arXiv subjects
Publications and source records attributed to Agnieszka Piliszek.
We find the asymptotic spectral distribution of random Kummer matrix. Then we formulate and prove a~free analogue of HV independence property, which is known for classical Kummer and Gamma random variables and for Kummer and Wishart matrices. We also prove a related characterization of free--Kummer and free--Poisson (Marchenko-Pastur) non--commutative random variables.
We generalize the following univariate characterization of the Kummer and Gamma distributions to the cone of symmetric positive definite matrices: let $X$ and $Y$ be independent, non-degenerate random variables valued in $(0, \infty)$, then $U= Y/(1+X)$ and $V = X(1+U)$ are independent if and only if $X$ follows the Kummer distribution and $Y$ follows the the Gamma distribution with appropriate parameters. We solve a related functional equation in the cone of symmetric positive definite matrices, which is our first main result and apply its solution to prove the characterization of Wishart and matrix-Kummer distributions, which is our second main result.
If $X$ and $Y$ are independent random variables with distributions $μ$ and $ν$ then $U=ψ(X,Y)$ and $V=ϕ(X,Y)$ are also independent for some $ψ$ and $ϕ$. Properties of this type are known for many important probability distributions $μ$ and $ν$. Also related characterization questions have been widely investigated: Let $X$ and $Y$ be independent and let $U$ and $V$ be independent. Are the distributions of $X$ and $Y$ $μ$ and $ν$, respectively? Recently two new properties and characterizations of this kind involving the Kummer distribution appeared in the literature. For independent $X$ and $Y$ with gamma and Kummer distributions Koudou and Vallois observed that $U=(1+(X+Y)^{-1})/(1+X^{-1})$ and $V=X+Y$ are also independent, and Hamza and Vallois observed that $U=Y/(1+X)$ and $V=X(1+Y/(1+X))$ are independent. In 2011 and 2012 Koudou, Vallois characterizations related to the first property were proved, while the characterizations in the second setting have been recently given in Piliszek, Wesołowski (2016). In both cases technical assumptions on smoothness properties of densities of $X$ and $Y$ were needed. In 2015, the assumption of independence of $U$ and $V$ in the first setting was weakened to constancy of regressions of $U$ and $U^{-1}$ given $V$ with no density assumptions. However, the additional assumption $\mathbb{E} X^{-1}<\infty$ was introduced. In the present paper we provide a complete answer to the characterization question in both settings without any additional technical assumptions regarding smoothness or existence of moments. The approach is, first, via characterizations exploiting some conditions imposed on regressions of $U$ given $V$, which are weaker than independence, but for which moment assumptions are necessary. Second, using a technique of change of measure we show that the moment assumptions can be avoided.
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 $Φ_r$. For a random vector ${\bf S}$ having a $p$-variate tree-Kummer distribution and any root $r$, we prove that $Φ_r({\bf S})$ 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 $Φ_r({\bf S})$ 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.