具有独立坐标的随机点凸包期望测度的阈值
概率论
2023-09-26 v2 泛函分析
度量几何
摘要
设 μ \mu μ R {\mathbb R} R N > n N>n N > n R n {\mathbb R}^n R n N N N X ⃗ 1 , … , X ⃗ N \vec{X}_1,\ldots ,\vec{X}_N X 1 , … , X N μ \mu μ μ \mu μ Λ μ ∗ \Lambda_{\mu}^{\ast} Λ μ ∗ lim x ↑ x ∗ − ln μ ( [ x , ∞ ) ) Λ μ ∗ ( x ) = 1 , \lim\limits_{x\uparrow x^{\ast}}\dfrac{-\ln \mu ([x,\infty ))}{\Lambda_{\mu}^{\ast}(x)}=1, x ↑ x ∗ lim Λ μ ∗ ( x ) − ln μ ([ x , ∞ )) = 1 , x ∗ = sup { x ∈ R : μ ( [ x , ∞ ) ) > 0 } x^{\ast}=\sup\{x\in\mathbb{R}\colon \mu([x,\infty))>0\} x ∗ = sup { x ∈ R : μ ([ x , ∞ )) > 0 } K N : = c o n v { X ⃗ 1 , … , X ⃗ N } K_N:={\rm conv}\bigl\{\vec{X}_1,\ldots,\vec{X}_N\bigr\} K N := conv { X 1 , … , X N } R n {\mathbb R}^n R n μ n \mu_n μ n ν p n = ν p ⊗ n \nu_p^n=\nu_p^{\otimes n} ν p n = ν p ⊗ n p ≥ 1 p\geq 1 p ≥ 1 ( 2 γ p ) − n exp ( − ∥ x ∥ p p ) (2\gamma_p)^{-n}\exp(-\|x\|_p^p) ( 2 γ p ) − n exp ( − ∥ x ∥ p p ) ∥ ⋅ ∥ p \|\cdot\|_p ∥ ⋅ ∥ p ℓ p n \ell_p^n ℓ p n γ p = Γ ( 1 + 1 / p ) \gamma_p=\Gamma(1+1/p) γ p = Γ ( 1 + 1/ p )
引用
@article{arxiv.2303.02465,
title = {Threshold for the expected measure of the convex hull of random points with independent coordinates},
author = {Minas Pafis},
journal= {arXiv preprint arXiv:2303.02465},
year = {2023}
}