Geodesic
On Random Variables with the Same Distribution Type as Their Random sum
Publications de l'Institut Mathématique, _N_S_35 (1984) no. 49, p. 161

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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 $\nu$.
Classification : 60E99 
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     author = {Slobodanka Janji\'c},
     title = {On {Random} {Variables} with the {Same} {Distribution} {Type} as {Their} {Random} sum},
     journal = {Publications de l'Institut Math\'ematique},
     pages = {161 },
     publisher = {mathdoc},
     volume = {_N_S_35},
     number = {49},
     year = {1984},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/PIM_1984_N_S_35_49_a21/}
}
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Slobodanka Janjić. On Random Variables with the Same Distribution Type as Their Random sum. Publications de l'Institut Mathématique, _N_S_35 (1984) no. 49, p. 161 . http://geodesic.mathdoc.fr/item/PIM_1984_N_S_35_49_a21/