Let \(\{Y_{n}\), \(n \geq 1\}\) 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 \(k,n \to \infty\) such that \(k=o\left(n \right)\),...

Lower deviations for branching processes with immigration