Heavy tailed branching process with immigration

In this paper we analyze a branching process with immigration defined recursively by $X_t=\theta_t\circ X_{t-1}+B_t$ for a sequence $(B_t)$ of i.i.d. random variables and random mappings $ \theta_t\circ x:=\theta_t(x)=\sum_{i=1}^xA_i^{(t)}, $ with $(A_i^{(t)})_{i\in \mathbb{N}_0}$ being a sequence of $\mathbb{N}_0$-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 behaviour 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 Fr\'echet limiting distribution.