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

Abstract

Abstract. The paper develops a rather unexpected parallel to the multivariate Matsumoto-Yor (MY) property on trees considered in [19]. 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 [10]. 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.

Change of measure technique in characterizations of the gamma and Kummer distributions

“…However the characterization given in [24] did not need any assumptions regarding density and its smoothness. Unfortunately, in the multivariate characterization given in [29] (Th. 4) we assumed that random vector X has density which is continuously differentiable.…”

“…In order to state an improved version of Th. 4 from [29], first, we have to recall some definitions.…”

“…are independent, U ∼ K(b, a − b, c) and V ∼ G(a, c). A related characterization assuming smoothness of densities (as well as its multivariate version with transformations defined in the language of directed trees) has been obtained in [29] (some preliminary observations have been made also in [15]). In Section 2 we provide regression version of the characterization, Th.…”

“…For independent X and Y with gamma and Kummer distributions Koudou and Vallois in [17] observed that U = (1 + (X + Y ) −1 )/(1 + X −1 ) and V = X + Y are also independent, and Hamza and Vallois in [13] observed that U = Y /(1 + X) and V = X(1 + Y /(1 + X)) are independent. In [16] and [17] characterizations related to the first property were proved, while the characterizations in the second setting have been recently given in [29]. These results were not fully satisfactory since in both cases technical assumptions on smoothness properties of densities of X and Y were needed.…”

A Matsumoto–Yor characterization for Kummer and Wishart random matrices

“…Theorem 5.3 Let X and Y be two independent random matrices valued in S + with continuous densities, which are strictly positive on S + . The random matrices U and V defined in (20) are independent if and only if X follows the matrix Kummer distribution K(a, b, σ) and Y the Wishart distribution γ(b − a, σ) for some a, b, σ with a > r−1 2 , b − a > r−1 2 and σ ∈ S + r .…”