## Probability Seminar: Konstantin Tikhomirov

- Date: 09/30/2015
- Time: 15:00

*University of Alberta*

University of British Columbia

The Smallest Singular Value of Random Matrices with Independent Entries

We consider a classical problem of estimating
the smallest singular value of random rectangular and square matrices with
independent identically distributed entries. The novelty of our results lies in
very weak, or nonexisting, moment assumptions on the distribution of the
entries. We prove that, given a sufficiently “tall” $N \times n$ rectangular
matrix $A = (a_{ij} )$ with i.i.d. entries satisfying the condition
$\sup_{\lambda \in \mathbb{R}} \mathbb{P} \left\{ \lvert a_{ij} − \lambda
\rvert \le 1 \right\} \le 1/2$, the smallest singular value $s_n (A)$ satisfies
$s_n(A) \gtrsim \sqrt{N}$ with probability very close to one.

Our second theorem is an extension of the fundamental result of Bai and
Yin from the early 1990’s. Let $\{a_{ij} \}^\infty_{i,j=1}$ be an infinite
double array of i.i.d. random variables with zero mean and unit variance, and
let $(N_m )_{m=1}^\infty$ be an integer sequence satisfying $\lim_{m \to
\infty} \frac{N_m}{m} = r\) for some \(r \in (1, \infty)$. Then, denoting by
$A_m$ the $N_m \times m$ top-left corner of the array $\{a_{ij}\}$, we
have

\[ \lim_{m \to \infty} \frac{s_m(A_m)}{\sqrt{N_m}} = \sqrt{r}-1
\hspace{3mm}\mbox{ almost surely}.\]

This 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.

Location: ESB 2012