Markov and almost Markov properties in one, two or more directions

TL;DR

This paper reviews and compares Markov and generalized dependence properties in multiple directions, discussing extensions like g-measures, variable length chains, and generalized Gibbs measures, highlighting their theoretical relationships and differences.

Contribution

It provides a comprehensive comparison of Markov properties and their generalizations in multiple directions, including new insights into various extended models.

Findings

Comparison of Markov and generalized dependence properties

Analysis of extensions like g-measures and variable neighborhood models

Discussion of properties and relationships among extended Markov models

Abstract

In this review-type paper written at the occasion of the Oberwolfach workshop {\em One-sided vs. Two-sided stochastic processes} (february 22-29, 2020), we discuss and compare Markov properties and generalisations thereof in more directions, as well as weaker forms of conditional dependence, again either in one or more directions. In particular, we discuss in both contexts various extensions of Markov chains and Markov fields and their properties, such as $g$-measures, Variable Length Markov Chains, Variable Neighbourhood Markov Fields, Variable Neighbourhood (Parsimonious) Random Fields, and Generalized Gibbs Measures.