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

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.

💡 Research Summary

The paper revisits the long‑standing claim that any higher‑order Markov chain can be treated as a first‑order chain by augmenting the state space. Starting from a rigorous definition, an (m‑1)‑th order chain on a finite state set S={1,…,n} is described by a transition tensor P=