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