Regression version of the Matsumoto-Yor type characterization of the gamma and Kummer distributions

In this paper we study a Matsumoto-Yor type property for the gamma and Kummer inde- pendent variables discovered in Koudou and Vallois (2012). We prove that constancy of regressions of U = (1 + 1/(X + Y ))=(1 + 1/X) given V = X + Y and of 1/U given V , where X and Y are indepen- dent 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.