Lower deviations for branching processes with immigration

Let $\{Y_n$, $n\ge 1\}$ be a critical branching process with immigration having finite variance for the offspring number of particles and finite mean for the immigrating number of particles. In this paper, we study lower deviation probabilities for $Y_n$. More precisely, assuming that $k,n\to \mathrm{\infty }$ such that $k=o\left(n\right)$, we investigate the asymptotics of $\mathbf{P}(Y_n\le k)$ and $\mathbf{P}(Y_n=k)$. Our results clarify the role of the moment conditions in the local limit theorem for $Y_n$ proven by Mellein.