Lower deviations for supercritical branching processes with immigration

Abstract

for a supercritical branching processes with immigration {Zn}, it is known that under suitable conditions on the offspring and immigration distributions, Zn/mn converges almost surely to a finite and strictly positive limit, where m is the offspring mean. We are interested in the limiting properties of ℙ(Zn = kn) with kn = o(mn) as n → ∞. We give asymptotic behavior of such lower deviation probabilities in both Schroder and Bottcher cases, unifying and extending the previous results for Galton-Watson processes in literature.