Let $A=(a_{ ij})$ be an $n\times n$ random matrix with i.i.d. entries such that $\mathbb{ E} a_{ 11} = 0$ and $\mathbb{ E} { a_{ 11}}^2 = 1$. We prove that for any $\delta>0$ there is $L>0$ depending only on $\delta$, and a subset $\mathcal{ N}$ of $B_2^n$ of cardinality at most $\exp(\delta n)$ such that with probability very close to one we have $$A(B_2^n)\subset \bigcup_{ y\in A(\mathcal{ N})}\bigl(y+L\sqrt{ n}B_2^n\bigr).$$ As an application, we show that for some $L'>0$ and $u\in[0,1)$ depending only on the distribution law of $a_{ 11}$, the smallest singular value $s_n$ of the matrix $A$ satisfies $\mathbb{ P}\{ s_n(A)\le \varepsilon n^{ -1/2}\}\le L'\varepsilon+u^n$ for all $\varepsilon>0$. The latter result generalizes a theorem of Rudelson and Vershynin which was proved for random matrices with subgaussian entries.
Categories: math.PR