On the Matsumoto–Yor type regression characterization of the gamma and Kummer distributions

Abstract

In this paper we study a Matsumoto–Yor type property for the gamma and Kummer independent variables discovered by Koudou and Vallois (2012). We prove that constancy of regressions of U=(1+(X+Y)−1)/(1+X−1) given V=X+Y and of U−1 given V, where X and Y are independent and positive random variables, characterizes the gamma and Kummer distributions. This result completes characterizations by independence of U and V obtained, under smoothness assumptions for densities, in Koudou and Vallois (2011, 2012). Since we work with differential equations for the Laplace transforms, no density assumptions are needed.