LAUSR

For a supercritical branching processes with immigration { Z n }, it is known that under suitable conditions on the offspring and immigration distributions, Z n / m n 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( Z n = k n ) with k n = o ( m n ) as n → ∞. We give asymptotic behavior of such lower deviation probabilities in both Schröder and Böttcher cases, unifying and extending the previous results for Galton-Watson processes in literature.