Heavy tailed branching process with immigration

TL;DR

This paper studies a branching process with immigration, analyzing its tail behavior, limit theorems for sums, and maxima, especially under heavy-tailed conditions, providing insights into its long-term distributional properties.

Contribution

It characterizes the tail behavior of the stationary distribution and establishes CLT and Fréchet limit laws for sums and maxima in heavy-tailed branching processes.

Findings

Stationary distribution exhibits heavy-tailed behavior.

CLT holds for partial sums of the process.

Partial maxima follow a Fréchet distribution.

Abstract

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.