Eigenvalue Distributions of Sums and Products of Large Random Matrices via Incremental Matrix Expansions

TL;DR

This paper introduces an incremental matrix expansion method to derive asymptotic eigenvalue distributions of large random matrices, aligning with free probability results and extending to non-free matrices, with applications in multiuser CDMA systems.

Contribution

It presents a novel incremental matrix expansion approach for eigenvalue distributions, applicable to sums and products of large matrices, including non-free cases, and demonstrates practical relevance in communication systems.

Findings

Derived asymptotic eigenvalue distributions matching free probability results.

Extended the method to non-free matrix sums.

Applied results to analyze multiuser CDMA system performance.

Abstract

This paper uses an incremental matrix expansion approach to derive asymptotic eigenvalue distributions (a.e.d.'s) of sums and products of large random matrices. We show that the result can be derived directly as a consequence of two common assumptions, and matches the results obtained from using R- and S-transforms in free probability theory. We also give a direct derivation of the a.e.d. of the sum of certain random matrices which are not free. This is used to determine the asymptotic signal-to-interference-ratio of a multiuser CDMA system with a minimum mean-square error linear receiver.