A CLT for empirical processes involving time-dependent data

TL;DR

This paper establishes conditions under which the empirical process of time-dependent stochastic processes satisfies the CLT uniformly, with applications to classical processes and counterexamples like Brownian motion.

Contribution

It provides sufficient conditions for the CLT to hold for empirical processes involving time-dependent data, extending classical results to new settings.

Findings

CLT holds for certain classical processes

Counterexample shows CLT fails for Brownian motion tied down at zero

Provides a framework for analyzing empirical processes with time dependence

Abstract

For stochastic processes $\{X_t:t\in E\}$, we establish sufficient conditions for the empirical process based on $\{I_{X_t\le y}-\operatorname{Pr}(X_t\le y):t\in E,y\in\mathbb{R}\}$ to satisfy the CLT uniformly in $t\in E,y\in\mathbb{R}$. Corollaries of our main result include examples of classical processes where the CLT holds, and we also show that it fails for Brownian motion tied down at zero and $E=[0,1]$.