ON THE MEASURE CONCENTRATION OF INFINITELY DIVISIBLE DISTRIBUTIONS  

作　　者：Jing ZHANG Zechun HU Wei SUN 张静;胡泽春;孙玮

机构地区：[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

出　　处：《Acta Mathematica Scientia》2025年第2期473-492,共20页数学物理学报(B辑英文版)

基　　金：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);the Natural Sciences and Engineering Research Council of Canada(4394-2018).

摘　　要：Let I be the set of all infinitely divisible random variables with finite second moments,I_(0)={X∈I;Var(X)>0},P_(I)=inf_(x∈I)P{|X-E[X]|≤√Var(X)}and P_(I_(0))=inf P{|X-E[X]|<√Var(X)}.Firstly,we prove that P_(I)≥P_(I_(0))>0.Secondly,we find_(x∈I_(0))the exact values of inf P{|X-E[X]|≤√Var(X)}and inf P{|X-E[X]|<√Var(X)}for the cases that 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_(I)≤e^(-1)^(∞)∑_(k=0)1/2^(2k)(k!)^(2)≈0.46576 and P_(I_(0))≤e^(-1)≈0.36788.

关 键 词：measure concentration infinitely divisible distribution geometric distribution Poisson distribution Berry-Esseen theorem 

分 类 号：O211[理学—概率论与数理统计]