distribution:
∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝐾𝑠,𝛼𝛼(𝛼+1)/22𝑡−𝛼/2 ∑ 𝐶𝑖,𝑖
𝛼/2
𝑖:𝐶𝑖,𝑖>0
Applications to Random Matrices
For 𝐴𝑡= {𝑥 ∈ 𝑀(𝑛,ℝ):det𝑥 > 𝑡}:
Weibull-like distribution:
∫ ∏ 𝑤𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ {𝑐𝑒−𝑡𝛼/𝑛𝑡(𝛼(𝑛2−1)−2𝑛2)/2𝑛 if 𝛼 ≠ 2
𝑐𝑒−𝑛𝑡2/2𝑡(𝑛2−𝑛−2)/2 if 𝛼 = 2
Student distribution:
∫ ∏ 𝑠𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ 𝑐(log𝑡)𝑛−1
𝑡𝛼
Conditional Distributions and Gibbs Conditioning
Key question: Distribution of 𝑛 points in ℝ𝑛 given the volume of their parallelogram.
Examples of Conditional Distributions
Example 1: Standard Normal Variables
For 𝑋,𝑌 independent standard normal, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (1/2,1/2) in probability
Example 2: Exponential Variables
For 𝑋,𝑌 independent exponential, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → Uniform on {𝛼(1,0)+(1−𝛼)(0,1):𝛼 ∈ [0,1]}
Example 3: Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (𝑃𝑥+𝑃𝑦)/2
where 𝑃𝑥,𝑃𝑦 are Pareto distributions on x and y axes respectively.
Example 4: Product of Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
(Distribution to be analyzed)
Key insights:
• Limiting conditional distributions relate to tail behavior
• Geometry of level sets vs. conditioning set is crucial
• Naive interpretation of pictures can be misleading
Key motivation: Study behavior of integrals of the form
∫𝑓
𝐴(𝑥)𝑑𝑥, 𝐴 ⊂ ℝ𝑑
for sets 𝐴 far away from the origin.
Applications
• Statistics and probability theory
• Other areas of mathematics where such integrals arise
Challenges
• Difficult to compute numerically for high dimensions (10-50)
• Quadrature methods ineffective
• Monte-Carlo methods fail for small integral values (10−2, 10−3)
Examples in Probability and Statistics
Density Functions
Two example density functions on ℝ:
1. Weibull-like: 𝑤𝛼(𝑥)= 𝐾𝑤,𝛼exp(−|𝑥|𝛼/𝛼) 2. Student: 𝑠𝛼(𝑥)= 𝐾𝑠,𝛼(1+𝑥2/𝛼)−(𝛼+1)/2
Asymptotic Behavior of Integrals
For domain 𝐴𝑡= {𝑥 ∈ ℝ𝑑:⟨𝐶𝑥,𝑥⟩ > 𝑡}:
For 𝛼 ≠ 2:
∫ ∏𝑤𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(𝛼,𝑑,𝐶)𝑒−𝑡𝛼/2/𝑐2𝑡−(𝛼−2)𝑑/(4−𝛼)
For 𝛼 = 2:
∫ ∏𝑤2
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(2,𝑑,𝐶)𝑒−𝑡/𝜆𝑡(𝑘−1)/2
where 𝑘 is the dimension of the eigensubspace for the largest eigenvalue 𝜆 of 𝐶𝑇+𝐶.
For Student distribution:
∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝐾𝑠,𝛼𝛼(𝛼+1)/22𝑡−𝛼/2 ∑ 𝐶𝑖,𝑖
𝛼/2
𝑖:𝐶𝑖,𝑖>0
Applications to Random Matrices
For 𝐴𝑡= {𝑥 ∈ 𝑀(𝑛,ℝ):det𝑥 > 𝑡}:
Weibull-like distribution:
∫ ∏ 𝑤𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ {𝑐𝑒−𝑡𝛼/𝑛𝑡(𝛼(𝑛2−1)−2𝑛2)/2𝑛 if 𝛼 ≠ 2
𝑐𝑒−𝑛𝑡2/2𝑡(𝑛2−𝑛−2)/2 if 𝛼 = 2
Student distribution:
∫ ∏ 𝑠𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ 𝑐(log𝑡)𝑛−1
𝑡𝛼
Conditional Distributions and Gibbs Conditioning
Key question: Distribution of 𝑛 points in ℝ𝑛 given the volume of their parallelogram.
