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 Wesołowski (2004). The parallel concerns a multivariate version of the Kummer distribution, which is generated by a tree. Given a tree of size pp, we direct it by choosing a vertex, say rr, as a root. With such a directed tree we associate a map ΦrΦr. For a random vector S having a pp-variate tree-Kummer distribution and any root rr, we prove that Φr(S) has independent components. Moreover, we show that if S is a random vector in (0,∞)p(0,∞)p and for any leaf rr of the tree the components of Φr(S) are independent, then one of these components has a Gamma distribution and the remaining p−1p−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 XX and YY are independent random variables having Kummer and Gamma distributions (with suitably related parameters) and T:(0,∞)2→(0,∞)2T:(0,∞)2→(0,∞)2 is the involution defined by T(x,y)=(y/(1+x),x+xy/(1+x))T(x,y)=(y/(1+x),x+xy/(1+x)), then the random vector T(X,Y)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.