Necessary and Sufficient Conditions for Almost Sure Convergence of the Largest Eigenvalue of a Wigner Matrix

Abstract

Let $W = (X_{ij}; 1 \leq i, j < \infty)$ be an infinite matrix. Suppose $W$ is symmetric, entries on the diagonal are $\operatorname{iid}$, entries off the diagonal are $\operatorname{iid}$ and they are independent. Then it is proved that the necessary and sufficient conditions for $\lambda_{\max}((1/\sqrt{n})W_n) \rightarrow a \mathrm{a.s.}$ are (1) $E(X^+_{11})^2 < \infty$; (2) $EX^4_{12} < \infty$; (3) $EX_{12} \leq 0$; (4) $a = 2\sigma, \sigma^2 = EX^2_{12}$. Here $W_n = (X_{ij}; 1 \leq i, j \leq n)$.