Variable Length Memory Chains: Characterization of stationary probability measures
| Autor: | Brigitte Chauvin, Peggy Cénac, Camille Noûs, Nicolas Pouyanne, Frédéric Paccaut |
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| Přispěvatelé: | Institut de Mathématiques de Bourgogne [Dijon] (IMB), Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université de Bourgogne (UB), Laboratoire de Mathématiques de Versailles (LMV), Université de Versailles Saint-Quentin-en-Yvelines (UVSQ)-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Cogitamus Laboratory, Laboratoire Amiénois de Mathématique Fondamentale et Appliquée - UMR CNRS 7352 (LAMFA), Université de Picardie Jules Verne (UPJV)-Centre National de la Recherche Scientifique (CNRS) |
| Rok vydání: | 2021 |
| Předmět: |
Statistics and Probability
Pure mathematics Longest Internal Suffix Stationary distribution Markov chain 60J05 60C05 60G10 Probability (math.PR) 010102 general mathematics 01 natural sciences Measure (mathematics) Variable Length Memory Chains 010104 statistics & probability Probability theory Convergence of random variables FOS: Mathematics Countable set State space Renewal theory [MATH]Mathematics [math] 0101 mathematics stable context trees semi-Markov chains Mathematics - Probability stationary probability measure Mathematics |
| Zdroj: | Bernoulli Bernoulli, Bernoulli Society for Mathematical Statistics and Probability, 2021, 27 (3), pp.2011-2039. ⟨10.3150/20-BEJ1299⟩ |
| ISSN: | 1350-7265 |
| DOI: | 10.3150/20-bej1299 |
| Popis: | Variable Length Memory Chains (VLMC), which are generalizations of finite order Markov chains, turn out to be an essential tool to modelize random sequences in many domains, as well as an interesting object in contemporary probability theory. The question of the existence of stationary probability measures leads us to introduce a key combinatorial structure for words produced by a VLMC: the Longest Internal Suffix. This notion allows us to state a necessary and sufficient condition for a general VLMC to admit a unique invariant probability measure. This condition turns out to get a much simpler form for a subclass of VLMC: the stable VLMC. This natural subclass, unlike the general case, enjoys a renewal property. Namely, a stable VLMC induces a semi-Markov chain on an at most countable state space. Unfortunately, this discrete time renewal process does not contain the whole information of the VLMC, preventing the study of a stable VLMC to be reduced to the study of its induced semi-Markov chain. For a subclass of stable VLMC, the convergence in distribution of a VLMC towards its stationary probability measure is established. Finally, finite state space semi-Markov chains turn out to be very special stable VLMC, shedding some new light on their limit distributions. Comment: arXiv admin note: text overlap with arXiv:1807.01075 |
| Databáze: | OpenAIRE |
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