Asymptotic Products of Independent Gaussian Random Matrices with Correlated Entries

In this work we address the problem of determining the asymptotic spectral measure of the product of independent, Gaussian random matrices with correlated entries, as the dimension and the number of multiplicative terms goes to infinity. We show for a fixed number of multiplicative terms $n$ the spectral measure of the product converges in distribution to a compactly supported measure $nu_{n}$ as the dimension of the matrices $N-infty$. We also show that the sequence of measures $nu_{n}$ converges in distribution to a compactly supported measure $nu_{n}-nu$ as the number of multiplicative terms increases to infinity. Moreover, we deduce an exact closed-form expression for the measure $nu$. This problem is fundamentally important for the analysis of the wireless network with fading when a single source and destination pair is considered. For instance, it was previously studied that if one is interested in exploring the performance in a layered relay network having a single source destination pair, then the effective channel of the system is given by the product of independent Gaussian matrices.