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

TL;DR

The paper addresses critical Markov branching processes with infinite offspring variance and time-inhomogeneous immigration occurring at Poisson jump times with intensity $r(t)\to 0$ and infinite immigrant mean, focusing on the probability $P_t$ that the process remains positive and on limit theorems under suitable normalizations. The authors develop a detailed framework using generating functions, regular variation, and inverse-slowly varying functions to derive precise asymptotics for $P_t$ and to classify a spectrum of limiting distributions that depend on the offspring tail index $\gamma$, immigration tail index $\alpha$, and Poisson-decay exponent $\theta$. They obtain eight distinct limit regimes, including conditional limits with atoms at zero or infinity and non-conditional stable-type limits, illustrating rich interplay between reproduction, immigration, and time-inhomogeneity. The results extend Sevastyanov-type theory to infinite-variance offspring and non-homogeneous immigration, with potential applications to cellular populations and stem-cell dynamics where immigration is controlled by a decaying Poisson mechanism.

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.

Theorems & Definitions (23)

• Proposition 3.1
• proof
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Theorem 4.1
• proof
• Theorem 4.2
• proof