Sample Covariance Matrices of Heavy-Tailed Distributions

Let p > 2, B >= 1, N >= n and let X be a centered n-dimensional random vector with the identity covariance matrix such that sup(a is an element of Sn-1) E vertical bar < X, a >(p) <= B. Further, let X-1, X-2,..., X-N be independent copies of X, and is an element of(N) := 1/N Sigma(N)(i=1) XiXiT be the sample covariance matrix. We prove that, under a mild assumption on the Euclidean norm of X, the sample covariance matrix approximates the actual covariance matrix in the operator norm with any given precision as long as the size of the sample is a large multiple of n: P{parallel to Sigma(N) - Id(n)parallel to(2 -> 2) <= delta} approximate to 1 for any fixed delta > 0, whenever N >= Ln for L = L(delta, p, B) > 0.