The limiting spectral distribution of large dimensional general information-plus-noise type matrices

TL;DR

This paper derives the limiting spectral distribution of large information-plus-noise matrices, showing convergence of eigenvalue distributions to a deterministic limit characterized by a system of equations.

Contribution

It establishes the spectral distribution limit for a broad class of large information-plus-noise matrices with general structures, extending previous results.

Findings

Eigenvalue distributions converge weakly to a deterministic limit.

The limit is characterized by a system of equations for the Stieltjes transform.

Results apply to matrices with general non-random and random components.

Abstract

Let $ X_{n} $ be $ n\times N $ random complex matrices, $R_{n}$ and $T_{n}$ be non-random complex matrices with dimensions $n\times N$ and $n\times n$, respectively. We assume that the entries of $ X_{n} $ are independent and identically distributed, $ T_{n} $ are nonnegative definite Hermitian matrices and $T_{n}R_{n}R_{n}^{*}= R_{n}R_{n}^{*}T_{n} $. The general information-plus-noise type matrices are defined by $C_{n}=\frac{1}{N}T_{n}^{\frac{1}{2}} \left( R_{n} +X_{n}\right) \left(R_{n}+X_{n}\right)^{*}T_{n}^{\frac{1}{2}} $. In this paper, we establish the limiting spectral distribution of the large dimensional general information-plus-noise type matrices $C_{n}$. Specifically, we show that as $n$ and $N$ tend to infinity proportionally, the empirical distribution of the eigenvalues of $C_{n}$ converges weakly to a non-random probability distribution, which is characterized in terms of a system of equations of its Stieltjes transform.