From convergence in distribution to uniform convergence

TL;DR

This paper establishes conditions under which convergence in distribution of probability measures implies uniform convergence of their quantile functions, enabling precise approximation of eigenvalues of large matrices.

Contribution

It provides new conditions linking distribution convergence to uniform convergence of quantile functions, extending classical theorems to stronger uniform results.

Findings

Conditions for passing from distribution to uniform convergence

Application to eigenvalues of large Hermitian matrices

Extension of Szegő-type theorems to uniform convergence

Abstract

We present conditions that allow us to pass from the convergence of probability measures in distribution to the uniform convergence of the associated quantile functions. Under these conditions, one can in particular pass from the asymptotic distribution of collections of real numbers, such as the eigenvalues of a family of $n$-by-$n$ matrices as $n$ goes to infinity, to their uniform approximation by the values of the quantile function at equidistant points. For Hermitian Toeplitz-like matrices, convergence in distribution is ensured by theorems of the Szeg\H{o} type. Our results transfer these convergence theorems into uniform convergence statements.