  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.
