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

Kosto V. Mitov, Nikolay M. Yanev

TL;DR

The paper analyzes a class of critical Markov branching processes with infinite offspring variance and non-homogeneous Poisson immigration. By leveraging functional equations for generating functions and regular variation tools, it derives the asymptotics of the non-visitation probability $P\{Y(t)>0\}$ and identifies four distinct limit-distribution regimes under different normalizations, ranging from stable laws to a Uniform$(0,1)$ limit. The results hinge on the tail indices $\alpha$ (immigration) and $\gamma$ (offspring) and the slowly varying components, linking the immigration tail to the limit laws and revealing how heavy-tailed immigration shapes long-time behavior. These findings advance understanding of critical branching dynamics with time-inhomogeneous immigration and infinite-variance offspring, with potential applications to biological population dynamics and cellular systems where immigration occurs in a nonstationary manner.

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.

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

TL;DR

The paper analyzes a class of critical Markov branching processes with infinite offspring variance and non-homogeneous Poisson immigration. By leveraging functional equations for generating functions and regular variation tools, it derives the asymptotics of the non-visitation probability and identifies four distinct limit-distribution regimes under different normalizations, ranging from stable laws to a Uniform limit. The results hinge on the tail indices (immigration) and (offspring) and the slowly varying components, linking the immigration tail to the limit laws and revealing how heavy-tailed immigration shapes long-time behavior. These findings advance understanding of critical branching dynamics with time-inhomogeneous immigration and infinite-variance offspring, with potential applications to biological population dynamics and cellular systems where immigration occurs in a nonstationary manner.

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.
Paper Structure (5 sections, 2 theorems, 83 equations)

This paper contains 5 sections, 2 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Description of the models
  3. Basic assumptions and preliminary results
  4. Probability for non-visiting the state zero
  5. Limit distributions

Key Result

Theorem 4.1

Assume the conditions(rt-3-g), (infinite-var), and (im-fin) hold. (i) If $Q(t)\rightarrow \infty ,t\rightarrow \infty ,$ then (ii) If $Q=\int_{0}^{\infty }q(u)du<\infty$ then

Theorems & Definitions (4)

  • Definition 2.1
  • Theorem 4.1
  • proof
  • Theorem 5.1