A Limit Theorem for the Norm of Random Matrices

Abstract

This paper establishes an almost sure limit for the operator norm of rectangular random matrices: Suppose $\{v_{ij}\}i = 1,2, \cdots, j = 1,2, \cdots$ are zero mean i.i.d. random variables satisfying the moment condition $E|\nu_{11}|^n \leqslant n^{\alpha n}$ for all $n \geqslant 2$, and some $\alpha$. Let $\sigma^2 = Ev^2_{11}$ and let $V_{pn}$ be the $p \times n$ matrix $\{v_{ij}\}_{1\leqslant i\leqslant p; 1\leqslant j\leqslant n}$. If $p_n$ is a sequence of integers such that $p_n/n \rightarrow y$ as $n \rightarrow \infty$, for some $0 < y < \infty$, then $1/n|V_{p_nn}V^T_{p_nn}| \rightarrow (1 + y^{\frac{1}{2}})^2\sigma^2$ almost surely, where $|A|$ denotes the operator ("induced") norm of $A$. Since $1/n|V_{p_nn}V^T_{p_nn}|$ is the maximum eigenvalue of $1/nV_{p_nn}V^T_{p_nn}$, the result relates to studies on the spectrum of symmetric random matrices.