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arXiv · 1309.5711

On minimal singular values of random matrices with correlated entries

Friedrich Götze·Alexey Naumov·Alexander Tikhomirov
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Abstract

Let $\mathbf X$ be a random matrix whose pairs of entries $X_{jk}$ and $X_{kj}$ are correlated and vectors $ (X_{jk},X_{kj})$, for $1\le j 0$ and $Q\ge 0$. Let $s_n(\mathbf X+\mathbf M_n)$ denote the least singular value of the matrix $\mathbf X+\mathbf M_n$. It is shown that there exist positive constants $A$ and $B$ depending on $K,Q,ρ$ only such that $$ \mathbb{P}(s_n(\mathbf X+\mathbf M_n)\le n^{-A})\le n^{-B}. $$ As an application of this result we prove the elliptic law for this class of matrices with non identically distributed correlated entries.

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Friedrich Götze, Alexey Naumov, Alexander Tikhomirov. 2013-09-23. On minimal singular values of random matrices with correlated entries. https://arxiv.org/abs/1309.5711

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