Fluctuations of an improved population eigenvalue estimator in sample covariance matrix models

TL;DR

This paper establishes a central limit theorem for an improved estimator of population eigenvalues in high-dimensional sample covariance matrices, especially when sample size is limited, with applications to wireless sensor networks.

Contribution

It introduces a new CLT for a consistent eigenvalue estimator with large multiplicities, providing exact and approximate variance expressions in limited sample scenarios.

Findings

Theoretical CLT for the estimator is proven.

Derived exact and asymptotic variance formulas.

Simulations confirm the theoretical results.

Abstract

This article provides a central limit theorem for a consistent estimator of population eigenvalues with large multiplicities based on sample covariance matrices. The focus is on limited sample size situations, whereby the number of available observations is known and comparable in magnitude to the observation dimension. An exact expression as well as an empirical, asymptotically accurate, approximation of the limiting variance is derived. Simulations are performed that corroborate the theoretical claims. A specific application to wireless sensor networks is developed.