关于无穷可分分布的浓度度量
概率论
2023-10-18 v2
摘要
令I {\cal I} I I 0 = { X ∈ I : V a r ( X ) > 0 } {\cal I}_0=\{X\in{\cal I}:{\rm Var}(X)>0\} I 0 = { X ∈ I : Var ( X ) > 0 } P I = inf X ∈ I P { ∣ X − E [ X ] ∣ ≤ V a r ( X ) } P_{\cal I}=\inf_{X\in{\cal I}}P\{|X-E[X]|\le \sqrt{{\rm Var}(X)}\} P I = inf X ∈ I P { ∣ X − E [ X ] ∣ ≤ Var ( X ) } P I 0 = inf X ∈ I 0 P { ∣ X − E [ X ] ∣ < V a r ( X ) } P_{{\cal I}_0}=\inf_{X\in{\cal I}_0} P\{|X-E[X]|< \sqrt{{\rm Var}(X)}\} P I 0 = inf X ∈ I 0 P { ∣ X − E [ X ] ∣ < Var ( X ) } P I ≥ P I 0 > 0 P_{{\cal I}}\ge P_{{\cal I}_0}>0 P I ≥ P I 0 > 0 J \cal J J inf X ∈ J P { ∣ X − E [ X ] ∣ ≤ V a r ( X ) } \inf_{X\in{\cal J}}P\{|X-E[X]|\le \sqrt{{\rm Var}(X)}\} inf X ∈ J P { ∣ X − E [ X ] ∣ ≤ Var ( X ) } inf X ∈ J P { ∣ X − E [ X ] ∣ < V a r ( X ) } \inf_{X\in\cal J} P\{|X-E[X]|< \sqrt{{\rm Var}(X)}\} inf X ∈ J P { ∣ X − E [ X ] ∣ < Var ( X ) } P I ≤ e − 1 ∑ k = 0 ∞ 1 2 2 k ( k ! ) 2 ≈ 0.46576 P_{\cal I}\le e^{-1}\sum_{k=0}^{\infty}\frac{1}{2^{2k}(k!)^2}\approx 0.46576 P I ≤ e − 1 ∑ k = 0 ∞ 2 2 k ( k ! ) 2 1 ≈ 0.46576 P I 0 ≤ e − 1 ≈ 0.36788 P_{{\cal I}_0}\le e^{-1}\approx 0.36788 P I 0 ≤ e − 1 ≈ 0.36788
引用
@article{arxiv.2310.03471,
title = {On the measure concentration of infinitely divisible distributions},
author = {Jing Zhang and Ze-Chun Hu and Wei Sun},
journal= {arXiv preprint arXiv:2310.03471},
year = {2023}
}