Abstract

Let \(Z\) be a geometrically composed random variable of the i.i.d. (independent, identically distributed) random variables \(X_1,X_2, \dots\) with parameter \(p\) \((0<p<1)\). Extending Rényi’s characteristic theorem and a result of his own [Vietnam J. Math. 25, No. 3, 267–269 (1997; Zbl 0905.62010)], the author proves the following property: under certain conditions, the fact that \(\sqrt pZ\) and \(X_i\) \((i=1,2,\dots)\) are identically distributed random variables is a characteristic property of the set of all two-sided exponential distributions.