A Limit Theorem for the Norm of Random Matrices

This paper establishes an almost sure limit for the operator norm of rectangular random matrices: Suppose ᵢⱼ = 1,2, ⋯, j = 1,2, ⋯ are zero mean i.i.d. random variables satisfying the moment condition E|ν₁₁|ⁿ nα n for all n 2, and some α. Let σ² = Ev²₁₁ and let Vₚₙ be the p × n matrix ᵢⱼ\1 i p; 1 j n. If pₙ is a sequence of integers such that pₙ/n → y as n → ∞, for some 0 < y < ∞, then 1/n|VpₙnVTpₙn| → (1 + y1/2)²σ² almost surely, where $|A|$ denotes the operator ("induced") norm of A. Since 1/n|VpₙnVTpₙn| is the maximum eigenvalue of 1/nVpₙnVTpₙn, the result relates to studies on the spectrum of symmetric random matrices.