On Null-homology and stationary sequences

TL;DR

This paper provides a new proof of Schmidt's coboundary theorem, clarifies null-homology concepts in stationary sequences, and explores their applications in ergodic theory and Markov processes.

Contribution

It offers a concise proof of a key criterion for null-homology in stationary processes and examines its implications in various probabilistic and ergodic contexts.

Findings

New proof of Schmidt's coboundary theorem

Comparison between null-homology and strict-sense null-homology

Application of null-homology concepts to Markov random walks

Abstract

The concept of homology, originally developed as a useful tool in algebraic topology, has by now become pervasive in quite different branches of mathematics. The notion particularly appears quite naturally in ergodic theory in the study of measure-preserving transformations arising from various group actions or, equivalently, the study of stationary sequences when adopting a probabilistic perspective as in this paper. Our purpose is to give a new and relatively short proof of the coboundary theorem due to Schmidt (1977) which provides a sharp criterion that determines (and rules out) when two stationary processes belong to the same \emph{null-homology equivalence class}. We also discuss various aspects of null-homology within the class of Markov random walks, compare null-homology with a formally stronger notion which we call {\it strict-sense null-homology}. Finally, we also discuss some concrete cases where the notion of null-homology turns up in a relevant manner.