Ordinal Pattern Dependence in the Context of Long-Range Dependence

TL;DR

This paper explores how ordinal pattern dependence behaves in long-range dependent time series, deriving limit distributions for estimators and highlighting differences based on dependence structures.

Contribution

It extends ordinal pattern dependence analysis to long-range dependent series, providing new limit theorems and including mixed dependence cases.

Findings

Limit distributions depend on the dependence structure.

Central and non-central limit theorems are established.

Simulation confirms theoretical results.

Abstract

Ordinal pattern dependence is a multivariate dependence measure based on the co-movement of two time series. In strong connection to ordinal time series analysis, the ordinal information is taken into account to derive robust results on the dependence between the two processes. This article deals with ordinal pattern dependence for long-range dependent time series including mixed cases of short- and long-range dependence. We investigate the limit distributions for estimators of ordinal pattern dependence. In doing so we point out the differences that arise for the underlying time series having different dependence structures. Depending on these assumptions, central and non-central limit theorems are proven. The limit distributions for the latter ones can be included in the class of multivariate Rosenblatt processes. Finally, a simulation study is provided to illustrate our theoretical findings.