Variation comparison between infinitely divisible distributions and the normal distribution (Q6640092)

Given a standard normal random variable \(Z\), the authors show that the inequality\N\[\NP\left(|X-EX|\leq\sqrt{\text{Var}(X)}\right)\geq P\left(|Z|\leq1\right)\N\]\Nholds for certain infinitely divisible random variables \(X\). In particular, this inequality holds if \(X\) has a Laplace, Gumbel, Logistic, Pareto, infinitely divisible Weibull, Log-normal, Student's t or Inverse Gaussian distribution, though it does not necessarily hold if \(X\) has a Weibull distribution which is not infinitely divisible. Several cases where \(X\) has a discrete infinitely divisible distribution are investigated numerically, including the Geometric, Negative Binomial and Poisson distributions. In the latter two cases it is shown that a continuity correction needs to be incorporated for such an inequality to hold.