On the Measure Concentration of Infinitely Divisible Distributions

Let I be the set of all infinitely divisible random variables with finite second moments, I-0={X is an element of I:Var(X)>0},P-I=inf (X is an element of I) P{|X-E[X]|<=root Var(X)} and P-I0=inf(X is an element of I0)P{|X-E[X]|<root Var(X)}. Firstly, we prove that P-I >= P-I0>0. Secondly, we find the exact values of inf(X is an element of J)P{|X-E[X]|<=root Var(X)} and inf(X is an element of J)P{|X-E[X]|<root Var(X)} for the cases that 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-I <= e(-1)& sum;(infinity)(k=0)1/2(2k)(k!)(2)approximate to 0.46576 and PI0 <= e(-1)approximate to 0.36788.