GIG random matrices and a Yang-Baxter extension of the Matsumoto-Yor property

TL;DR

This paper extends the Matsumoto-Yor property to matrix GIG distributions, characterizes these distributions via a matrix Yang-Baxter map, and demonstrates the map's integrability and independence-preserving properties.

Contribution

It introduces a matrix variate extension of a Yang-Baxter map that preserves independence of GIG matrices and characterizes these distributions through this property.

Findings

Matrix GIG distributions are characterized by the independence property.

The extended map is a parametric Yang-Baxter map.

The map preserves independence of GIG random matrices.

Abstract

Sasada and Uozumi, \cite{SasUoz2024}, identified independence preserving $[2:2]$ quadrirational parametric Yang-Baxter maps, see \eqref{YBEQ}, on $(0,\infty)$. In particular, the map denoted there by $H_{III,B}^{(\alpha,\beta)}$, see \eqref{CS}, was connected to the independence preserving property of the GIG distributions on $(0,\infty)$. Remarkably, the property appears also naturally in probabilistic integrable models of discrete Korteweg de Vries type, as observed by Croydon and Sasada, \cite{CroSas2020}. In the case of $(\alpha,\beta)=(1,0)$ the independence reduces to the classical Matsumoto-Yor property, \cite{MatYor2001}. In \cite{LetWes2024} we proposed an extension of $H_{III,B}^{(\alpha,\beta)}$ to a map on the cone of symmetric positive definite matrices of a fixed dimension, showing that such extended map preserves independence of GIG random matrices. In the present paper we prove two results: (i) the matrix GIG distributions are characterized by the independence property governed by this map; (ii) the matrix variate extension of $H_{III,B}^{(\alpha,\beta)}$ we use, is a parametric Yang-Baxter map.