Covariance Matrix Estimation from Correlated Sub-Gaussian Samples

TL;DR

This paper provides non-asymptotic error bounds for estimating covariance matrices from correlated sub-Gaussian samples, revealing that a number of samples proportional to the dimension is sufficient under certain correlation conditions.

Contribution

It introduces a theoretical analysis of covariance estimation from correlated sub-Gaussian samples with explicit error bounds and conditions for sample sufficiency.

Findings

Error bounds depend on dimension, sample size, and correlation pattern.

O(n) samples are sufficient under specific correlation conditions.

Numerical simulations confirm theoretical predictions.

Abstract

This paper studies the problem of estimating a covariance matrix from correlated sub-Gaussian samples. We consider using the correlated sample covariance matrix estimator to approximate the true covariance matrix. We establish non-asymptotic error bounds for this estimator in both real and complex cases. Our theoretical results show that the error bounds are determined by the signal dimension $n$, the sample size $m$ and the correlation pattern $\textbf{B}$. In particular, when the correlation pattern $\textbf{B}$ satisfies $tr(\textbf{B})=m$, $||\textbf{B}||_{F}=O(m^{1/2})$, and $||\textbf{B}||=O(1)$, these results reveal that $O(n)$ samples are sufficient to accurately estimate the covariance matrix from correlated sub-Gaussian samples. Numerical simulations are presented to show the correctness of the theoretical results.