Operator $\ell^\infty \to \ell^\infty$ norm of products of random matrices

TL;DR

This paper analyzes the asymptotic behavior of the $\, ext{ell}^ ext{infty} o ext{ell}^ extfty$ operator norm of products of independent random matrices with i.i.d. entries, revealing a specific growth rate depending on matrix size and product length.

Contribution

It provides the first detailed asymptotic characterization of the operator norm for products of i.i.d. random matrices with certain moment conditions, including non-square and non-centered cases.

Findings

Operator norm of product behaves like $n^{(p+1)/2} imes ext{constant}$

Results hold for matrices with finite fourth moment and i.i.d. entries

Includes non-square and non-centered matrix products

Abstract

We study the $\ell^\infty \to \ell^\infty$ operator norm of products of independent random matrices with independent and identically distributed entries. For $n$-by-$n$ matrices whose entries are centered, have unit variance, and have a finite moment of order $4\alpha$ for some $\alpha > 1$, we find that the operator norm of the product of $p$ matrices behaves asymptotically like $n^{\frac {p+1}{2}}\sqrt{2/\pi}$. The case of products of possibly non-square matrices with possibly non-centered entries is also covered.