Large sample correlation matrices: a comparison theorem and its applications

TL;DR

This paper establishes a comparison theorem for large sample correlation matrices, providing spectral norm approximations, eigenvalue distribution insights, and estimators for population correlations in high-dimensional settings.

Contribution

It introduces a new comparison theorem for high-dimensional sample correlation matrices and applies it to eigenvalue analysis and estimation of population correlations.

Findings

Diagonal of sample covariance approximates population covariance in spectral norm.

Identifies the limiting spectral distribution of the sample correlation matrix.

Provides an estimator for the population correlation matrix and its eigenvalues.

Abstract

In this paper, we show that the diagonal of a high-dimensional sample covariance matrix stemming from $n$ independent observations of a $p$-dimensional time series with finite fourth moments can be approximated in spectral norm by the diagonal of the population covariance matrix. We assume that $n,p\to \infty$ with $p/n$ tending to a constant which might be positive or zero. As applications, we provide an approximation of the sample correlation matrix ${\mathbf R}$ and derive a variety of results for its eigenvalues. We identify the limiting spectral distribution of ${\mathbf R}$ and construct an estimator for the population correlation matrix and its eigenvalues. Finally, the almost sure limits of the extreme eigenvalues of ${\mathbf R}$ in a generalized spiked correlation model are analyzed.