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 \cite{MW04}. 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 ${\Phi }_r$. For a random vector $\mathbf{S}$ having a $p$-variate tree-Kummer distribution and any root $r$, we prove that ${\Phi }_r(\mathbf{S})$ has independent components. Moreover, we show that if $\mathbf{S}$ is a random vector in $(0,\infty {)}^p$ and for any leaf $r$ of the tree the components of ${\Phi }_r(\mathbf{S})$ 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 \cite{HV15}. It states that if $X$ and $Y$ are independent random variables having Kummer and Gamma distributions (with suitably related parameters) and $T:(0,\infty {)}^2\to (0,\infty {)}^2$ 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.

17 pages, 2 figures

References in corpus (4)

• Parameter Priors for Directed Acyclic Graphical Models and the Characterization of Several Probability Distributions
• Independence properties of the Matsumoto--Yor type
• The Matsumoto--Yor Property and Its Converse on Symmetric Cones
• Kernel Oriented Generator Distribution

Cited by in corpus (3)

• Change of measure technique in characterizations of the Gamma and Kummer distributions
• A Matsumoto-Yor characterization for Kummer and Wishart random matrices
• Regression conditions that characterizes free--Poisson and free--Kummer distributions