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 $\xi$ or immigration $\eta$ is regularly varying. The tail distribution of $S_n$ is heavily dependent on that of $\xi$ and $\eta$, and a precise large deviation probability for $S_{ n}$ is specified. (i)When the tail of offspring $\xi$ is lighter than immigration $\eta$, uniformly for $x\geq x_{ n}$, $P(S_{ n}-ES_{ n}>x)\sim c_{ 1}nP(\eta>x)$ 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 $\eta$ is not heavier than offspring $\xi$, uniformly for $x\geq x_{ n}$,$P(S_{ n} ES_{ n}>x)\sim c_{ 2}nP(\xi>x)$ 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.
Categories: math.PR