Singular value correlation functions for products of Wishart random matrices

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.