Continuity properties of a factor of Markov chains

TL;DR

This paper investigates the conditions under which a simplified Markov chain, derived by merging symbols, maintains continuous transition probabilities relative to its past, providing insights into the chain's structural properties.

Contribution

It establishes necessary and sufficient conditions for the continuity of transition probabilities in a reduced Markov chain obtained by symbol merging.

Findings

Identifies conditions for transition probability continuity.

Provides a theoretical framework for simplified Markov chains.

Enhances understanding of chain structure after symbol merging.

Abstract

Starting from a Markov chain with a finite alphabet, we consider the chain obtained when all but one symbol are undistinguishable for the practitioner. We study necessary and sufficient conditions for this chain to have continuous transition probabilities with respect to the past.