Near-Epoch Dependence in Riesz Spaces

TL;DR

This paper extends the concept of near-epoch dependence to processes in Riesz spaces and demonstrates that such processes, when combined with mixing conditions, are mixingales, enabling a law of large numbers.

Contribution

It introduces the notion of near-epoch dependence in Riesz spaces and links it with mixingale properties under strong or uniform mixing conditions.

Findings

Near-epoch dependent processes are mixingales under certain mixing conditions.

Application to autoregressive processes of order 1 in Riesz spaces.

Establishment of a law of large numbers for these processes.

Abstract

The abstraction of the study of stochastic processes to Banach lattices and vector lattices has received much attention by Grobler, Kuo, Labuschagne, Stoica, Troitsky and Watson over the past fifteen years. By contrast mixing processes have received very little attention. In particular mixingales were generalized to the Riesz space setting in {\sc W.-C. Kuo, J.J. Vardy, B.A. Watson,} Mixingales on Riesz spaces, {\em J. Math. Anal. Appl.}, \textbf{402} (2013), 731-738. The concepts of strong and uniform mixing as well as related mixing inequalities were extended to this setting in {\sc W.-C. Kuo, M.J. Rogans, B.A. Watson,} Mixing inequalities in Riesz spaces, {\em J. Math. Anal. Appl.}, \textbf{456} (2017), 992-1004. In the present work we formulate the concept of near-epoch dependence for Riesz space processes and show that if a process is near-epoch dependent and either strong or uniform mixing then the process is a mixingale, giving access to a law of large numbers. The above is applied to autoregessive processes of order 1 in Riesz spaces.