A few remarks on the operator norm of random Toeplitz matrices

TL;DR

This paper investigates the almost sure behavior of the operator norm of random Toeplitz matrices, providing laws of large numbers and concentration inequalities to understand their asymptotic properties.

Contribution

It introduces new concentration inequalities for empirical process suprema, refining recent results and applying them to analyze the operator norm of random Toeplitz matrices.

Findings

Law of large numbers for the normalized operator norm

Refined concentration inequalities for empirical processes

Asymptotic almost sure behavior of Toeplitz matrix norms

Abstract

We present some results concerning the almost sure behaviour of the operator norm or random Toeplitz matrices, including the law of large numbers for the norm, normalized by its expectation (in the i.i.d. case). As tools we present some concentration inequalities for suprema of empirical processes, which are refinements of recent results by Einmahl and Li.