Covariance estimation for distributions with ${2+\varepsilon}$ moments

We study the minimal sample size $N=N(n)$ that suffices to estimate the covariance matrix of an $n$-dimensional distribution by the sample covariance matrix in the operator norm, with an arbitrary fixed accuracy. We establish the optimal bound $N=O(n)$ for every distribution whose $k$-dimensional marginals have uniformly bounded $2+\varepsilon$ moments outside the sphere of radius $O(\sqrt{k})$. In the specific case of log-concave distributions, this result provides an alternative approach to the Kannan–Lovasz–Simonovits problem, which was recently solved by Adamczak et al. [J. Amer. Math. Soc. 23 (2010) 535–561]. Moreover, a lower estimate on the covariance matrix holds under a weaker assumption—uniformly bounded $2+\varepsilon$ moments of one-dimensional marginals. Our argument consists of randomizing the spectral sparsifier, a deterministic tool developed recently by Batson, Spielman and Srivastava [SIAM J. Comput. 41 (2012) 1704–1721]. The new randomized method allows one to control the spectral edges of the sample covariance matrix via the Stieltjes transform evaluated at carefully chosen random points.