The sector constants of continuous state branching processes with immigration

TL;DR

This paper investigates the mathematical properties of continuous state branching processes with immigration, focusing on their stationary distributions, sector conditions, and connections to noncommutative probability, expanding understanding of their long-term behavior.

Contribution

It characterizes stationary distributions as generalized gamma convolutions and analyzes the sector condition using the Thorin measure, linking stochastic processes with noncommutative probability.

Findings

Gamma distributions are the only reversible stationary distributions.

Generalized gamma convolutions can be stationary distributions with suitable mechanisms.

The strong sector condition is characterized via the Thorin measure.

Abstract

Continuous state branching processes with immigration are studied. We are particularly concerned with the associated (non-symmetric) Dirichlet form. After observing that gamma distributions are only reversible distributions for this class of models, we prove that every generalized gamma convolution is a stationary distribution of the process with suitably chosen branching mechanism and with continuous immigration. For such non-reversible processes, the strong sector condition is discussed in terms of a characteristic called the Thorin measure. In addition, some connections with notion from noncommutative probability theory will be pointed out through calculations involving the Stieltjes transform.