A unifying approach to branching processes in a varying environment
Götz Kersting
Abstract
Abstract Branching processes $(Z_n)_{n \ge 0}$ 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 almost sure extinction, square integrability of the martingale $(Z_n/\mathrm E[Z_n])_{n \ge 0}$ , 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 $Z_n \gt 0$ . The theorems generalize/unify diverse results from the literature and lead to a classification of the processes.
Topics & Concepts
MathematicsMartingale (probability theory)Limit (mathematics)Branching processPopulationExponential functionCombinatoricsPure mathematicsDiscrete mathematicsApplied mathematicsMathematical analysisDemographySociologyStochastic processes and statistical mechanicsBayesian Methods and Mixture ModelsStatistical Methods and Bayesian Inference