Precise Large Deviations for the Total Population of Heavy-Tailed Subcritical Branching Processes with Immigration

In this article, we focus on the partial sum \(S_{n}=X_{1}+\cdots +X_{n}\) $S_n=X_1+\cdots +X_n$ of the subcritical branching process with immigration \(\{X_{n}\}_{n\in \mathbb {N_{+}}}\) ${\{X_n\}}_{n\in {\mathbb{N}}_{+}}$ , under the condition that one of the offspring \(\xi \) $\xi$ or immigration \(\eta \) $\eta$ is regularly varying. The tail distribution of \(S_n\) $S_n$ is heavily dependent on that of \(\xi \) $\xi$ and \(\eta \) $\eta$ , and a precise large deviation probability for \(S_{n}\) $S_n$ is specified. (i) When the tail of offspring \(\xi \) $\xi$ is “lighter” than immigration \(\eta \) $\eta$ , then uniformly for \(x\ge x_{n}\) $x\ge x_n$ we have \(P(S_{n}-ES_{n}>x)\sim c_{1}nP(\eta >x)\) $P(S_n-E{S}_n>x)\sim c_{1}nP(\eta >x)$ with some constant \(c_{1}\) $c_1$ and sequence \(\{x_{n}\}\) $\{x_n\}$ , where \(c_{1}\) $c_1$ is related only to the mean of offspring; (ii) when the tail of immigration \(\eta \) $\eta$ is not “heavier” than offspring \(\xi \) $\xi$ , then uniformly for \(x\ge x_{n}\) $x\ge x_n$ we have \(P(S_{n}-ES_{n}>x)\sim c_{2}nP(\xi >x)\) $P(S_n-E{S}_n>x)\sim c_{2}nP(\xi >x)$ with some constant \(c_{2}\) $c_2$ and sequence \(\{x_{n}\}\) $\{x_n\}$ , where \(c_{2}\) $c_2$ is related to both the mean of offspring and the mean of immigration.