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

The paper develops a rather unexpected parallel to the multivariate Matsumoto–Yor (MY) property on trees considered in Massam and Wesolowski (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 Φr. For a random vector S having a p-variate tree-Kummer distribution and any root r, we prove that Φr(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 Φr(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+xy/(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.

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