Examples of Conditional Distributions
Example 1: Standard Normal Variables
For 𝑋,𝑌 independent standard normal, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (1/2,1/2) in probability
Example 2: Exponential Variables
For 𝑋,𝑌 independent exponential, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → Uniform on {𝛼(1,0)+(1−𝛼)(0,1):𝛼 ∈ [0,1]}
Example 3: Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (𝑃𝑥+𝑃𝑦)/2
where 𝑃𝑥,𝑃𝑦 are Pareto distributions on x and y axes respectively.
Example 4: Product of Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
(Distribution to be analyzed)
Key insights:
• Limiting conditional distributions relate to tail behavior
• Geometry of level sets vs. conditioning set is crucial
• Naive interpretation of pictures can be misleading
Key motivation: Study behavior of integrals of the form
∫𝑓
𝐴(𝑥)𝑑𝑥, 𝐴 ⊂ ℝ𝑑
for sets 𝐴 far away from the origin.
Applications
• Statistics and probability theory
• Other areas of mathematics where such integrals arise
Challenges
• Difficult to compute numerically for high dimensions (10-50)
• Quadrature methods ineffective
• Monte-Carlo methods fail for small integral values (10−2, 10−3)
Examples in Probability and Statistics
Density Functions
Two example density functions on ℝ:
1. Weibull-like: 𝑤𝛼(𝑥)= 𝐾𝑤,𝛼exp(−|𝑥|𝛼/𝛼) 2. Student: 𝑠𝛼(𝑥)= 𝐾𝑠,𝛼(1+𝑥2/𝛼)−(𝛼+1)/2
Asymptotic Behavior of Integrals
For domain 𝐴𝑡= {𝑥 ∈ ℝ𝑑:⟨𝐶𝑥,𝑥⟩ > 𝑡}:
For 𝛼 ≠ 2:
∫ ∏𝑤𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(𝛼,𝑑,𝐶)𝑒−𝑡𝛼/2/𝑐2𝑡−(𝛼−2)𝑑/(4−𝛼)
For 𝛼 = 2:
∫ ∏𝑤2
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(2,𝑑,𝐶)𝑒−𝑡/𝜆𝑡(𝑘−1)/2
where 𝑘 is the dimension of the eigensubspace for the largest eigenvalue 𝜆 of 𝐶𝑇+𝐶.
For Student distribution:
∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝐾𝑠,𝛼𝛼(𝛼+1)/22𝑡−𝛼/2 ∑ 𝐶𝑖,𝑖
𝛼/2
𝑖:𝐶𝑖,𝑖>0
Applications to Random Matrices
For 𝐴𝑡= {𝑥 ∈ 𝑀(𝑛,ℝ):det𝑥 > 𝑡}:
Weibull-like distribution:
∫ ∏ 𝑤𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ {𝑐𝑒−𝑡𝛼/𝑛𝑡(𝛼(𝑛2−1)−2𝑛2)/2𝑛 if 𝛼 ≠ 2
𝑐𝑒−𝑛𝑡2/2𝑡(𝑛2−𝑛−2)/2 if 𝛼 = 2
Student distribution:
∫ ∏ 𝑠𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ 𝑐(log𝑡)𝑛−1
𝑡𝛼
Conditional Distributions and Gibbs Conditioning
Key question: Distribution of 𝑛 points in ℝ𝑛 given the volume of their parallelogram.
