Limit theorems for critical branching processes in a finite state space Markovian environment
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912163009921024 |
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| author | Grama, Ion Lauvergnat, Ronan Page, Émile Le |
| author_facet | Grama, Ion Lauvergnat, Ronan Page, Émile Le |
| contents | Let $(Z_n)_{n\geq 0}$ be a critical branching process in a random environment defined by a Markov chain $(X_n)_{n\geq 0}$ with values in a finite state space $\mathbb X$. Let $ S_n = \sum_{k=1}^n \ln f_{X_k}'(1)$ be the Markov walk associated to $(X_n)_{n\geq 0}$, where $f_i$ is the offspring generating function when the environment is $i \in \mathbb X$. Conditioned on the event $\{ Z_n>0\}$, we show the non degeneracy of limit law of the normalized number of particles ${Z_n}/{e^{S_n}}$ and determine the limit of the law of $\frac{S_n}{\sqrt{n}} $ jointly with $X_n$. Based on these results we establish a Yaglom-type theorem which specifies the limit of the joint law of $ \log Z_n$ and $X_n$ given $Z_n>0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_15585 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Limit theorems for critical branching processes in a finite state space Markovian environment Grama, Ion Lauvergnat, Ronan Page, Émile Le Probability 60J80, 60F05, 60J10 Let $(Z_n)_{n\geq 0}$ be a critical branching process in a random environment defined by a Markov chain $(X_n)_{n\geq 0}$ with values in a finite state space $\mathbb X$. Let $ S_n = \sum_{k=1}^n \ln f_{X_k}'(1)$ be the Markov walk associated to $(X_n)_{n\geq 0}$, where $f_i$ is the offspring generating function when the environment is $i \in \mathbb X$. Conditioned on the event $\{ Z_n>0\}$, we show the non degeneracy of limit law of the normalized number of particles ${Z_n}/{e^{S_n}}$ and determine the limit of the law of $\frac{S_n}{\sqrt{n}} $ jointly with $X_n$. Based on these results we establish a Yaglom-type theorem which specifies the limit of the joint law of $ \log Z_n$ and $X_n$ given $Z_n>0$. |
| title | Limit theorems for critical branching processes in a finite state space Markovian environment |
| topic | Probability 60J80, 60F05, 60J10 |
| url | https://arxiv.org/abs/2412.15585 |