The Sequential Empirical Process of Nonlinear Long-Range Dependent Random Vectors

TL;DR

This paper investigates the asymptotic behavior of the sequential empirical process for multivariate long-range dependent Gaussian processes under different subordination types, revealing complex limiting processes involving Hermite processes.

Contribution

It extends the understanding of empirical processes for nonlinear long-range dependent vectors, characterizing their asymptotic limits under various subordination schemes.

Findings

Limiting processes are either a product of a deterministic function and a Hermite process or a sum of such processes.

The study provides a detailed description of the asymptotic behavior of the empirical process in the multivariate long-range dependence setting.

The results generalize known one-dimensional cases to multivariate contexts.

Abstract

Let $(G(X_j))_{j\geq1}$ be a multivariate subordinated Gaussian process, which exhibits long-range dependence. We study the asymptotic behaviour of the corresponding sequential empirical process under two different types of subordination. The limiting process is either a product of a deterministic function and a Hermite process as in the one-dimensional case or a sum of various processes of this kind.