On the measure concentration of infinitely divisible distributions (2310.03471v2)

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