Let ${\cal I}$ be the set of all infinitely divisible random variables with finite second moments, ${\cal I}_0=\{X\in{\cal I}:{\rm Var}(X)>0\}$, $P_{\cal I}=\inf\limits_{X\in{\cal I}}P\{|X-E[X]|\le \sqrt{{\rm Var}(X)}\}$ and $P_{{\cal I}_0}=\inf\limits_{X\in{\cal I}_0} P\{|X-E[X]|< \sqrt{{\rm Var}(X)}\}$. Firstly, we prove that $P_{{\cal I}}\ge P_{{\cal I}_0}>0$. Secondly, we find the exact values of $\inf\limits_{X\in{\cal J}}P\{|X-E[X]|\le \sqrt{{\rm Var}(X)}\}$ and $\inf\limits_{X\in\cal J} P\{|X-E[X]|< \sqrt{{\rm Var}(X)}\}$ for the cases that $\cal J$ is the set of all geometric random variables, symmetric geometric random variables, Poisson random variables and symmetric Poisson random variables, respectively. As a consequence, we obtain that $P_{\cal I}\le {\rm e}^{-1}\sum\limits_{k=0}^{\infty}\frac{1}{2^{2k}(k!)^2}\approx 0.46576$ and $P_{{\cal I}_0}\le {\rm e}^{-1}\approx 0.36788$.
Jing ZHANG
,
Zechun HU
,
Wei SUN
. ON THE MEASURE CONCENTRATION OF INFINITELY DIVISIBLE DISTRIBUTIONS[J]. Acta mathematica scientia, Series B, 2025
, 45(2)
: 473
-492
.
DOI: 10.1007/s10473-025-0211-x
[1] Ethier S N, Kurtz T G.Markov Processes: Characterization and Convergence. New York: John Wiley and Sons, 1986
[2] Feller W.An Introduction to Probability Theory and Its Applications, Volumn II. New York: John Wiley and Sons, 1970
[3] Hu Z C, Sun W. Hunt's hypothesis (H) andGetoor's conjecture for Lévy processes. Stochastic Process Appl, 2012, 122: 2319-2328
[4] Hu Z C, Sun W. Hunt's hypothesis (H) for Markov processes: survey and beyond. Acta Math Sin Engl Ser, 2021, 37: 491-508
[5] Nakamura T. A complete Riemann zeta distribution and the Riemann hypothesis. Bernoulli, 2015, 21: 604-617
[6] Nakamura T, Suzuki M. On infinitely divisible distributions related to the Riemann hypothesis. Statist Probab Lett, 2023, 201: 109889
[7] Sato K.Lévy Processes and Infinitely Divisible Distributions. Cambridge: Cambridge University Press, 1999
[8] Shiganov I S. Refinement of the upper bound of a constant in the remainder term of the central limit theorem. J Soviet Math, 1986, 35: 2545-2550
[9] Skorokhod A V. Limit theorems for stochastic processes. Theory Probab Appl, 1956, 1: 261-290
[10] Sun P, Hu Z C, Sun W. The infimum values of two probability functions for the Gamma distribution. J Inequal Appl, 2024, 2024: Art 5
[11] Sun P, Hu Z C, Sun W. Variation comparison between infinitely divisible distributions and the normal distribution. Stat Papers, 2024, 65: 4405-4429
[12] Sun P, Hu Z C, Sun W.Variation comparison between the $F$-distribution and the normal distribution. arXiv:2305.13615
