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

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Abstract

Let δ > 1 and β > 0 be some real numbers. We prove that there are positive u, v, N 0 depending only on β and δ with the following property: for any N,n such that N ≥ max(N 0, δ n ), any N × n random matrix A = (a ij ) with i.i.d. entries satisfying \({\sup _{\lambda \in \mathbb{R}}}P\left\{ {\left| {{a_{11}} - \lambda } \right| \leqslant 1} \right\} \leqslant 1 - \beta \) and any non-random N × n matrix B, the smallest singular value s n of A + B satisfies \(P\left\{ {{s_n}\left( {A + B} \right) \leqslant u\sqrt N } \right\} \leqslant \exp \left( { - vN} \right)\). The result holds without any moment assumptions on the distribution of the entries of A.

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  1. Department of Mathematics, University of Alberta Edmonton, Alberta, T6G 2G1, Canada

    Konstantin E. Tikhomirov

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  1. Konstantin E. Tikhomirov
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Correspondence to Konstantin E. Tikhomirov.

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Tikhomirov, K.E. The smallest singular value of random rectangular matrices with no moment assumptions on entries. Isr. J. Math. 212, 289–314 (2016). https://doi.org/10.1007/s11856-016-1287-8

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  • DOI: https://doi.org/10.1007/s11856-016-1287-8

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