Characterizations of the Poisson distribution based on random splitting and random expanding

AbstractLet X be a discrete random variable with parameter λ = E[X] < ∞ and denote B(n,r,a) = (nrnr)arr(1 − a)nrn−r. Let the distribution of X be compounded with B(n,r,a ). If the resulting distribution is governed by the same law as X, then a characterization of the Poisson distribution is obtained. An alternative proof of the Rao-Rubin characterization is provided