On generic chaining and the smallest singular value of random matrices with heavy tails

We present a very general chaining method which allows one to control the supremum of the empirical process sup(h is an element of H) vertical bar N-1 Sigma(N)(i=1) h(2)(X-i) - Eh(2)vertical bar rather general situations. We use this method to establish two main results. First, a quantitative (non-asymptotic) version of the celebrated Bai-Yin Theorem on the singular values of a random matrix with i.i.d. entries that have heavy tails, and second, a sharp estimate on the quadratic empirical process when H = {< t, center dot >: t is an element of T}, T subset of R '' and mu is an isotropic, unconditional, log-concave measure. (C) 2012 Elsevier Inc. All rights reserved.