Examples of Conditional Distributions
Example 1: Standard Normal Variables
For 𝑋,𝑌 independent standard normal, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (1/2,1/2) in probability
Example 2: Exponential Variables
For 𝑋,𝑌 independent exponential, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → Uniform on {𝛼(1,0)+(1−𝛼)(0,1):𝛼 ∈ [0,1]}
Example 3: Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (𝑃𝑥+𝑃𝑦)/2
where 𝑃𝑥,𝑃𝑦 are Pareto distributions on x and y axes respectively.
Example 4: Product of Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
(Distribution to be analyzed)
Key insights:
• Limiting conditional distributions relate to tail behavior
• Geometry of level sets vs. conditioning set is crucial
• Naive interpretation of pictures can be misleading
Key motivation: Study behavior of integrals of the form
∫𝑓
𝐴(𝑥)𝑑𝑥, 𝐴 ⊂ ℝ𝑑
for sets 𝐴 far away from the origin.
Applications
• Statistics and probability theory
• Other areas of mathematics where such integrals arise
Challenges
• Difficult to compute numerically for high dimensions (10-50)
• Quadrature methods ineffective
• Monte-Carlo methods fail for small integral values (10−2, 10−3)
Examples in Probability and Statistics
Density Functions
Two example density functions on ℝ:
1. Weibull-like: 𝑤𝛼(𝑥)= 𝐾𝑤,𝛼exp(−|𝑥|𝛼/𝛼) 2. Student: 𝑠𝛼(𝑥)= 𝐾𝑠,𝛼(1+𝑥2/𝛼)−(𝛼+1)/2
Asymptotic Behavior of Integrals
For domain 𝐴𝑡= {𝑥 ∈ ℝ𝑑:⟨𝐶𝑥,𝑥⟩ > 𝑡}:
For 𝛼 ≠ 2:
∫ ∏𝑤𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(𝛼,𝑑,𝐶)𝑒−𝑡𝛼/2/𝑐2𝑡−(𝛼−2)𝑑/(4−𝛼)
For 𝛼 = 2:
∫ ∏𝑤2
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(2,𝑑,𝐶)𝑒−𝑡/𝜆𝑡(𝑘−1)/2
where 𝑘 is the dimension of the eigensubspace for the largest eigenvalue 𝜆 of 𝐶𝑇+𝐶.
For Student distribution:
∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝐾𝑠,𝛼𝛼(𝛼+1)/22𝑡−𝛼/2 ∑ 𝐶𝑖,𝑖
𝛼/2
𝑖:𝐶𝑖,𝑖>0
Applications to Random Matrices
For 𝐴𝑡= {𝑥 ∈ 𝑀(𝑛,ℝ):det𝑥 > 𝑡}:
Weibull-like distribution:
∫ ∏ 𝑤𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ {𝑐𝑒−𝑡𝛼/𝑛𝑡(𝛼(𝑛2−1)−2𝑛2)/2𝑛 if 𝛼 ≠ 2
𝑐𝑒−𝑛𝑡2/2𝑡(𝑛2−𝑛−2)/2 if 𝛼 = 2
Student distribution:
∫ ∏ 𝑠𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ 𝑐(log𝑡)𝑛−1
𝑡𝛼
Conditional Distributions and Gibbs Conditioning
Key question: Distribution of 𝑛 points in ℝ𝑛 given the volume of their parallelogram.
Examples of Conditional Distributions
Example 1: Standard Normal Variables
For 𝑋,𝑌 independent standard normal, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (1/2,1/2) in probability
Example 2: Exponential Variables
For 𝑋,𝑌 independent exponential, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → Uniform on {𝛼(1,0)+(1−𝛼)(0,1):𝛼 ∈ [0,1]}
Example 3: Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (𝑃𝑥+𝑃𝑦)/2
where 𝑃𝑥,𝑃𝑦 are Pareto distributions on x and y axes respectively.
