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 $p_0=0$, $p_1\ne 0$, $q_0=0$ (that is, when $\text{essinf}({\mathcal{Z}}_n)$ grows linearly in $n$), we establish the asymptotics of the left tail $\mathbb{P}\{\mathcal{W}<\epsilon \}$, as $\epsilon \downarrow 0$, 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 $\{\mathcal{W}<\epsilon \}$, as $\epsilon \downarrow 0$. 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.