The limit of the smallest singular value of random matrices with i.i.d. entries

@article{Tikhomirov2014TheLO,
  title={The limit of the smallest singular value of random matrices with i.i.d. entries},
  author={Konstantin E. Tikhomirov},
  journal={arXiv: Probability},
  year={2014}
}
Let $\{a_{ij}\}$ $(1\le i,j<\infty)$ be i.i.d. real valued random variables with zero mean and unit variance and let an integer sequence $(N_m)_{m=1}^\infty$ satisfy $m/N_m\longrightarrow z$ for some $z\in(0,1)$. For each $m\in{\mathbb N}$ denote by $A_m$ the $N_m\times m$ random matrix $(a_{ij})$ $(1\le i\le N_m,1\le j\le m)$ and let $s_{m}(A_m)$ be its smallest singular value. We prove that the sequence $\bigl({N_m}^{-1/2} s_{m}(A_m)\bigr)_{m=1}^\infty$ converges to $1-\sqrt{z}$ almost surely… Expand
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