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

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TAIL BOUNDS FOR ALL EIGENVALUES OF A SUM OF RANDOM MATRICES (2011, methodology): “x, so it is most useful when the eigenvalues of the covariance matrix satisfy decay conditions such as those given in the statement of Theorem 1.1. If such conditions are not satised, the results of [ALPTJ11] on the convergence of empirical covariance matrices of isotropic log-concave random vectors”
The smallest singular value of random rectangular matrices with no moment assumptions on entries (2014, background): “Later, the log-concavity was replaced by weaker assumptions (see, in particular, [2], [17], [9], [4]).”
Sample covariance matrices of heavy-tailed distributions (2016, background): “The next lemma, stated in [10] (the argument appeared already in [2]), will be very helpful for us.”