ON THE MEASURE CONCENTRATION OF INFINITELY DIVISIBLE DISTRIBUTIONS

ON THE MEASURE CONCENTRATION OF INFINITELY DIVISIBLE DISTRIBUTIONS

ON THE MEASURE CONCENTRATION OF INFINITELY DIVISIBLE DISTRIBUTIONS

ON THE MEASURE CONCENTRATION OF INFINITELY DIVISIBLE DISTRIBUTIONS

ON THE MEASURE CONCENTRATION OF INFINITELY DIVISIBLE DISTRIBUTIONS

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doi:10.1007/s10473-025-0211-x

数学物理学报(英文版) ›› 2025, Vol. 45 ›› Issue (2): 473-492.doi: 10.1007/s10473-025-0211-x

Jing Zhang¹, Zechun Hu², Wei Sun^3,*

1. School of Mathematics and Statistics, Hainan Normal University, Haikou 571158, China;
2. College of Mathematics, Sichuan University, Chengdu 610065, China;
3. Department of Mathematics and Statistics, Concordia University, Montreal H3G 1M8, Canada

收稿日期:2023-10-18 修回日期:2024-01-08 出版日期:2025-03-25 发布日期:2025-05-08

Jing Zhang¹, Zechun Hu², Wei Sun^3,*

1. School of Mathematics and Statistics, Hainan Normal University, Haikou 571158, China;
2. College of Mathematics, Sichuan University, Chengdu 610065, China;
3. Department of Mathematics and Statistics, Concordia University, Montreal H3G 1M8, Canada

Received:2023-10-18 Revised:2024-01-08 Online:2025-03-25 Published:2025-05-08
Contact: *Wei Sun, E-mail: wei.sun@concordia.ca
About author:Jing Zhang, E-mail: zh_jing0820@hotmail.com;Zechun Hu, E-mail: zchu@scu.edu.cn
Supported by:
This work was supported by the National Natural Science Foundation of China (12161029, 12171335), the National Natural Science Foundation of Hainan Province (121RC149), the Science Development Project of Sichuan University (2020SCUNL201) and the Natural Sciences and Engineering Research Council of Canada (4394-2018).

摘要： Let ${\cal I}$ be the set of all infinitely divisible random variables with finite second moments, ${\cal I}_0=\{X\in{\cal I}:{\rm Var}(X)>0\}$ , $P_{\cal I}=\inf\limits_{X\in{\cal I}}P\{|X-E[X]|\le \sqrt{{\rm Var}(X)}\}$ and $P_{{\cal I}_0}=\inf\limits_{X\in{\cal I}_0} P\{|X-E[X]|< \sqrt{{\rm Var}(X)}\}$ . Firstly, we prove that $P_{{\cal I}}\ge P_{{\cal I}_0}>0$ . Secondly, we find the exact values of $\inf\limits_{X\in{\cal J}}P\{|X-E[X]|\le \sqrt{{\rm Var}(X)}\}$ and $\inf\limits_{X\in\cal J} P\{|X-E[X]|< \sqrt{{\rm Var}(X)}\}$ for the cases that $\cal J$ is the set of all geometric random variables, symmetric geometric random variables, Poisson random variables and symmetric Poisson random variables, respectively. As a consequence, we obtain that $P_{\cal I}\le {\rm e}^{-1}\sum\limits_{k=0}^{\infty}\frac{1}{2^{2k}(k!)^2}\approx 0.46576$ and $P_{{\cal I}_0}\le {\rm e}^{-1}\approx 0.36788$ .

关键词: measure concentration, infinitely divisible distribution, geometric distribution, Poisson distribution, Berry-Esseen theorem

Abstract: Let ${\cal I}$ be the set of all infinitely divisible random variables with finite second moments, ${\cal I}_0=\{X\in{\cal I}:{\rm Var}(X)>0\}$ , $P_{\cal I}=\inf\limits_{X\in{\cal I}}P\{|X-E[X]|\le \sqrt{{\rm Var}(X)}\}$ and $P_{{\cal I}_0}=\inf\limits_{X\in{\cal I}_0} P\{|X-E[X]|< \sqrt{{\rm Var}(X)}\}$ . Firstly, we prove that $P_{{\cal I}}\ge P_{{\cal I}_0}>0$ . Secondly, we find the exact values of $\inf\limits_{X\in{\cal J}}P\{|X-E[X]|\le \sqrt{{\rm Var}(X)}\}$ and $\inf\limits_{X\in\cal J} P\{|X-E[X]|< \sqrt{{\rm Var}(X)}\}$ for the cases that $\cal J$ is the set of all geometric random variables, symmetric geometric random variables, Poisson random variables and symmetric Poisson random variables, respectively. As a consequence, we obtain that $P_{\cal I}\le {\rm e}^{-1}\sum\limits_{k=0}^{\infty}\frac{1}{2^{2k}(k!)^2}\approx 0.46576$ and $P_{{\cal I}_0}\le {\rm e}^{-1}\approx 0.36788$ .

Key words: measure concentration, infinitely divisible distribution, geometric distribution, Poisson distribution, Berry-Esseen theorem

中图分类号:

60E07

Jing Zhang, Zechun Hu, Wei Sun. ON THE MEASURE CONCENTRATION OF INFINITELY DIVISIBLE DISTRIBUTIONS[J]. 数学物理学报(英文版), 2025, 45(2): 473-492.

Jing Zhang, Zechun Hu, Wei Sun. ON THE MEASURE CONCENTRATION OF INFINITELY DIVISIBLE DISTRIBUTIONS[J]. Acta mathematica scientia,Series B, 2025, 45(2): 473-492.

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$F$ -distribution and the normal distribution. arXiv:2305.13615

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