Precise Large Deviations For The Total Population Of Heavy-Tailed Subcritical Branching Process With Immigration

Abstract

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.