Robust covariance estimation under $L_{4}-L_{2}$ norm equivalence

Abstract

Let $X$ be a centered random vector taking values in $\mathbb{R}^{d}$ and let $\Sigma=\mathbb{E}(X\otimes X)$ be its covariance matrix. We show that if $X$ satisfies an $L_{4}-L_{2}$ norm equivalence (sometimes referred to as the bounded kurtosis assumption), there is a covariance estimator $\hat{\Sigma}$ that exhibits almost the same performance one would expect had $X$ been a Gaussian vector. The procedure also improves the current state-of-the-art regarding high probability bounds in the sub-Gaussian case (sharp results were only known in expectation or with constant probability).

In both scenarios the new bounds do not depend explicitly on the dimension $d$, but rather on the effective rank of the covariance matrix $\Sigma$.