The smallest singular value of random rectangular matrices with no moment assumptions on entries

Konstantin E. Tikhomirov

doi:10.1007/s11856-016-1287-8

The smallest singular value of random rectangular matrices with no moment assumptions on entries

Konstantin E. Tikhomirov

Mathematics

Research output: Contribution to journal › Article › peer-review

27 Scopus citations

Abstract

Let δ > 1 and β > 0 be some real numbers. We prove that there are positive u, v, N₀ depending only on β and δ with the following property: for any N,n such that N ≥ max(N₀, δ_n), any N × n random matrix A = (a_ij) with i.i.d. entries satisfying (Formula presented.) and any non-random N × n matrix B, the smallest singular value s_n of A + B satisfies (Formula presented.). The result holds without any moment assumptions on the distribution of the entries of A.

Original language	English (US)
Pages (from-to)	289-314
Number of pages	26
Journal	Israel Journal of Mathematics
Volume	212
Issue number	1
DOIs	https://doi.org/10.1007/s11856-016-1287-8
State	Published - May 1 2016

All Science Journal Classification (ASJC) codes

General Mathematics

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10.1007/s11856-016-1287-8

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abstract = "Let δ > 1 and β > 0 be some real numbers. We prove that there are positive u, v, N0 depending only on β and δ with the following property: for any N,n such that N ≥ max(N0, δn), any N × n random matrix A = (aij) with i.i.d. entries satisfying (Formula presented.) and any non-random N × n matrix B, the smallest singular value sn of A + B satisfies (Formula presented.). The result holds without any moment assumptions on the distribution of the entries of A.",

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N2 - Let δ > 1 and β > 0 be some real numbers. We prove that there are positive u, v, N0 depending only on β and δ with the following property: for any N,n such that N ≥ max(N0, δn), any N × n random matrix A = (aij) with i.i.d. entries satisfying (Formula presented.) and any non-random N × n matrix B, the smallest singular value sn of A + B satisfies (Formula presented.). The result holds without any moment assumptions on the distribution of the entries of A.

AB - Let δ > 1 and β > 0 be some real numbers. We prove that there are positive u, v, N0 depending only on β and δ with the following property: for any N,n such that N ≥ max(N0, δn), any N × n random matrix A = (aij) with i.i.d. entries satisfying (Formula presented.) and any non-random N × n matrix B, the smallest singular value sn of A + B satisfies (Formula presented.). The result holds without any moment assumptions on the distribution of the entries of A.

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The smallest singular value of random rectangular matrices with no moment assumptions on entries

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