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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2601.06674 |
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| _version_ | 1866911614535467008 |
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| author | Gallesco, Christophe Oliveira, Caio Teodore Genovese Huss Takahashi, Daniel Yasumasa |
| author_facet | Gallesco, Christophe Oliveira, Caio Teodore Genovese Huss Takahashi, Daniel Yasumasa |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_06674 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Reduction and classification of higher-order Markov chains Gallesco, Christophe Oliveira, Caio Teodore Genovese Huss Takahashi, Daniel Yasumasa Statistics Theory Probability 60J10 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. |
| title | Reduction and classification of higher-order Markov chains |
| topic | Statistics Theory Probability 60J10 |
| url | https://arxiv.org/abs/2601.06674 |