Long-time behaviour of Galton-Watson systems with circular mechanism

TL;DR

This paper investigates the long-term behavior of Galton-Watson branching processes with a circular mechanism, identifying three types of limit behaviors and their rates under different moment conditions.

Contribution

It explicitly characterizes three types of limit behaviors and their convergence rates for Galton-Watson systems with circular mechanisms.

Findings

One limit behavior has geometric rate.

Two other behaviors have supergeometric rates.

Results depend on moment conditions of branching rates.

Abstract

This paper concentrates on the limit behavior of discrete-time branching process with circular mechanism. Three types of limit behaviour of discrete-time branching process with circular mechanism are given explicitly under various moment conditions on branching rates. It is proved that the rate of the first one is geometric, while the other two are supergeometric.