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arXiv · 2601.06674

Reduction and classification of higher-order Markov chains

Christophe Gallesco·Caio Teodore Genovese Huss Oliveira·Daniel Yasumasa Takahashi
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Abstract

We study the class structure of finite-alphabet Markov chains with arbitrary memory length. To capture the structural constraints induced by prohibited transitions, we introduce the skeleton of a higher-order transition kernel, defined as a reduced set of contexts encoding all essential zero-probability patterns. To each skeleton we associate a binary transition matrix. We show that the communicating class structure of this matrix completely determines the recurrent classes of the original higher-order Markov chain, along with their periods. As a consequence, simple criteria for essential irreducibility and periodicity follow directly from the skeleton, without constructing the full first-order representation on the enlarged state space. From a practical perspective, this approach can yield significant computational gains. An example illustrates how the skeleton may have substantially smaller order than the original chain.

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Christophe Gallesco, Caio Teodore Genovese Huss Oliveira, Daniel Yasumasa Takahashi. 2026-01-10. Reduction and classification of higher-order Markov chains. https://doi.org/10.1016/j.spl.2026.110787

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