Can a Higher Order Markov Chain Be Treated as a First Order Markov Chain?

TL;DR

This paper explores the relationship between higher order and first order Markov chains, providing a comprehensive analysis and illustrative examples to address whether higher order chains can be treated as first order.

Contribution

It offers the first detailed analysis of how higher order Markov chains can be represented as first order chains, with illustrative examples and theoretical insights.

Findings

Higher order Markov chains can be represented as first order chains

Provides a comprehensive analysis of the relationship between the two types

Includes illustrative examples to clarify the concepts

Abstract

It is well known that any higher order Markov chain can be associated with a first order Markov chain. In this primarily expository article, we present the first fairly comprehensive analysis of the relationship between higher order and first order Markov chains, together with illustrative examples. Our main objective is to address the central question as posed in the title.