We extend to the matrix setting a recent result of Srivastava-Vershynin 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 assumption. Beside giving examples, we discuss the notion of log-concave matrices and give estimates on the smallest and largest eigenvalues of a sum of such matrices.
Pierre Youssef. "Estimating the covariance of random matrices." Electron. J. Probab. 18 1 - 26, 2013. https://doi.org/10.1214/EJP.v18-2579