Non-Markovian chains with long-range dependence and their scaling limits
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| Format: | Preprint |
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2026
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| _version_ | 1866915819954372608 |
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| author | Facciaroni, Lorenzo Ricciuti, Costantino Scalas, Enrico |
| author_facet | Facciaroni, Lorenzo Ricciuti, Costantino Scalas, Enrico |
| contents | There is a well-established theory linking certain semi-Markov chains and continuous-time random walks to time-fractional equations and anomalous diffusion. In this work, we go beyond the semi-Markov framework by considering some non-Markovian chains, which exhibit long-memory behaviour, due to stochastic dependence among their waiting times. Particular attention is devoted to the so-called para-Markov chains. Their waiting times share the same marginal distributions as those of the above mentioned semi-Markov chains, but they are dependent; their joint distribution is of Schur-constant type and is closely related to complete Bernstein functions and De Finetti's theorems. A second model that we focus on is given by time-changed Markov chains, where the random time is the inverse of an increasing stable process. This generalizes well-known semi-Markov models available in the literature, which typically focus solely on the inverse of the Levy stable subordinator. The above mentioned models are unified by a general theory of time change of Markov chains. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_23049 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Non-Markovian chains with long-range dependence and their scaling limits Facciaroni, Lorenzo Ricciuti, Costantino Scalas, Enrico Probability There is a well-established theory linking certain semi-Markov chains and continuous-time random walks to time-fractional equations and anomalous diffusion. In this work, we go beyond the semi-Markov framework by considering some non-Markovian chains, which exhibit long-memory behaviour, due to stochastic dependence among their waiting times. Particular attention is devoted to the so-called para-Markov chains. Their waiting times share the same marginal distributions as those of the above mentioned semi-Markov chains, but they are dependent; their joint distribution is of Schur-constant type and is closely related to complete Bernstein functions and De Finetti's theorems. A second model that we focus on is given by time-changed Markov chains, where the random time is the inverse of an increasing stable process. This generalizes well-known semi-Markov models available in the literature, which typically focus solely on the inverse of the Levy stable subordinator. The above mentioned models are unified by a general theory of time change of Markov chains. |
| title | Non-Markovian chains with long-range dependence and their scaling limits |
| topic | Probability |
| url | https://arxiv.org/abs/2602.23049 |