Which distributions have the Matsumoto-Yor property?

Abstract

For four types of functions $ξ : ]0,∞[→ ]0,∞[$, we characterize the law of two independent and positive r.v.'s $X$ and $Y$ such that $U:=ξ(X+Y)$ and $V:=ξ(X)-ξ(X+Y)$ are independent. The case $ξ(x)=1/x$ has been treated by Letac and Wesolowski (2000). As for the three other cases, under the weak assumption that $X$ and $Y$ have density functions whose logarithm is locally integrable, we prove that the distribution of $(X,Y)$ is unique. This leads to Kummer, gamma and beta distributions. This improves the result obtained in [1] where more regularity was required from the densities.