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

Gérard Letac, Mauro Piccioni, Jacek Wesołowski

TL;DR

This work extends the Matsumoto–Yor independence property to the matrix variate setting by employing a matrix MGIG framework on the SPD cone $Ω_+$. Through a functional equation approach linked to the Jacobian of a matrix Yang–Baxter map $φ^{(α,β)}$, the authors prove that the MGIG law is characterized by an independence property induced by $φ^{(α,β)}$, and that the corresponding matrix map is a parametric Yang–Baxter map. They explicitly identify the density parameters $a,b$ and the common scalar $λ$ by analyzing log-densities, derivatives, and scaling relations, culminating in a full characterization: $X∼MGIG(λ,α a,b)$, $Y∼MGIG(-λ,a,β b)$, $U∼MGIG(λ,α b,a)$, and $V∼MGIG(-λ,b,β a)$. The results bridge probabilistic independence properties with integrable systems in the matrix setting and open paths to symmetric-cone extensions and broader noncommutative generalizations.

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 $H_{III,B}^{(α,β)}$, 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 $(α,β)=(1,0)$ the independence reduces to the classical Matsumoto-Yor property, \cite{MatYor2001}. In \cite{LetWes2024} we proposed an extension of $H_{III,B}^{(α,β)}$ 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 $H_{III,B}^{(α,β)}$ we use, is a parametric Yang-Baxter map.

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

TL;DR

This work extends the Matsumoto–Yor independence property to the matrix variate setting by employing a matrix MGIG framework on the SPD cone . Through a functional equation approach linked to the Jacobian of a matrix Yang–Baxter map , the authors prove that the MGIG law is characterized by an independence property induced by , and that the corresponding matrix map is a parametric Yang–Baxter map. They explicitly identify the density parameters and the common scalar by analyzing log-densities, derivatives, and scaling relations, culminating in a full characterization: , , , and . The results bridge probabilistic independence properties with integrable systems in the matrix setting and open paths to symmetric-cone extensions and broader noncommutative generalizations.

Abstract

Sasada and Uozumi, \cite{SasUoz2024}, identified independence preserving quadrirational parametric Yang-Baxter maps, see \eqref{YBEQ}, on . In particular, the map denoted there by , see \eqref{CS}, was connected to the independence preserving property of the GIG distributions on . 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 the independence reduces to the classical Matsumoto-Yor property, \cite{MatYor2001}. In \cite{LetWes2024} we proposed an extension of 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 we use, is a parametric Yang-Baxter map.
Paper Structure (14 sections, 4 theorems, 116 equations)

This paper contains 14 sections, 4 theorems, 116 equations.

Table of Contents

  1. INTRODUCTION
  2. CHARACTERIZATION AND ITS PROOF
  3. A functional equation on $\Omega_+$ and its counterpart on $(0,\infty)$.
  4. Solution of equation \ref{['HYPSTR']} for the quadruple $(\mathfrak h^{(w)}_W)_{W\in\{X,Y,U,V\}}$.
  5. Scaling arguments of $\mathfrak a$, $\mathfrak b$, $\mathfrak l$ and $\mathfrak C_W$, $W\in\{X,Y,U,V\}$.
  6. Separate equations for $\mathfrak a$, $\mathfrak b$, $\mathfrak l$ and $\mathfrak C_W$, $W\in\{X,Y,U,V\}$.
  7. Some observations about derivatives $D_x$ and $D_y$.
  8. Identification of $\mathfrak a$.
  9. Identification of $\mathfrak b$.
  10. Identification of $\mathfrak l$.
  11. Identification of $\mathfrak C_W$, $W\in\{X,Y,U,V\}$.
  12. Identification of distributions of $U$, $V$, $X$ and $Y$.
  13. BIBLIOGRAPHICAL COMMENTS AND OPEN PROBLEMS
  14. Proof of Theorem \ref{['YBTH']}

Key Result

Theorem 1.1

LetWes2024 Let $X$ and $Y$ be independent $\Omega_+$-valued random matrices with distributions $\mathrm{MGIG}(\lambda,\alpha a,b)$ and $\mathrm{MGIG}(\lambda,\beta b,a)$, respectively. Let $\phi^{(\alpha,\beta)}:\Omega_+^2\to\Omega_+^2$ be defined by and let Then $U$ and $V$ are independent, $U\sim\mathrm{MGIG}(\lambda,\alpha b,a)$ and $V=\mathrm{MGIG}(\lambda,\beta a,b)$.

Theorems & Definitions (7)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1
  • Remark 2.2
  • proof : Proof of Theorem \ref{['charac']}.
  • proof