On a class of critical Markov branching processes with non-homogeneous Poisson immigration

TL;DR

This paper investigates a class of critical Markov branching processes with infinite variance offspring and non-homogeneous Poisson immigration, analyzing their asymptotic behavior and limit distributions under specific conditions.

Contribution

It introduces a novel analysis of critical branching processes with infinite variance and non-homogeneous Poisson immigration, deriving their asymptotic properties and limit distributions.

Findings

Asymptotic probability of non-visiting zero is characterized.

Proper normalization leads to convergence to specific limit distributions.

Results depend on offspring and immigration distribution characteristics.

Abstract

The paper studies a class of critical Markov branching processes with infinite variance of the offspring distribution. The processes admit also an immigration component at the jump-points of a non-homogeneous Poisson process, assuming that the mean number of immigrants is infinite and the intensity of the Poisson process converges to a constant. The asymptotic behavior of the probability for non-visiting zero is obtained. Proper limit distributions are proved, under suitable normalization of the sample paths, depending on the offspring distribution and the distribution of the immigrants.