[Paper Review] Matrix Dufresne Identity

Brian Rider, Benedek Valkó|arXiv (Cornell University)|Sep 5, 2014

Random Matrices and Applications21 references3 citations

TL;DR

This paper establishes a matrix extension of the classical Dufresne identity, proving that the inverse Wishart distribution arises as the law of the integrated matrix geometric Brownian motion $ A_{ au}^{(-\mu)} = \int_0^\infty M_s M_s^T ds $, where $ M_t $ is a drifted Brownian motion on $ GL_r(\mathbb{R}) $. The key result shows that for $ 2\mu > r-1 $, $ A_{\infty}^{(-\mu)} \stackrel{\text{law}}{=} \gamma_{2\mu}^{-1} $, the standard inverse Wishart distribution with parameter $ 2\mu $, generalizing scalar Dufresne identities to matrix processes.

ABSTRACT

We prove a version of the classical Dufresne identity for matrix processes. In particular, we show that the inverse Wishart laws on the space of positive definite r x r matrices can be realized by the infinite time horizon integral of M_t times its transpose in which t -> M_t is a drifted Brownian motion on the general linear group. This solves a problem in the study of spiked random matrix ensembles which served as the original motivation for this result. Various known extensions of the Dufresne identity (and their applications) are also shown to have analogs in this setting. For example, we identify matrix valued diffusions built from M_t which generalize in a natural way the scalar processes figuring into the geometric Levy and Pitman theorems of Matsumoto and Yor.

Motivation & Objective

• To extend the classical Dufresne identity from scalar to matrix-valued processes, particularly in the context of spiked random matrix ensembles.
• To establish a matrix version of the Dufresne identity involving the law of the integral $ \int_0^\infty M_s M_s^T ds $, where $ M_t $ is a drifted Brownian motion on $ GL_r(\mathbb{R}) $.
• To generalize scalar processes from Matsumoto and Yor's geometric Lévy and Pitman theorems to matrix-valued diffusions.
• To identify a partial Burke-type property for the matrix process and explore its implications in random matrix theory.

Proposed method

• Define the matrix geometric Brownian motion $ M_t $ via the Itô SDE $ dM_t = M_t dB_t + (\frac{1}{2} + \mu)M_t dt $, with $ M_0 = I $, where $ B_t $ is an $ r \times r $ matrix of independent Brownian motions.
• Introduce the additive functional $ A_t^{(\mu)} = \int_0^t M_s M_s^T ds $, the matrix analog of the scalar running integral $ a_t^{(\mu)} $.
• Prove that for $ 2\mu > r-1 $, the law of $ A_\infty^{(-\mu)} $ is the standard inverse Wishart distribution with parameter $ 2\mu $, using properties of the multiplicative and independent increments of $ M_t $.
• Establish a matrix version of the Pitman-type identity: $ \{1/A_t^{(-\mu)}\} \stackrel{\text{law}}{=} \{1/A_t^{(\mu)} + 1/\tilde{A}_\infty^{(-\mu)}\} $, where $ \tilde{A}_\infty^{(-\mu)} $ is an independent copy.
• Use the eigenvalue decomposition and recursive structure of the Wishart distribution to derive a recursive integral representation for the density of $ A_\infty^{(-\mu)} $, involving Macdonald functions and multivariate Bessel functions.
• Apply asymptotic analysis and scaling limits to show convergence to the scalar Matsumoto–Yor results in the $ r=1 $ case, confirming consistency with known identities.

Experimental results

Research questions

• RQ1 Does the classical Dufresne identity for scalar geometric Brownian motion extend to matrix-valued processes via the integral of $ M_t M_t^T $?
• RQ2 Can the inverse Wishart distribution be realized as the law of $ \int_0^\infty M_s M_s^T ds $ for a matrix-valued diffusion $ M_t $?
• RQ3 Do matrix analogs of the geometric Lévy and Pitman theorems exist, and how do they relate to the matrix Dufresne identity?
• RQ4 What is the asymptotic behavior of the eigenvalues of $ A_t^{(\mu)} $, and how does it relate to the eigenvalue structure of the Wishart distribution?
• RQ5 Can a partial Burke-type property be established for the matrix process $ M_t $, generalizing the scalar case?

Key findings

• The matrix Dufresne identity holds: for $ 2\mu > r-1 $, the random matrix $ A_\infty^{(-\mu)} = \int_0^\infty M_s M_s^T ds $ has the standard inverse Wishart distribution with parameter $ 2\mu $, i.e., $ A_\infty^{(-\mu)} \stackrel{\text{law}}{=} \gamma_{2\mu}^{-1} $.
• The condition $ 2\mu > r-1 $ ensures almost sure finiteness of $ A_\infty^{(-\mu)} $ and non-degeneracy of the inverse Wishart distribution.
• The process $ M_t $ on $ GL_r(\mathbb{R}) $ has independent multiplicative increments and is rotationally invariant, enabling the derivation of the identity via distributional symmetry.
• A matrix version of the Pitman-type identity is established: $ \{1/A_t^{(-\mu)}\} \stackrel{\text{law}}{=} \{1/A_t^{(\mu)} + 1/\tilde{A}_\infty^{(-\mu)}\} $, where $ \tilde{A}_\infty^{(-\mu)} $ is an independent copy.
• The eigenvalue dynamics of $ A_t^{(\mu)} $ are analyzed via a change of variables to logarithmic coordinates $ y_i = \log z_i $, leading to a diffusion with drift involving ratios of Macdonald functions.
• In the limit $ r=1 $, the matrix process recovers the scalar Matsumoto–Yor identities, confirming consistency and validating the generalization.