Estimating the covariance of random matrices (Q389013)

The author extends to the matrix setting a recent result of \textit{N. Srivastava} and \textit{R. Vershynin} [Ann. Probab. 41, No. 5, 3081--3111 (2013; Zbl 1293.62121)] about estimating the covariance matrix of a random vector. The result can be interpreted as a quantified version of the law of large numbers for positive semi-definite matrices which verify some regularity assumptions. Namely, for independent copies \(B_1,\dots,B_N\) of a random positive semi-definite \(n\times n\)-matrix \(B\) with \(\operatorname{E} B=I_n\) (where \(I_n\) is the identity matrix) which satisfies certain regularity assumption with parameter \(\eta\), the author proves that NEWLINE\[NEWLINE \operatorname{E} \left\| \frac 1 N\sum_{i=1}^N B_i - I_n \right\| \leq \varepsilon, NEWLINE\]NEWLINE where \(N=C(\eta) n \varepsilon^{-2-\frac 2 {\eta}}\). The author gives examples and discusses the notion of log-concave matrices. Estimates on the smallest and largest eigenvalues of a sum of such matrices are provided.