Asymptotic Eigenvalue Moments of Wishart-Type Random Matrix Without Ergodicity in One Channel Realization

TL;DR

This paper derives asymptotic eigenvalue moments for large Wishart-type random matrices with random variance profiles, using noncrossing partition theory, addressing both ergodic and nonergodic cases in one channel realization.

Contribution

It provides explicit formulas for asymptotic eigenvalue moments of Wishart-type matrices with random variance profiles, extending analysis to nonergodic scenarios using noncrossing partition theory.

Findings

Derived expressions for AEM in terms of variance profile

AEM depend on variance profile in nonergodic case

Formulas applicable to ergodic matrices with known variance profiles

Abstract

Consider a random matrix whose variance profile is random. This random matrix is ergodic in one channel realization if, for each column and row, the empirical distribution of the squared magnitudes of elements therein converges to a nonrandom distribution. In this paper, noncrossing partition theory is employed to derive expressions for several asymptotic eigenvalue moments (AEM) related quantities of a large Wishart-type random matrix $\bb H\bb H^\dag$ when $\bb H$ has a random variance profile and is nonergodic in one channel realization. It is known the empirical eigenvalue moments of $\bb H\bb H^\dag$ are dependent (or independent) on realizations of the variance profile of $\bb H$ when $\bb H$ is nonergodic (or ergodic) in one channel realization. For nonergodic $\bb H$, the AEM can be obtained by i) deriving the expression of AEM in terms of the variance profile of $\bb H$, and then ii) averaging the derived quantity over the ensemble of variance profiles. Since the AEM are independent of the variance profile if $\bb H$ is ergodic, the expression obtained in i) can also serve as the AEM formula for ergodic $\bb H$ when any realization of variance profile is available.