Markovian paths to extinction

Abstract

Subcritical Markov branching processes {Z t } die out sooner or later, say at time T < ∞. We give results for the path to extinction {Z uT , 0 ≤ u ≤ 1} that include its finite dimensional distributions and the asymptotic behaviour of x u−1 Z uT , as Z 0=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 Z T-u , u > 0, and show that as Z 0= x → ∞ this process converges to a Markov process {Y u }, which we describe completely. Emphasis is on continuous time processes, those in discrete time displaying a more complex behaviour, related to Martin boundary theory.