Regeneration of branching processes with immigration in varying environments
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866929604304830464 |
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| author | Li, Baozhi Sun, Hongyan Wang, Hua-Ming |
| author_facet | Li, Baozhi Sun, Hongyan Wang, Hua-Ming |
| contents | In this paper, we consider certain linear-fractional branching processes with immigration in varying environments. For $n\ge0,$ let $Z_n$ counts the number of individuals of the $n$-th generation, which excludes the immigrant which enters into the system at time $n.$ We call $n$ a regeneration time if $Z_n=0.$ We give first a criterion for the finiteness or infiniteness of the number of regeneration times. Then, we construct some concrete examples to exhibit the strange phenomena caused by the so-called varying environments. It may happen that the process is extinct but there are only finitely many regeneration times. Also, when there are infinitely many regeneration times, we show that for each $\varepsilon>0,$ the number of regeneration times in $[0,n]$ is no more than $(\log n)^{1+\varepsilon}$ as $n\rightarrow\infty.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_00490 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Regeneration of branching processes with immigration in varying environments Li, Baozhi Sun, Hongyan Wang, Hua-Ming Probability 60J80, 60J10 In this paper, we consider certain linear-fractional branching processes with immigration in varying environments. For $n\ge0,$ let $Z_n$ counts the number of individuals of the $n$-th generation, which excludes the immigrant which enters into the system at time $n.$ We call $n$ a regeneration time if $Z_n=0.$ We give first a criterion for the finiteness or infiniteness of the number of regeneration times. Then, we construct some concrete examples to exhibit the strange phenomena caused by the so-called varying environments. It may happen that the process is extinct but there are only finitely many regeneration times. Also, when there are infinitely many regeneration times, we show that for each $\varepsilon>0,$ the number of regeneration times in $[0,n]$ is no more than $(\log n)^{1+\varepsilon}$ as $n\rightarrow\infty.$ |
| title | Regeneration of branching processes with immigration in varying environments |
| topic | Probability 60J80, 60J10 |
| url | https://arxiv.org/abs/2205.00490 |