Sharp lower bounds on the least singular value of a random matrix without the fourth moment condition

Abstract

We obtain non-asymptotic lower bounds on the least singular value of ${\mathbf X}_{pn}^\top/\sqrt{n}$, where ${\mathbf X}_{pn}$ is a $p\times n$ random matrix whose columns are independent copies of an isotropic random vector $X_p$ in $ {\mathbb R}^p$. We assume that there exist $M>0$ and $\alpha\in\leqslant M/t^{2+\alpha}$ for all $t>0$ and any unit vector $v\in{\mathbb R}^p$. These bounds depend on $y=p/n,$ $\alpha$, $M$ and are asymptotically optimal up to a constant factor.