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

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)U=(1+(X+Y)−1)/(1+X−1) given V=X+YV=X+Y and of U−1U−1 given VV, where XX and YY are independent and positive random variables, characterizes the gamma and Kummer distributions. This result completes characterizations by independence of UU and VV 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.