Characterization of beta distribution on symmetric cones

TL;DR

This paper extends a known characterization of beta distributions to the setting of symmetric cones, identifying conditions under which certain transformations of independent variables are also independent.

Contribution

It generalizes the classical beta distribution characterization to symmetric cones, broadening understanding of distribution properties in higher-dimensional algebraic structures.

Findings

Independence of transformed variables characterizes beta distributions on symmetric cones.

The characterization involves specific parameter conditions for the distributions.

The results unify and extend classical beta distribution properties to a broader mathematical setting.

Abstract

In the paper we generalize the following characterization of beta distribution to the symmetric cone setting: let $X$ and $Y$ be independent, non-degenerate random variables with values in $(0,1)$, then $U=1-XY$ and $V=\frac{1-X}{U}$ are independent if and only if there exist positive numbers $p_i$, $i=1,2,3$, such that $X$ and $Y$ follow beta distributions with parameters $(p_1+p_3,p_2)$ and $(p_3,p_1)$, respectively.