Sample covariance matrices and special symmetric partitions

TL;DR

This paper extends the theory of sample covariance matrices by analyzing their spectral distributions under various patterns and partitions, revealing connections to symmetric partitions and broadening the scope beyond classical results.

Contribution

It introduces new results on the limiting spectral distribution of patterned sample covariance matrices using complex partition structures, including symmetric partitions.

Findings

Limiting spectral distribution exists under certain conditions.

Moments are characterized by advanced partition structures.

Results apply to various patterned matrices like circulant and Toeplitz.

Abstract

Suppose $X_p$ is a real $p \times n$ matrix with independent entries and consider the (unscaled) sample covariance matrix $S_p=X_pX_p^T$. The Marchenko-Pastur law was discovered as the limit of the bulk distribution of the sample covariance matrix in 1967. There have been extensions of this result in several directions. In this paper, we consider an extension that handles several of the existing ones as well as generates new results. We show that under suitable assumptions on the entries of $X_p$, the limiting spectral distribution exists in probability or almost surely. The moments are described by a set of partitions that are beyond pair partitions and non-crossing partitions and are also related to special symmetric partitions, which are known to appear in the limiting spectral distribution of Wigner-type matrices. Similar results hold for other patterned matrices such as reverse circulant, circulant, Toeplitz, and Hankel matrices.