Multi-type branching processes with time dependent branching rates

TL;DR

This paper establishes criteria for almost-sure extinction and the asymptotic distribution of the total number of particles in multi-type branching processes with time-dependent rates, under mild assumptions.

Contribution

It introduces new criteria for extinction and asymptotic behavior in multi-type branching processes with non-constant rates, extending existing theory.

Findings

Criteria for almost-sure extinction under mild conditions

Conditions for the total number of particles to converge to an exponential distribution

Analysis applicable to processes with time-varying branching rates

Abstract

Under mild non-degeneracy assumptions on branching rates in each generation, we provide a criterion for almost-sure extinction of a multi-type branching process with time-dependent branching rates. We also provide a criterion for the total number of particles (conditioned on survival and divided by the expectation of the resulting random variable) to approach an exponential random variable as time goes to infinity.