Coverings of random ellipsoids, and invertibility of matrices with i.i.d. heavytailed entries
Abstract
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.
 Publication:

arXiv eprints
 Pub Date:
 August 2015
 arXiv:
 arXiv:1508.06690
 Bibcode:
 2015arXiv150806690R
 Keywords:

 Mathematics  Probability