A limit property of the geometric distribution

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