# Quantitative estimates of the convergence of the empirical covariance matrix in log-concave ensembles

@article{Adamczak2009QuantitativeEO, title={Quantitative estimates of the convergence of the empirical covariance matrix in log-concave ensembles}, author={Radosław Adamczak and Alexander E. Litvak and Alain Pajor and Nicole Tomczak-Jaegermann}, journal={Journal of the American Mathematical Society}, year={2009}, volume={23}, pages={535-561} }

Let K be an isotropic convex body in Rn. Given e > 0, how many independent points Xi uniformly distributed on K are neededfor the empirical covariance matrix to approximate the identity up to e with overwhelming probability? Our paper answers this question from [12]. More precisely, let X ∈ Rn be a centered random vector with a log-concave distribution and with the identity as covariance matrix. An example of such a vector X is a random point in an isotropic convex body. We show that for any e…

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