On Null-Homology and Stationary Sequences

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 (Cocycles on ergodic transformation groups. Macmillan lectures in mathematics, vol 1, Macmillan Company of India, Ltd., Delhi, 1977) which provides a sharp criterion that determines (and rules out) when two stationary processes belong to the same null-homology equivalence class. We also discuss various aspects of null-homology within the class of Markov random walks and compare null-homology with a formally stronger notion which we call strict-sense null-homology. Finally, we also discuss some concrete cases where the notion of null-homology turns up in a relevant manner.