The Lukacs-Olkin-Rubin theorem on symmetric cones through Gleason's theorem

TL;DR

This paper extends the Lukacs characterization of the Wishart distribution to non-octonion symmetric cones of rank greater than 2, using Gleason's theorem to weaken smoothness assumptions.

Contribution

It introduces a new solution to the Olkin-Baker functional equation on symmetric cones, leveraging Gleason's theorem to relax regularity conditions.

Findings

Proves Lukacs characterization on certain symmetric cones

Provides a new approach to solving functional equations on cones

Weakens smoothness assumptions in distribution characterization

Abstract

We prove the Lukacs characterization of the Wishart distribution on non-octonion symmetric cones of rank greater than 2. We weaken the smoothness assumptions in the version of the Lukacs theorem of [Bobecka-Weso{\l}owski, Studia Math. 152 (2002), 147-160]. The main tool is a new solution of the Olkin-Baker functional equation on symmetric cones, under the assumption of continuity of respective functions. It was possible thanks to the use of Gleason's theorem.