The Annals of Probability

The Spectral Radius of Large Random Matrices

Stuart Geman

Abstract

Let $\{m_{ij}\}, i = 1,2,\ldots, j = 1,2,\ldots,$ be iid random variables with $Em_{11} = 0$ and $Em^2_{11} = \sigma^2$. For each $n$ define $M_n = \{m_{ij}\}_{1 \leq i, j \leq n}$, the $n \times n$ matrix whose $(i, j)$ component is $m_{ij}$. We show that $\lim \sup_{n \rightarrow \infty}\rho_n \leq \sigma$ a.s., where $\rho_n$ is the spectral radius of $M_n/\sqrt n$. Evidence from computer experiments indicates that in fact $\rho_n \rightarrow \sigma$ a.s.

Article information

Source
Ann. Probab. Volume 14, Number 4 (1986), 1318-1328.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
http://projecteuclid.org/euclid.aop/1176992372

Digital Object Identifier
doi:10.1214/aop/1176992372

Mathematical Reviews number (MathSciNet)
MR866352

Zentralblatt MATH identifier
0605.60037

JSTOR