On a Generalisation of the Marcenko-Pastur Problem

TL;DR

This paper generalizes the Marcenko-Pastur problem by analyzing the spectrum of a class of generalized Wishart matrices, revealing new spectral behaviors depending on the correlation parameter c.

Contribution

It derives the spectral distribution of generalized Wishart matrices for arbitrary correlation c, extending classical results and connecting to Wigner semi-circle law in a special case.

Findings

Stieltjes transform satisfies a cubic equation.

Eigenvalue density converges to Wigner semi-circle when c=0 and T>>N.

Provides a unified framework for spectral analysis of correlated Wishart matrices.

Abstract

We study the spectrum of generalized Wishart matrices, defined as $\mathbf{F}=( X Y^\top + Y X^\top)/2T$, where $X$ and $Y$ are $N \times T$ matrices with zero mean, unit variance IID entries and such that $\mathbb{E}[X_{it} Y_{jt}]=c \delta_{i,j}$. The limit $c=1$ corresponds to the Marcenko-Pastur problem. For a general $c$, we show that the Stietjes transform of $\mathbf{F}$ is the solution of a cubic equation. In the limit $c=0$, $T \gg N$ the density of eigenvalues converges to the Wigner semi-circle.