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

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

2011-08-19

arxiv.org/abs/1108.3886v1

Mathematics

Probability

42 pages

Scientific paper

We present a very general chaining method which allows one to control the supremum of the empirical process $\sup_{h \in H} |N^{-1}\sum_{i=1}^N h^2(X_i)-\E h^2|$ in rather general situations. We use this method to establish two main results. First, a quantitative (non asymptotic) version of the classical 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 $H=\{\inr{t,\cdot} : t \in T\}$, $T \subset \R^n$ and $\mu$ is an isotropic, unconditional, log-concave measure.

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