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

TL;DR

This paper derives precise large deviation probabilities for the partial sums of a critical branching process with heavy-tailed offspring and immigration, revealing the influence of both factors on tail behavior.

Contribution

It provides the first detailed large deviation results for critical branching processes with heavy-tailed immigration and offspring, highlighting the combined effect on tail probabilities.

Findings

Large deviation probabilities are specified for the sum of the process.

The tail behavior depends on both offspring and immigration in the critical case.

Different from subcritical processes, an upper bound sequence is necessary.

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 $\xi$ 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.