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 N i=1 h2(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, · : t ∈ T }, T ⊂ Rn and μ is an isotropic, unconditional, log-concave measure. © 2012 Elsevier Inc. All rights reserved.