On the measure concentration of infinitely divisible distributions

Jing Zhang, Ze-Chun Hu, Wei Sun

Published 2023-10-05Version 1

Let ${\cal I}$ be the set of all infinitely divisible random variables\ with finite second moments, ${\cal I}_0=\{X\in{\cal I}:{\rm Var}(X)>0\}$, $P_{\cal I}=\inf_{X\in{\cal I}}P\{|X-E[X]|\le \sqrt{{\rm Var}(X)}\}$ and $P_{{\cal I}_0}=\inf_{X\in{\cal I}_0} P\{|X-E[X]|< \sqrt{{\rm Var}(X)}\}$. We prove that $P_{{\cal I}}\ge P_{{\cal I}_0}>0$. Further, we use geometric and Poisson distributions to investigate the values of $P_{\cal I}$ and $P_{{\cal I}_0}$. In particular, we show that $P_{\cal I}\le e^{-1}\sum_{k=0}^{\infty}\frac{1}{2^{2k}(k!)^2}\approx 0.46576$ and $P_{{\cal I}_0}\le e^{-1}\approx 0.36788$.