A unifying approach to branching processes in varying environments

2017/03/06 by Götz Kersting, Kersting, Götz · 8 citations
Chemistry · Computer Science · Mathematics · #60J80 #Applied mathematics #Bayesian Methods and Mixture Models #Branching (polymer chemistry) #Branching process #Chemistry #Combinatorics #Exponential function #FOS: Mathematics #Limit (mathematics) #Martingale (probability theory) #Mathematical analysis #Mathematics #Physics #Population #Probability (math.PR) #Pure mathematics #Statistical Methods and Bayesian Inference #Statistical physics #Stochastic processes and statistical mechanics #math.PR #msc:60J80

paper · pdf · doi:10.48550/arxiv.1703.01960

published in arXiv (Cornell University) (Cornell University) · 26 pages, improvements, in particular Theorem 4, version as to appear in the J. Appl. Probab. 57.1

openalex publication_date 2017/03/06 · arxiv created 2019/11/06 · arxiv updated 2019/11/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

Branching processes (Z_n)_{n ≥ 0} in a varying environment generalize the Galton-Watson process, in that they allow time-dependence of the offspring distribution. Our main results concern general criteria for a.s. extinction, square-integrability of the martingale (Z_n/\mathbf E[Z_n])_{n ≥ 0}, properties of the martingale limit W and a Yaglom type result stating convergence to an exponential limit distribution of the suitably normalized population size Z_n, conditioned on the event Z_n >0. The theorems generalize/unify diverse results from the literature and lead to a classification of the processes.

Citations

• Time-Inhomogeneous Branching Processes Conditioned on Non-Extinction

Cited by

• Probabilistic Proof of Conditional Limit Theorem for Critical Galton--Waston Process
• Inheritance entropy quantifies epigenetic regulation of cell-cycle exit in human bone marrow stromal cells
• Beyond Poisson Approximation: Sums of Markovian Bernoulli Variables with Applications to Brownian Motions and Branching Processes
• Numerical Aspects of Computing Possible Equilibria for Resource Dependent Branching Processes with Immigration

Related

• Yaglom's limit for critical Galton-Watson processes in varying environment: A probabilistic approach
• Large deviations for Branching Processes in Random Environment
• On the scaling limits of Galton Watson processes in varying environment
• Time-Inhomogeneous Branching Processes Conditioned on Non-Extinction
• Galton-Watson processes in varying environment and accessibility percolation