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

Shahar Mendelson*, Grigoris Paouris

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

20 Citations (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)3775-3811
Number of pages37
JournalJournal of Functional Analysis
Volume262
Issue number9
DOIs
Publication statusPublished - 1 May 2012
Externally publishedYes

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