Sample covariance matrices of heavy-tailed distributions

2016/06/11 by Tikhomirov, Konstantin · 1 citation
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1606.03557

Abstract

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∈ S^n-1}\mathrm E|⟨ X,a⟩|^p≤ B. Further, let X₁,X₂,…,X_N be independent copies of X, and Σ_N:=(1)/(N)∑_i=1^N X_i X_i^T be the sample covariance matrix. We prove that K^-1‖Σ_N-I_n‖_{2→ 2}≤(1)/(N)max_{i≤ N}‖X_i‖² +((n)/(N))^1-2/plog⁴(N)/(n)+((n)/(N))^1-2/min(p,4) with probability at least 1-(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.

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