Kummer and gamma laws through independences on trees - another parallel to the Matsumoto-Yor property

Agnieszka Piliszek, Jacek Wesołowski

Published 2015-10-31Version 1

The paper provides new results on the characterization of product of gamma and Kummer distributions which parallel ones known for the Matsumoto-Yor (MY) property. The departure point is an independence property discovered recently by \cite{HV15}. This property says that if $X$ and $Y$ are independent random variables of Kummer and gamma distribution (with suitable related parameters) and $T:(0,\infty)^2\to(0,\infty)^2$ is an involution defined by $${T(x,y)=\left(\tfrac{y}{1+x},x\left(1+\tfrac{y}{1+x}\right)\right)}$$ then the random vector $T(X,Y)$ has also independent components with Kummer and gamma distributions. We show, by a method inspired by a proof of a similar result for the MY property, that this independence property characterizes the gamma and Kummer law. The multivariate considerations parallel the approach to the MY property on trees which was developed in \cite{MW04}. For any tree of size $p$ and any fixed root $r$ of such a tree we design a transformation $\Phi_r$. For a random vector ${\bf S}$ having a $p$-variate tree-Kummer distribution (defined in Section \ref{TKum}) and any root $r$ of the tree we prove that $\Phi_r({\bf S})$ has independent components. Moreover, we show that if ${\bf S}$ is a random vector in $(0,\infty)^p$ and for any leaf $r$ of the tree the components of $\Phi_r({\bf S})$ are independent, then their distribution is a product of a gamma law and $p-1$ Kummer laws.