On the Weak Convergence of the Function-Indexed Sequential Empirical Process and its Smoothed Analogue under Nonstationarity

TL;DR

This paper establishes the weak convergence of function-indexed sequential empirical processes and their smoothed versions under nonstationary conditions, providing broad applicability to dependent time series.

Contribution

It introduces new conditions for asymptotic equicontinuity and proves weak convergence under mild assumptions for nonstationary, dependent series.

Findings

Weak convergence of sequential empirical process established

Applicable to nonstationary α-mixing series

Introduces a novel maximal inequality for nonmeasurable processes

Abstract

We study the sequential empirical process indexed by general function classes and its smoothed set-indexed analogue. Sufficient conditions for asymptotic equicontinuity are provided for nonstationary arrays of time series. This yields comprehensive general results that are applicable to various notions of dependence, which is exemplified in detail for nonstationary $\alpha$-mixing series. Especially, we obtain the weak convergence of the sequential process under essentially the same mild assumptions as known for the classical empirical process. Core ingredients of the proofs are a novel maximal inequality for nonmeasurable stochastic processes, uniform chaining arguments and, for the set-indexed smoothed process, uniform Lipschitz properties.