Spectral norm of random Toeplitz matrices

TL;DR

This paper proves that the largest eigenvalue of symmetric random Toeplitz matrices, scaled by , converges almost surely to 1, revealing spectral norm behavior under specific moment conditions.

Contribution

The paper establishes the almost sure convergence of the scaled spectral norm of symmetric random Toeplitz matrices under new moment conditions.

Findings

Largest eigenvalue scaled by  converges to 1 almost surely.

Spectral norm growth rate is characterized as  log(n).

Results hold under specific moment conditions on the entries.

Abstract

In this work, we consider symmetric random Toeplitz matrices $T_n$ generated by i.i.d. zero mean random variables ${X_k}$ satisfying the moment conditions: $E|X_k|^2=1$ and $\E|X_1|^n \le n^{\sqrt{n}}$ for all $n\ge 3$. We prove that the largest eigenvalue of $T_n$ scaled by $\sqrt{n log(n)}$ converges almost surely to $1$.