TY - JOUR

T1 - Coverings of random ellipsoids, and invertibility of matrices with i.i.d. heavy-tailed entries

AU - Rebrova, Elizaveta

AU - Tikhomirov, Konstantin

N1 - Funding Information:
∗ E. R. was partially supported by U.S. Air Force grant F035062. ∗∗K. T. was partially supported by PIMS Graduate Scholarship Excellence Award, Faculty of Science, UofA. Received January 25 ,2016 and in revised form February 13, 2017

PY - 2018/8/1

Y1 - 2018/8/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85049874717&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85049874717&partnerID=8YFLogxK

U2 - 10.1007/s11856-018-1732-y

DO - 10.1007/s11856-018-1732-y

M3 - Article

AN - SCOPUS:85049874717

VL - 227

SP - 507

EP - 544

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

SN - 0021-2172

IS - 2

ER -