Macroscale behavior of random lower triangular matrices

TL;DR

This paper investigates the large-scale behavior of random lower triangular matrices with iid entries, demonstrating their convergence to the Volterra operator in certain modes and providing insights into their moments.

Contribution

It introduces a novel analysis of the asymptotic behavior of random triangular matrices, connecting them to the Volterra operator and exploring convergence modes.

Findings

Matrices behave like probabilistic Riemann sums

Converge to the Volterra operator in specific modes

Provides discussion on moments of the matrices

Abstract

We analyze the macroscale behavior of random lower (and therefore upper) triangular matrices with entries drawn iid from a distribution with nonzero mean and finite variance. We show that such a matrix behaves like a probabilistic version of a Riemann sum and therefore in the limit behaves like the Volterra operator. Specifically, we analyze certain SOT-like and WOT-like modes of convergence for random lower triangular matrices to a scaled Volterra operator. We close with a brief discussion of moments.