A weak convergence result for sequential empirical processes under weak dependence

TL;DR

This paper establishes a weak convergence result for sequential empirical processes under weak dependence, removing the need for boundedness assumptions and applying to broader classes of functions.

Contribution

It proves a new weak convergence result for empirical processes with $eta$-mixing sequences, extending applicability beyond existing bounded function class assumptions.

Findings

Weak convergence proven for empirical processes under $eta$-mixing.

Results apply to unbounded function classes.

Provides examples in statistical applications.

Abstract

The purpose of this paper is to prove a weak convergence result for empirical processes indexed in general classes of functions and with an underlying $\alpha$-mixing sequence of random variables. In particular the uniformly boundedness assumption on the function class, which is required in most of the existing literature, is spared. Furthermore under strict stationarity a weak convergence result for the sequential empirical process indexed in function classes is obtained, as a direct consequence. Two examples in mathematical statistics, that cannot be treated with existing results, are given as possible applications.