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

Let $\{a_{ij}\}$ $(1\le i,j<\mathrm{\infty })$ be i.i.d. real valued random variables with zero mean and unit variance and let an integer sequence $(N_m{)}_{m=1}^{\mathrm{\infty }}$ satisfy $m\mathrm{/}{N}_m⟶z$ 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 $\bigl({N_m}^{-1/2} s_{m}(A_m)\bigr)_{m=1}^\infty$convergesto$1-\sqrt{z}$ almost surely. Our result does not require boundedness of any moments of $a_{ij}$'s higher than the $2$-nd and resolves a long standing question regarding the weakest moment assumptions on the distribution of the entries sufficient for the convergence to hold.