$XX^T$ Matrices With Independent Entries

TL;DR

This paper establishes general conditions for the existence of the limiting spectral distribution (LSD) of sample covariance matrices with independent entries, extending classical results and analyzing new cases involving structured matrices.

Contribution

It provides a unified framework for the LSD of $XX^T$ with independent entries and extends results to structured matrices like circulant and Toeplitz, with new insights into the associated partition structures.

Findings

LSD exists under broad conditions for matrices with independent entries.

Partition structures related to symmetric partitions determine the moments of the LSD.

Generalizes previous results for structured matrices like circulant, Toeplitz, and Hankel.

Abstract

Let $S=XX^T$ be the (unscaled) sample covariance matrix where $X$ is a real $p \times n$ matrix with independent entries. It is well known that if the entries of $X$ are independent and identically distributed (i.i.d.) with enough moments and $p/n \to y\neq 0$, then the limiting spectral distribution (LSD) of $\frac{1}{n}S$ converges to a Mar$\check{\text{c}}$enko-Pastur law. Several extensions of this result are also known. We prove a general result on the existence of the LSD of $S$ in probability or almost surely, and in particular, many of the above results follow as special cases. At the same time several new LSD results also follow from our general result. The moments of the LSD are quite involved but can be described via a set of partitions. Unlike in the i.i.d. entries case, these partitions are not necessarily non-crossing, but are related to the special symmetric partitions which are known to appear in the LSD of (generalised) Wigner matrices with independent entries. We also investigate the existence of the LSD of $S_{A}=AA^T$ when $A$ is the $p\times n$ symmetric or the asymmetric version of any of the following four random matrices: reverse circulant, circulant, Toeplitz and Hankel. The LSD of $\frac{1}{n}S_{A}$ for the above four cases have been studied by Bose, Gangopadhyay and Sen in 2010, when the entries are i.i.d. We show that under some general assumptions on the entries of $A$, the LSD of $S_{A}$ exists and this result generalises the existing results significantly.