Small deviations of a Galton-Watson process with immigration

TL;DR

This paper analyzes the asymptotic behavior of a Galton-Watson process with immigration, focusing on the tail distribution of its martingale limit and the first generation exceeding the essential infimum, revealing new scaling laws and fluctuations.

Contribution

It provides new asymptotic results for the tail behavior of the martingale limit and the first exceeding generation in a Galton-Watson process with immigration, under specific offspring and immigration conditions.

Findings

Asymptotics of the left tail of the martingale limit $ ext{P}igrace{ ext{W}< extvarepsilonigrace}$ as $ extvarepsilon o 0$

Identification of the scale and fluctuations of the first generation $ extK$ exceeding the essential infimum

Comparison of results with standard Galton-Watson processes

Abstract

We consider a Galton-Watson process with immigration $(\mathcal{Z}_n)$, with offspring probabilities $(p_i)$ and immigration probabilities $(q_i)$. In the case when $p_0=0$, $p_1\neq 0$, $q_0=0$ (that is, when $\text{essinf} (\mathcal{Z}_n)$ grows linearly in $n$), we establish the asymptotics of the left tail $\mathbb{P}\{\mathcal{W}<\varepsilon\}$, as $\varepsilon\downarrow 0$, of the martingale limit $\mathcal{W}$ of the process $( \mathcal{Z}_n)$. Further, we consider the first generation $\mathcal{K}$ such that $\mathcal{Z}_{\mathcal{K}}>\text{essinf} (\mathcal{Z}_{\mathcal{K}})$ and study the asymptotic behaviour of $\mathcal{K}$ conditionally on $\{\mathcal{W}<\varepsilon\}$, as $\varepsilon\downarrow 0$. We find the scale at which $\mathcal{K}$ goes to infinity and describe the fluctuations of $\mathcal{K}$ around that scale. Finally, we compare the results with those for standard Galton-Watson processes.