Example 4: Product of Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
(Distribution to be analyzed)
Key insights:
• Limiting conditional distributions relate to tail behavior
• Geometry of level sets vs. conditioning set is crucial
• Naive interpretation of pictures can be misleading
Key motivation: Study behavior of integrals of the form
∫𝑓
𝐴(𝑥)𝑑𝑥, 𝐴 ⊂ ℝ𝑑
for sets 𝐴 far away from the origin.
Applications
• Statistics and probability theory
• Other areas of mathematics where such integrals arise
Challenges
• Difficult to compute numerically for high dimensions (10-50)
• Quadrature methods ineffective
• Monte-Carlo methods fail for small integral values (10−2, 10−3)
Examples in Probability and Statistics
Density Functions
Two example density functions on ℝ:
1. Weibull-like: 𝑤𝛼(𝑥)= 𝐾𝑤,𝛼exp(−|𝑥|𝛼/𝛼) 2. Student: 𝑠𝛼(𝑥)= 𝐾𝑠,𝛼(1+𝑥2/𝛼)−(𝛼+1)/2
Asymptotic Behavior of Integrals
For domain 𝐴𝑡= {𝑥 ∈ ℝ𝑑:⟨𝐶𝑥,𝑥⟩ > 𝑡}:
For 𝛼 ≠ 2:
∫ ∏𝑤𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(𝛼,𝑑,𝐶)𝑒−𝑡𝛼/2/𝑐2𝑡−(𝛼−2)𝑑/(4−𝛼)
For 𝛼 = 2:
∫ ∏𝑤2
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝑐(2,𝑑,𝐶)𝑒−𝑡/𝜆𝑡(𝑘−1)/2
where 𝑘 is the dimension of the eigensubspace for the largest eigenvalue 𝜆 of 𝐶𝑇+𝐶.
For Student distribution:
∫ ∏𝑠𝛼
𝑑
𝑖=1
𝐴𝑡(𝑥𝑖)𝑑𝑥 ∼ 𝐾𝑠,𝛼𝛼(𝛼+1)/22𝑡−𝛼/2 ∑ 𝐶𝑖,𝑖
𝛼/2
𝑖:𝐶𝑖,𝑖>0
Applications to Random Matrices
For 𝐴𝑡= {𝑥 ∈ 𝑀(𝑛,ℝ):det𝑥 > 𝑡}:
Weibull-like distribution:
∫ ∏ 𝑤𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ {𝑐𝑒−𝑡𝛼/𝑛𝑡(𝛼(𝑛2−1)−2𝑛2)/2𝑛 if 𝛼 ≠ 2
𝑐𝑒−𝑛𝑡2/2𝑡(𝑛2−𝑛−2)/2 if 𝛼 = 2
Student distribution:
∫ ∏ 𝑠𝛼
𝑛
𝑖,𝑗=1
𝐴𝑡(𝑥𝑖,𝑗)𝑑𝑥𝑖,𝑗 ∼ 𝑐(log𝑡)𝑛−1
𝑡𝛼
Conditional Distributions and Gibbs Conditioning
Key question: Distribution of 𝑛 points in ℝ𝑛 given the volume of their parallelogram.
Examples of Conditional Distributions
Example 1: Standard Normal Variables
For 𝑋,𝑌 independent standard normal, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (1/2,1/2) in probability
Example 2: Exponential Variables
For 𝑋,𝑌 independent exponential, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → Uniform on {𝛼(1,0)+(1−𝛼)(0,1):𝛼 ∈ [0,1]}
Example 3: Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋+𝑌 > 𝑡:
(𝑋,𝑌)/𝑡 → (𝑃𝑥+𝑃𝑦)/2
where 𝑃𝑥,𝑃𝑦 are Pareto distributions on x and y axes respectively.
Example 4: Product of Cauchy Variables
For 𝑋,𝑌 independent Cauchy, given 𝑋𝑌 > 𝑡:
(Distribution to be analyzed)
Key insights:
• Limiting conditional distributions relate to tail behavior
• Geometry of level sets vs. conditioning set is crucial
• Naive interpretation of pictures can be misleading
