Weak convergence of the integral of semi-Markov processes
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866910953738600448 |
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| author | Pedicone, Andrea Cinque, Fabrizio |
| author_facet | Pedicone, Andrea Cinque, Fabrizio |
| contents | We study the asymptotic properties, in the weak sense, of regenerative processes and Markov renewal processes. For the latter, we derive both renewal-type results, also concerning the related counting process, and ergodic-type ones, including the so-called phi-mixing property. This theoretical framework permits us to study the weak limit of the integral of a semi-Markov process, which can be interpret as the position of a particle moving with finite velocities taken for a random time according to the Markov renewal process underlying the semi-Markov one. Under mild conditions, we obtain the weak convergence to scaled Brownian motion. As a particular case, this result establishes the weak convergence of the classical generalized telegraph process. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_13274 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Weak convergence of the integral of semi-Markov processes Pedicone, Andrea Cinque, Fabrizio Probability 60F17, 60K15, 60F05 We study the asymptotic properties, in the weak sense, of regenerative processes and Markov renewal processes. For the latter, we derive both renewal-type results, also concerning the related counting process, and ergodic-type ones, including the so-called phi-mixing property. This theoretical framework permits us to study the weak limit of the integral of a semi-Markov process, which can be interpret as the position of a particle moving with finite velocities taken for a random time according to the Markov renewal process underlying the semi-Markov one. Under mild conditions, we obtain the weak convergence to scaled Brownian motion. As a particular case, this result establishes the weak convergence of the classical generalized telegraph process. |
| title | Weak convergence of the integral of semi-Markov processes |
| topic | Probability 60F17, 60K15, 60F05 |
| url | https://arxiv.org/abs/2505.13274 |