Variation comparison between infinitely divisible distributions and the normal distribution

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