On the Limit Distributions of Continuous-State Branching Processes with Immigration

TL;DR

This paper analyzes the long-term behavior of continuous-state branching processes with immigration, providing explicit descriptions of their limit distributions and characterizing their support, continuity, and boundary behavior.

Contribution

It explicitly determines the Levy-Khintchine triplet of the limit distribution for CBI-processes, generalizing known results on self-decomposable distributions.

Findings

Explicit Levy-Khintchine triplet for limit distribution

Characterization of support and absolute continuity

Asymptotic behavior at the boundary of support

Abstract

We consider the class of continuous-state branching processes with immigration (CBI-processes), introduced by Kawazu and Watanabe (1971) and their limit distributions as time tends to infinity. We determine the Levy-Khintchine triplet of the limit distribution and give an explicit description in terms of the characteristic triplets of the Levy subordinator and the spectrally positive Levy process, which describe the immigration resp. branching mechanism of the CBI-process. This representation allows us to describe the support of the limit distribution and characterise its absolute continuity and asymptotic behavior at the boundary of the support, generalizing several known results on self-decomposable distributions.