Variation comparison between infinitely divisible distributions and the normal distribution

Ping Sun, Ze-Chun Hu, Wei Sun

Published 2023-04-22Version 1

Let $X$ be a continuous random variable with finite second moment. We investigate the inequality: $P\{|X-E[X]|\le \sqrt{{\rm Var}(X)}\}\ge P\{|Z|\le 1\}$, where $Z$ is a standard normal random variable. We prove that this inequality holds for many familiar infinitely divisible distributions including the log-normal distribution, student's $t$-distribution and the inverse Gaussian distribution. Numerical results are given to show that the inequality with continuity correction also holds for some infinitely divisible discrete distributions.