On Random Variables with the Same Distribution Type as Their Random sum

Slobodanka Janjić

Let $\xi_1,\xi_2,\dots,\xi_n,\dots$ be a sequence of nonnegative, independent, equally distributed random variables with distribution function $F(x)$ and corresponding Laplace transform $f(t)$; let $\nu$ be integer-valued random variable independent of $\xi_n$, $n=1,2,\dots$, $p_n=P(\nu=n)$, $p_0=0$, $P(s)=\sum_{n=0}^\infty s^np_n$ -- its generating function. In this paper, solutions $(P,f)$ of the following functional equation are found: $$ P(f(t))= f(c_\nu t), $$ where $c_\nu$ is a real number depending on $\nu$.