Coverings of random ellipsoids, and invertibility of matrices with i.i.d. heavy-tailed entries

Elizaveta Rebrova, Konstantin Tikhomirov

Research output: Contribution to journalArticle

8 Scopus citations

Abstract

Let A = (aij) be an n × n random matrix with i.i.d. entries such that Ea11 = 0 and Ea11 2 = 1. We prove that for any δ > 0 there is L > 0 depending only on δ, and a subset N of B2 n of cardinality at most exp(δn) such that with probability very close to one we have A(B2n)∪y∈A(N)(y+LnB2n). In fact, a stronger statement holds true. As an application, we show that for some L' > 0 and u ∈ [0, 1) depending only on the distribution law of a11, the smallest singular value sn of the matrix A satisfies P{ sn(A) ≤ εn− 1 / 2} ≤ L' ε+ un for all ε > 0. The latter result generalizes a theorem of Rudelson and Vershynin which was proved for random matrices with subgaussian entries.

Original languageEnglish (US)
Pages (from-to)507-544
Number of pages38
JournalIsrael Journal of Mathematics
Volume227
Issue number2
DOIs
StatePublished - Aug 1 2018

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Fingerprint Dive into the research topics of 'Coverings of random ellipsoids, and invertibility of matrices with i.i.d. heavy-tailed entries'. Together they form a unique fingerprint.

  • Cite this