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 $W_n = (X_{ij}; 1 \\leq i, j \\leq n)$.