Poisson representations for tree-indexed Markov chains
Fuente:
arXiv
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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866916581809848320 |
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| author | Forsström, Malin Palö |
| author_facet | Forsström, Malin Palö |
| contents | In~\cite{fgs}, the class of Poisson representable processes was introduced. Several well-known processes were shown not to belong to this class, with examples including both the Curie Weiss model and the Ising model on $ \mathbb{Z}^2 $ for certain choices of parameters. Curiously, it was also shown that all positively associated $ \{ 0,1 \}$-valued Markov chains do belong to this class. In this paper, we interpolate between Markov chains and Ising models by considering tree-indexed Markov chains. In particular, we show that for any finite tree that is not a path, whether or not the corresponding tree-indexed Markov chain is representable always depends on the parameters. Moreover, we give an example of a family of infinite trees such that the corresponding tree-indexed Markov chains are representable for some non-trivial parameters. In addition, we give alternative proofs and arguments and also strengthen several of the results in~\cite{fgs}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_14428 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Poisson representations for tree-indexed Markov chains Forsström, Malin Palö Probability In~\cite{fgs}, the class of Poisson representable processes was introduced. Several well-known processes were shown not to belong to this class, with examples including both the Curie Weiss model and the Ising model on $ \mathbb{Z}^2 $ for certain choices of parameters. Curiously, it was also shown that all positively associated $ \{ 0,1 \}$-valued Markov chains do belong to this class. In this paper, we interpolate between Markov chains and Ising models by considering tree-indexed Markov chains. In particular, we show that for any finite tree that is not a path, whether or not the corresponding tree-indexed Markov chain is representable always depends on the parameters. Moreover, we give an example of a family of infinite trees such that the corresponding tree-indexed Markov chains are representable for some non-trivial parameters. In addition, we give alternative proofs and arguments and also strengthen several of the results in~\cite{fgs}. |
| title | Poisson representations for tree-indexed Markov chains |
| topic | Probability |
| url | https://arxiv.org/abs/2501.14428 |