On limiting probability distributions of higher order Markov chains
Lixing Han, XU Jian-hong · Linear and Multilinear Algebra · 2026
The limiting probability distribution is one of the key characteristics of a Markov chain since it shows its long-term behaviour. 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.