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

Abstract

Let $\delta>1$ and $\beta>0$ be some real numbers. We prove that there are positive $u,v,N_0$ depending only on $\beta$ and $\delta$ with the following property: for any $N,n$ such that $N\ge \max(N_0,\delta n)$, any $N\times n$ random matrix $A=(a_{ij})$ with i.i.d. entries satisfying $\sup\limits_{\lambda\in {\mathbb R}}{\mathbb P}\bigl\{|a_{11}-\lambda|\le 1\bigr\}\le 1-\beta$ and any non-random $N\times n$ matrix $B$, the smallest singular value $s_n$ of $A+B$ satisfies ${\mathbb P}\bigl\{s_n(A+B)\le u\sqrt{N}\bigr\}\le \exp(-vN)$. The result holds without any moment assumptions on distribution of the entries of $A$.