Small deviations of a Galton-Watson process with immigration

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.