Robust covariance estimation under L₄ - L₂ norm equivalence

Let X be a centered random vector taking values in R^d and let σ = E(X ⊗ X) be its covariance matrix. We show that if X satisfies an L₄ - L₂ norm equivalence (sometimes referred to as the bounded kurtosis assumption), there is a covariance estimator σ 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 σ.