On Moments of Infinitely Divisible Distribution Functions

Abstract

Let $F(x)$ be an infinitely divisible distribution function with a Levy-Khintchine function $G(u)$ and let $p$ be any positive number. It is shown that $F(x)$ has an absolute moment of the $p$th order if and only if $G(u)$ has an absolute moment of the $p$th order, and $F(x)$ has an exponential moment of the $p$th order if and only if $G(u)$ has an exponential moment of the $p$th order. This result generalizes a theorem of J. M. Shapiro. Other related results are also obtained.