A note on a characterization of the Poisson and uniform distributions

It is shown that if U, X are independent random variables, X≥0, U is uniformly distributed in (0,1), and X satisfies the equation UX+2∼U+X, then X−2 has the Poisson distribution with the parameter equal one. The above equation also characterizes the uniform distribution if X−2 is a Poisson random variable. Moreover, a multivariate generalization is given. Proceedings of the XVI Seminar on Stability Problems for Stochastic Models, Part I, Eger, Hungary, 1994.