Sample covariance matrices of heavy-tailed distributions

Abstract

Let $p>2$, $B\ge 1$, $N\ge n$ and let $X$ be a centered $n$-dimensional random vector with the identity covariance matrix such that $\underset{a\in S^{n-1}}{\sup }\mathrm{E}|⟨X,a⟩{|}^p\le B$. Further, let $X_1,X_2,\dots ,X_N$ be independent copies of $X$, and ${\Sigma }_N:=\frac{1}{N}{\sum }_{i=1}^N{X}_i{X_i}^T$ be the sample covariance matrix. We prove that $$K^{-1}\Vert {\Sigma }_N-I_n{\Vert }_{2\to 2}\le \frac{1}{N}\underset{i\le N}{\max }\Vert X_i{\Vert }^2+(\frac{n}{N}{)}^{1-2/p}{\log }^4\frac{N}{n}+(\frac{n}{N}{)}^{1-2/\min (p,4)}$$ with probability at least $1-\frac{1}{n}$, where $K>0$ depends only on $B$ and $p$. In particular, for all $p>4$ we obtain a quantitative Bai-Yin type theorem.