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 - E [ X ] | ≤ Var ( X ) } ≥ P { | Z | ≤ 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.