On Limiting Probability Distributions of Higher Order Markov Chains

TL;DR

This paper investigates the long-term behavior of higher order Markov chains by establishing properties and conditions for the existence of their exact limiting probability distributions, extending known results from first order chains.

Contribution

It provides new theoretical conditions for the existence of exact limiting distributions in higher order Markov chains, extending and complementing prior approximation-based results.

Findings

Established a sufficient condition for the existence of limiting distributions.

Extended properties known for first order chains to higher order chains.

Provided illustrative examples demonstrating the theoretical results.

Abstract

The limiting probability distribution is one of the key characteristics of a Markov chain since it shows its long-term behavior. In this paper, for a higher order Markov chain, we establish some properties related to its exact limiting probability distribution, including a sufficient condition for the existence of such a distribution. Our results extend the corresponding conclusions on first order chains. Besides, they complement the existing results concerning higher order chains which rely on approximation schemes or two-phase power iterations. Several illustrative example are also given.