Small deviations of a Galton–Watson process with immigration

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 $p_{0}=0$, $p_{1}\neq0$, $q_{0}=0$ (that is, when $\operatorname{essinf}(\mathcal{Z}_{n})$ grows linearly in $n$), we establish the asymptotics of the left tail $\mathbb{P}\{\mathcal{W}<\varepsilon\}$, as $\varepsilon\downarrow0$, of the martingale limit $\mathcal{W}$ of the process $(\mathcal{Z}_{n})$. Further, we consider the first generation $\mathcal{K}$ such that $\mathcal{Z}_{\mathcal{K}}\operatorname{essinf}(\mathcal{Z}_{\mathcal{K}})$ and study the asymptotic behaviour of $\mathcal{K}$ conditionally on $\{\mathcal{W}<\varepsilon\}$, as $\varepsilon\downarrow 0$. We find the growth scale and the fluctuations of $\mathcal{K}$ and compare the results with those for standard Galton–Watson processes.