Precise Large Deviations For The Total Population Of Heavy-Tailed Subcritical Branching Process With Immigration

TL;DR

This paper derives precise large deviation probabilities for the total population in a subcritical branching process with immigration, highlighting how tail behaviors of offspring and immigration influence the tail distribution of the sum.

Contribution

It provides exact asymptotic formulas for large deviations of the total population, considering different tail heaviness scenarios of offspring and immigration distributions.

Findings

When offspring tail is lighter, deviations are driven by immigration tail.

When immigration tail is not heavier, deviations are driven by offspring tail.

Explicit constants relate to offspring and immigration means.

Abstract

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.