On a Special Family of Critical Markov Branching Processes with Infinite Mean Immigration

Abstract

This paper investigates the impact of immigration with an infinite mean, driven by a discrete-stable compound Poisson process, on a critical branching environment with infinite variance of reproduction. Our objective is to derive an explicit expression for the probability generating function (p.g.f.) of the number of particles present at any time

. The resulting explicit form of the p.g.f. is expressed in terms of the Gauss hypergeometric function and, in particular, is shown to correspond to Linnik distribution. Moreover, the availability of well-established numerical methods for evaluating these special functions in the literature ensures the practical applicability of the theoretical results.