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 suph∈H|N-1∑i=1Nh2(Xi)-Eh2| in 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, {dot operator}. 〉. :. t∈. T}, T⊂Rn and μ is an isotropic, unconditional, log-concave measure.