Transformation of the discrete stable process via branching reproduction environment
Maroussia Slavtchova-Bojkova, Penka Mayster
TL;DR
This paper addresses a continuous-time MBP with homogeneous immigration where arrivals follow a discrete-stable process, and reproduction exhibits infinite variance. It derives explicit probability generating functions for the population size by transforming the immigration flux through the branching environment, revealing connections to the discrete-Linnik distribution when immigration and reproduction exponents coincide. Using Laplace-transform techniques and careful normalization, the authors establish detailed asymptotic limit laws that depend on the relative values of the stability exponents $\gamma$ (immigration) and $\beta$ (reproduction), including convergences to $\Psi(\lambda)=(1+\lambda^{\beta})^{-\theta A}$ or $\Psi(\lambda)=\exp(-\theta A\lambda^{\gamma})$ in various regimes, and identify degeneracies under other parameter configurations. The results extend classical Sevastyanov theory to settings with infinite mean and infinite variance, linking discrete-stable immigration to Linnik-type limiting distributions and clarifying the impact of immigration-transformation in branching processes.
Abstract
The current paper focuses on studying the impact of immigration with an infinite mean, driven by a discrete-stable compound Poisson process, when it is entering the branching environment with infinite variance of reproduction. Our goal is to determine the explicit form of the probability generating function and subsequently to analyze the probability of extinction, aiming to understand the long-term behavior of such processes.