Sharp bounds on the rate of convergence of the empirical covariance matrix

Details Sharp bounds on the rate of convergence of the empirical covariance matrix Sharp bounds on the rate of convergence of the empirical covariance matrix

2010-12-01

arxiv.org/abs/1012.0294v2

Mathematics

Probability

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Let $X_1,..., X_N\in\R^n$ be independent centered random vectors with log-concave distribution and with the identity as covariance matrix. We show that with overwhelming probability at least $1 - 3 \exp(-c\sqrt{n}\r)$ one has $ \sup_{x\in S^{n-1}} \Big|\frac{1/N}\sum_{i=1}^N (||^2 - \E||^2\r)\Big| \leq C \sqrt{\frac{n/N}},$ where $C$ is an absolute positive constant. This result is valid in a more general framework when the linear forms $()_{i\leq N, x\in S^{n-1}}$ and the Euclidean norms $(|X_i|/\sqrt n)_{i\leq N}$ exhibit uniformly a sub-exponential decay. As a consequence, if $A$ denotes the random matrix with columns $(X_i)$, then with overwhelming probability, the extremal singular values $\lambda_{\rm min}$ and $\lambda_{\rm max}$ of $AA^\top$ satisfy the inequalities $ 1 - C\sqrt{{n/N}} \le {\lambda_{\rm min}/N} \le \frac{\lambda_{\rm max}/N} \le 1 + C\sqrt{{n/N}} $ which is a quantitative version of Bai-Yin theorem \cite{BY} known for random matrices with i.i.d. entries.

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