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

Let W = (Xᵢⱼ; 1 ≤ i, j < ∞) be an infinite matrix. Suppose W is symmetric, entries on the diagonal are iid, entries off the diagonal are iid and they are independent. Then it is proved that the necessary and sufficient conditions for λₘₐₓ((1/√n)Wₙ) → a a.s. are (1) E(X⁺₁₁)² < ∞; (2) EX⁴₁₂ < ∞; (3) EX₁₂ ≤ 0; (4) a = 2σ, σ² = EX²₁₂. Here Wₙ = (Xᵢⱼ; 1 ≤ i, j ≤ n).