A Limit Property of the Geometric Distribution

Abstract

Let random variables $X^{\ast}, X$ have discrete distributions on the nonnegative integers and let \begin{eqnarray*} {\bf P}\{X=k\}=c\sum^{\infty}_{j=k}{\bf P}\{X^{\ast}=j\},\qq k=0,1,2,\dots, \end{eqnarray*} with c a proper constant. Repeated summations of this type are investigated. The limit distribution is geometric for a wide class of parent distributions.