MOMENTS AND LARGE DEVIATIONS FOR SUPERCRITICAL BRANCHING PROCESSES WITH IMMIGRATION IN RANDOM ENVIRONMENTS

Let (Zn) be a branching process with immigration in a random environment ζ,where ζ is an independent and identically distributed sequence of random variables.We show asymptotic properties for all the moments of Zn and describe the decay rates of the n-step transition probabilities.As applications,a large deviation principle for the sequence log Zn is established,and related large deviations are also studied.