Markov branching process with infinite variance and non-homogeneous immigration with infinite mean

TL;DR

This paper analyzes a critical Markov branching process with infinite variance and non-homogeneous immigration, revealing its asymptotic behavior and limiting distributions under complex conditions.

Contribution

It introduces a new model combining infinite variance offspring with infinite mean immigration and derives its asymptotic properties.

Findings

Asymptotic probability for non-visiting zero is characterized.

Limiting distributions depend on offspring and immigration distributions.

Behavior varies with the intensity of the immigration Poisson process.

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 zero. The asymptotic behavior of the probability for non-visiting zero is obtained. Limiting distributions are proved, under suitable normalization of the sample paths, depending on the offspring distribution, on the distribution of the immigrants and on the intensity of the Poisson process.