A Limit Theorem for the Norm of Random Matrices

TLDR

The study builds on spectral theory of symmetric random matrices, where the operator norm equals the largest eigenvalue of the sample covariance matrix. The authors aim to prove an almost sure limit for the operator norm of rectangular random matrices with i.i.d. entries satisfying a moment growth condition. They consider zero‑mean i.i.d.

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.