Heavy-Tailed Branching Process with Immigration

In this article, we analyze a branching process with immigration defined recursively by X t = θ t ○ X t−1 + B t for a sequence (B t ) of i.i.d. random variables and random mappings , with being a sequence of ℕ0-valued i.i.d. random variables independent of B t . We assume that one of generic variables A and B has a regularly varying tail distribution. We identify the tail behavior of the distribution of the stationary solution X t . We also prove CLT for the partial sums that could be further generalized to FCLT. Finally, we also show that partial maxima have a Fréchet limiting distribution.