Precise Large Deviations for the Total Population of Heavy-tailed Critical Branching Processes with Immigration
Jiayan Guo, Wenming Hong
TL;DR
This work derives precise large deviation asymptotics for the total population S_n of a heavy-tailed, critical branching process with immigration, showing P(S_n>x) scales as n x^{−δ/(1+ν)} L(x) uniformly over an admissible x-range. The proof hinges on decomposing S_n into a dominant part S_{n,1} and a negligible part S_{n,2}, leveraging regular variation of the stationary distribution X and the tails of the underlying process. The results illuminate how both offspring and immigration jointly govern extreme totals in the critical regime, in contrast to the subcritical case where a single mechanism typically dominates. The findings provide explicit upper and lower bounds via sequences x_n and y_n, clarifying the range of uniformity for the large deviation approximation and enriching the theory of heavy-tailed branching processes with immigration.
Abstract
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 $ξ$ is regularly varying with index $ν+1$ and the immigration $η$ is regularly varying with index $δ$ $(0\leq ν<δ<1)$. The precise large deviation probabilities for $S_{n}$ are specified, that is, for some appropriate sequences $\{x_{n}\}$ and $\{y_{n}\}$, uniformly for $x_{n}\leq x\leq y_{n}$, $P(S_{n}>x)\sim nx^{-δ/(1+ν)}L(x)$, 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.