On the measure concentration of infinitely divisible distributions

TL;DR

This paper investigates the probability concentration of infinitely divisible distributions with finite second moments, establishing positive lower bounds and exact values for specific classes like geometric and Poisson distributions.

Contribution

It proves positive lower bounds for measure concentration in infinitely divisible distributions and computes exact probabilities for key subclasses.

Findings

Established that $P_{{ m I}} ext{ and }P_{{ m I}_0}$ are strictly positive.

Derived exact probability bounds for geometric and Poisson distributions.

Provided upper bounds for measure concentration in the class of infinitely divisible distributions.

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 $P_{{\cal I}_0}\le e^{-1}\approx 0.36788$.