Rates of convergence for laws of the spectral maximum of free random variables

TL;DR

This paper establishes the rate at which the spectral maximum of free random variables converges to the free extreme value distribution, providing quantitative bounds in the Kolmogorov distance.

Contribution

It introduces explicit convergence rates for the spectral maximum of free random variables towards the free extreme value distribution.

Findings

Derived explicit convergence rates in Kolmogorov distance.

Quantified the speed of convergence for spectral maxima.

Extended understanding of free extreme value theory.

Abstract

Let $\{X_n\}_n$ be a sequence of freely independent, identically distributed non-commutative random variables. Consider a sequence $\{W_n\}_n$ of the renormalized spectral maximum of random variables $X_1,\cdots, X_n$. It is known that the renormalized spectral maximum $W_n$ converges to the free extreme value distribution under certain conditions on the distribution function. In this paper, we provide a rate of convergence in the Kolmogorov distance between a distribution function of $W_n$ and the free extreme value distribution.