Variation comparison between infinitely divisible distributions and the normal distribution

2023/04/22 by Sun, Ping, Hu, Ze-Chun, Sun, Wei
#60E15 #62G32 #90C15 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2304.11459

Abstract

Let X be a random variable with finite second moment. We investigate the inequality: P\|X-E[X]|≤ √\rm 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.

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