On the measure concentration of infinitely divisible distributions

2023/10/05 by Zhang, Jing, Hu, Ze-Chun, Sun, Wei
#60E07 #60E15 #62G32 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2310.03471

Abstract

Let \cal I be the set of all infinitely divisible random variables with finite second moments, \cal I₀=\X∈\cal I:\rm Var(X)>0\, P_{\cal I}=inf_X∈\cal IP\|X-E[X]|≤ √\rm Var(X)\ and P_\cal I₀=inf_X∈\cal I₀ P\|X-E[X]|< √\rm Var(X)\. Firstly, we prove that P_{\cal I}≥ P_\cal I₀>0. Secondly, we find the exact values of inf_X∈\cal JP\|X-E[X]|≤ √\rm Var(X)\ and inf_{X∈\cal J} P\|X-E[X]|< √\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}≤ e^-1∑_k=0^∞\frac12^2k(k!)²≈ 0.46576 and P_\cal I₀≤ e^-1≈ 0.36788.

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