Variation comparison between infinitely divisible distributions and the normal distribution

Abstract: Let $X$ be a 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 continuous distributions including the Laplace, Gumbel, Logistic, Pareto, infinitely divisible Weibull, log-normal, student's $t$ and inverse Gaussian distributions. Numerical results are given to show that the inequality with continuity correction also holds for some infinitely divisible discrete distributions.