ON A CLASS OF CHARACTERIZATION PROBLEMS FOR RANDOM CONVEX COMBINATIONS

LUDWIG BARINGHAUS AND RUDOLF GRÜBEL

Institut für Mathematische Stochastik, Universität Hannover, Postfach 6009, 30060 Hannover, Germany

(Received October 20, 1995; revised November 8, 1996)

Abstract. We consider stochastic equations of the form X =_{d} W_{1}X +W_{2}X′, where (W_{1}, W_{2}), X and X′ are independent, ‘=_{d}’ denotes equality in distribution, EW_{1} + EW_{2} = 1 and X =_{d} X′. We discuss existence, uniqueness and stability of the solutions, using contraction arguments and an approach based on moments. The case of {0, 1}-valued W_{1} and constant W_{2} leads to a characterization of exponential distributions.

Key words and phrases: Stochastic difference equations, exponential distributions, characterization problems, contractions.