Limiting Empirical Spectral Distribution for Products of Rectangular Matrices

TL;DR

This paper investigates the limiting spectral distribution of products of multiple independent rectangular Gaussian matrices as their dimensions grow, providing a comprehensive description of the asymptotic behavior.

Contribution

It offers a complete characterization of the limiting empirical spectral distribution for products of rectangular matrices with diverging dimensions, extending previous results.

Findings

Derived the limiting spectral distribution for the product matrices

Included examples illustrating the theoretical results

Analyzed the case where the number of matrices diverges

Abstract

In this paper, we consider $m$ independent random rectangular matrices whose entries are independent and identically distributed standard complex Gaussian random variables and assume the product of the $m$ rectangular matrices is an $n$ by $n$ square matrix. We study the limiting empirical spectral distributions of the product where the dimension of the product matrix goes to infinity, and $m$ may change with the dimension of the product matrix and diverge. We give a complete description for the limiting distribution of the empirical spectral distributions for the product matrix and illustrate some examples.