Mixing inequalities in Riesz spaces

TL;DR

This paper extends the theory of mixing inequalities for stochastic processes within Riesz spaces, focusing on alpha and phi mixing coefficients, and generalizes $L^1$ and $L^ty$ spaces to support these developments.

Contribution

It introduces new mixing inequalities for processes in Riesz spaces and extends the study of generalized $L^1$ and $L^ty$ spaces for this setting.

Findings

Derived mixing inequalities for alpha and phi mixing processes.

Extended the framework of generalized $L^1$ and $L^ty$ spaces.

Provided a unified approach to stochastic processes in Riesz spaces.

Abstract

Various topics in stochastic processes have been considered in the abstract setting of Riesz spaces, for example martingales, martingale convergence, ergodic theory, AMARTS, Markov processes and mixingales. Here we continue the relaxation of conditional independence begun in the study of mixingales and study mixing processes. The two mixing coefficients which will be considered are the $\alpha$ (strong) and $\varphi$ (uniform) mixing coefficients. We conclude with mixing inequalities for these types of processes. In order to facilitate this development, the study of generalized $L^1$ and $L^\infty$ spaces begun by Kuo, Labuschagne and Watson will be extended.