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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2503.05575 |
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| _version_ | 1866917947829649408 |
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| author | André, Mathilde Duchamps, Jean-Jil |
| author_facet | André, Mathilde Duchamps, Jean-Jil |
| contents | We investigate Kesten-Stigum-like results for multi-type Galton-Watson processes with a countable number of types in a general setting, allowing us in particular to consider processes with an infinite total population at each generation. Specifically, a sharp $L\log L$ condition is found under the only assumption that the mean reproduction matrix is positive recurrent in the sense of Vere-Jones (1967). The type distribution is shown to always converge in probability in the recurrent case, and under conditions covering many cases it is shown to converge almost surely. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_05575 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sharp $L \log L$ condition for supercritical Galton-Watson processes with countable types André, Mathilde Duchamps, Jean-Jil Probability We investigate Kesten-Stigum-like results for multi-type Galton-Watson processes with a countable number of types in a general setting, allowing us in particular to consider processes with an infinite total population at each generation. Specifically, a sharp $L\log L$ condition is found under the only assumption that the mean reproduction matrix is positive recurrent in the sense of Vere-Jones (1967). The type distribution is shown to always converge in probability in the recurrent case, and under conditions covering many cases it is shown to converge almost surely. |
| title | Sharp $L \log L$ condition for supercritical Galton-Watson processes with countable types |
| topic | Probability |
| url | https://arxiv.org/abs/2503.05575 |