Singular value correlation functions for products of Wishart random matrices

TL;DR

This paper derives the joint probability distribution and correlation functions of singular values for products of Gaussian Wishart matrices, extending classical models and applying results to multi-layered MIMO communication channels.

Contribution

It provides explicit formulas for singular value correlations of Wishart matrix products using biorthogonal polynomials and Meijer G-functions, generalizing previous models.

Findings

Derived joint probability distribution for singular values of matrix products.

Explicit correlation kernels in terms of hypergeometric and Meijer G-functions.

Applied results to ergodic mutual information in multi-layered MIMO channels.

Abstract

Consider the product of $M$ quadratic random matrices with complex elements and no further symmetry, where all matrix elements of each factor have a Gaussian distribution. This generalises the classical Wishart-Laguerre Gaussian Unitary Ensemble with M=1. In this paper we first compute the joint probability distribution for the singular values of the product matrix when the matrix size $N$ and the number $M$ are fixed but arbitrary. This leads to a determinantal point process which can be realised in two different ways. First, it can be written as a one-matrix singular value model with a non-standard Jacobian, or second, for $M\geq2$, as a two-matrix singular value model with a set of auxiliary singular values and a weight proportional to the Meijer $G$-function. For both formulations we determine all singular value correlation functions in terms of the kernels of biorthogonal polynomials which we explicitly construct. They are given in terms of hypergeometric and Meijer $G$-functions, generalising the Laguerre polynomials. Our investigation was motivated from applications in telecommunication of multi-layered scattering MIMO channels. We present the ergodic mutual information for finite-$N$ for such a channel model with $M-1$ layers of scatterers as an example.