Markovian paths to extinction

Abstract

Subcritical Markov branching processes {Zt} die out sooner or later, say at time T < ∞. We give results for the path to extinction {ZuT, 0 ≤ u ≤ 1} that include its finite dimensional distributions and the asymptotic behaviour of xu−1ZuT, as Z0=x → ∞. The limit reflects an interplay of branching and extreme value theory. Then we consider the population on the verge of extinction, as modelled by ZT-u, u > 0, and show that as Z0= x → ∞ this process converges to a Markov process {Yu}, which we describe completely. Emphasis is on continuous time processes, those in discrete time displaying a more complex behaviour, related to Martin boundary theory.