A unifying approach to branching processes in varying environments

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 a.s. extinction, square-integrability of the martingale $(Z_n/\mathbf{E}[Z_n]{)}_{n\ge 0}$, 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 $Z_n>0$. The theorems generalize/unify diverse results from the literature and lead to a classification of the processes.

26 pages, improvements, in particular Theorem 4, version as to appear in the J. Appl. Probab. 57.1

References in corpus (1)

• Time-Inhomogeneous Branching Processes Conditioned on Non-Extinction

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