Variation comparison between infinitely divisible distributions and the normal distribution

Let $X$ be a random variable with finite second moment. We investigate the inequality: $P\{\mathrm{\mid }X-E[X]\mathrm{\mid }\le \sqrt{\mathrm{V}\mathrm{a}\mathrm{r}(X)}\}\ge P\{\mathrm{\mid }Z\mathrm{\mid }\le 1\}$, 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.