Central limit theorems for local empirical processes near boundaries of sets

TL;DR

This paper establishes a uniform central limit theorem for local empirical processes near set boundaries, using measure differentiation, with applications in statistical analysis.

Contribution

It introduces a framework for analyzing local empirical processes near boundaries, extending CLT results to neighborhoods that shrink with sample size.

Findings

Uniform CLT holds for local empirical processes near boundaries.

Differentiation of sets in measure is key to the analysis.

Includes examples and applications in statistics.

Abstract

We define the local empirical process, based on $n$ i.i.d. random vectors in dimension $d$, in the neighborhood of the boundary of a fixed set. Under natural conditions on the shrinking neighborhood, we show that, for these local empirical processes, indexed by classes of sets that vary with $n$ and satisfy certain conditions, an appropriately defined uniform central limit theorem holds. The concept of differentiation of sets in measure is very convenient for developing the results. Some examples and statistical applications are also presented.