Precise Large Deviations For The Total Population Of Heavy-Tailed Subcritical Branching Process With Immigration
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911869679173632 |
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| author | Guo, Jiayan Hong, Wenming |
| author_facet | Guo, Jiayan Hong, Wenming |
| contents | In this article we focus on the partial sum $S_{n}=X_{1}+\cdots+X_{n}$ of the subcritical branching process with immigration $\{X_{n}\}_{n\in\mathbb{N_{+}}}$, under the condition that one of the offspring $ξ$ or immigration $η$ is regularly varying. The tail distribution of $S_n$ is heavily dependent on that of $ξ$ and $η$, and a precise large deviation probability for $S_{n}$ is specified. (i)When the tail of offspring $ξ$ is lighter than immigration $η$, uniformly for $x\geq x_{n}$, $P(S_{n}-ES_{n}>x)\sim c_{1}nP(η>x)$ with some constant $c_{1}$ and sequence $\{x_{n}\}$, where $c_{1}$ is only related to the mean of offspring; (ii) When the tail of immigration $η$ is not heavier than offspring $ξ$, uniformly for $x\geq x_{n}$,$P(S_{n} ES_{n}>x)\sim c_{2}nP(ξ>x)$ with some constant $c_{2}$ and sequence $\{x_{n}\}$, where $c_{2}$ is related to both the mean of offspring and the mean of immigration. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_04073 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Precise Large Deviations For The Total Population Of Heavy-Tailed Subcritical Branching Process With Immigration Guo, Jiayan Hong, Wenming Probability In this article we focus on the partial sum $S_{n}=X_{1}+\cdots+X_{n}$ of the subcritical branching process with immigration $\{X_{n}\}_{n\in\mathbb{N_{+}}}$, under the condition that one of the offspring $ξ$ or immigration $η$ is regularly varying. The tail distribution of $S_n$ is heavily dependent on that of $ξ$ and $η$, and a precise large deviation probability for $S_{n}$ is specified. (i)When the tail of offspring $ξ$ is lighter than immigration $η$, uniformly for $x\geq x_{n}$, $P(S_{n}-ES_{n}>x)\sim c_{1}nP(η>x)$ with some constant $c_{1}$ and sequence $\{x_{n}\}$, where $c_{1}$ is only related to the mean of offspring; (ii) When the tail of immigration $η$ is not heavier than offspring $ξ$, uniformly for $x\geq x_{n}$,$P(S_{n} ES_{n}>x)\sim c_{2}nP(ξ>x)$ with some constant $c_{2}$ and sequence $\{x_{n}\}$, where $c_{2}$ is related to both the mean of offspring and the mean of immigration. |
| title | Precise Large Deviations For The Total Population Of Heavy-Tailed Subcritical Branching Process With Immigration |
| topic | Probability |
| url | https://arxiv.org/abs/2405.04073 |