The Function-Indexed Sequential Empirical Process under Long-Range Dependence

TL;DR

This paper investigates the asymptotic behavior of the sequential empirical process for multivariate long-range dependent Gaussian data, showing convergence to Hermite processes under certain entropy conditions.

Contribution

It extends the understanding of empirical processes to long-range dependent Gaussian sequences, identifying conditions for weak convergence to Hermite processes.

Findings

Weak convergence to Hermite processes established

Entropy condition identified for convergence

Applicable to multivariate long-range dependent data

Abstract

Let $(X_j)_{j\geq1}$ be a multivariate long-range dependent Gaussian process. We study the asymptotic behavior of the corresponding sequential empirical process indexed by a class of functions. If some entropy condition is satisfied we have weak convergence to a linear combination of Hermite processes.