Uniform approximation in classical weak convergence theory

TL;DR

This paper discusses the need for stronger uniform convergence results in classical weak convergence theory to establish asymptotic normality uniformly over families of probability measures.

Contribution

It compiles results on uniform approximation in weak convergence, extending classical theory to cover stronger, uniform convergence scenarios.

Findings

Provides conditions for uniform asymptotic normality

Extends classical weak convergence results

Focuses on weak convergence in ^d with continuous limits

Abstract

A common statistical task lies in showing asymptotic normality of certain statistics. In many of these situations, classical textbook results on weak convergence theory suffice for the problem at hand. However, there are quite some scenarios where stronger results are needed in order to establish an asymptotic normal approximation uniformly over a family of probability measures. In this note we collect some results in this direction. We restrict ourselves to weak convergence in $\mathbb R^d$ with continuous limit measures.