Kummer and gamma laws through independences on trees—Another parallel with the Matsumoto–Yor property

Abstract

The paper develops a rather unexpected parallel to the multivariate Matsumoto–Yor (MY) property on trees considered in Massam and Wesołowski (2004). The parallel concerns a multivariate version of the Kummer distribution, which is generated by a tree. Given a tree of size p , we direct it by choosing a vertex, say r , as a root. With such a directed tree we associate a map Φ r . For a random vector S having a p -variate tree-Kummer distribution and any root r , we prove that Φ r ( S ) has independent components. Moreover, we show that if S is a random vector in ( 0 , ∞ ) p and for any leaf r of the tree the components of Φ r ( S ) are independent, then one of these components has a Gamma distribution and the remaining p − 1 components have Kummer distributions. Our point of departure is a relatively simple independence property due to Hamza and Vallois (2016). It states that if X and Y are independent random variables having Kummer and Gamma distributions (with suitably related parameters) and T : ( 0 , ∞ ) 2 → ( 0 , ∞ ) 2 is the involution defined by T ( x , y ) = ( y / ( 1 + x ) , x + x y / ( 1 + x ) ) , then the random vector T ( X , Y ) has also independent components with Kummer and gamma distributions. By a method inspired by a proof of a similar result for the MY property, we show that this independence property characterizes the gamma and Kummer laws.