Precise Large Deviations for the Total Population of Heavy-tailed Critical Branching Processes with Immigration

We focus on the partial sum $S_{n}=X_{1}+\cdots+X_{n}$ of the critical branching process with immigration $\{X_{n}\}$, when the offspring $\xi$ is regularly varying with index $\nu+1$ and the immigration $\eta$ is regularly varying with index $\delta$ $(0\leq \nu<\delta<1)$. The precise large deviation probabilities for $S_{n}$ are specified, that is, for some appropriate sequences $\{x_{n}\}$ and $\{y_{n}\}$, uniformly for $x_{n}\leq x\leq y_{n}$, $P(S_{n}>x)\sim nx^{-\delta/(1+\nu)}L(x)$, where $L(x)$ is a slowly varying function. Different from that of the subcritical case, here the upper bound $y_n$ is needed. Essentially, this is because the tail probability of the stationary distribution is determined by the offspring or the immigration in the subcritical case. But it is determined by both when the process is critical.