On tail behaviour of stationary second-order Galton-Watson processes with immigration

TL;DR

This paper investigates the tail behavior of stationary second-order Galton-Watson processes with immigration, establishing conditions for regular variation of their distributions and their marginals, and analyzing the tail behavior of processes without immigration.

Contribution

It provides new sufficient conditions for the regular variation of stationary distributions in second-order Galton-Watson processes with immigration and characterizes tail behavior without immigration.

Findings

Stationary distributions can be regularly varying under certain conditions.

Distribution of the process at fixed times is regularly varying if initial sizes are.

Conditions are derived for tail behavior in processes without immigration.

Abstract

A second-order Galton-Watson process with immigration can be represented as a coordinate process of a 2-type Galton-Watson process with immigration. Sufficient conditions are derived on the offspring and immigration distributions of a second-order Galton-Watson process with immigration under which the corresponding 2-type Galton-Watson process with immigration has a unique stationary distribution such that its common marginals are regularly varying. In the course of the proof sufficient conditions are given under which the distribution of a second-order Galton-Watson process (without immigration) at any fixed time is regularly varying provided that the initial sizes of the population are independent and regularly varying.