Limit Theorems for Branching Processes with Immigration in a Random Environment

2020/02/03 by Bojan Basrak, Basrak, Bojan, Péter Kevei +1 · 1 citation
Mathematics · Physics and Astronomy · #60F05 #60J80 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.2002.00634

openalex publication_date 2020/02/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We investigate subcritical Galton-Watson branching processes with immigration in a random environment. Using Goldie's implicit renewal theory we show that under general Cramér condition the stationary distribution has a power law tail. We determine the tail process of the stationary Markov chain, prove point process convergence, and convergence of the partial sums. The original motivation comes from Kesten, Kozlov and Spitzer seminal 1975 paper, which connects a random walk in a random environment model to a special Galton-Watson process with immigration in a random environment. We obtain new results even in this very special setting.

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• Heavy-tailed critical Galton--Watson processes with immigration

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