The asymptotic form of the extinction probabilities for supercritical multitype branching processes

Abstract

We consider a positively regular branching process for which there is a finite number T of types. It is assumed that ϱ, the dominant characteristic root of the matrix of first moments, is larger than 1. Then if q( n) i is the probability that the line descended from an ancestor of type i has become extinct by generation n, q i = lim n →∞ q i ( n) , and q i > 0, i = 1,…, T, it can be shown that lim n→∞ (q i − q n i) ϱ ̌ n = d i > 0 , where 0 < ϱ <1. A consequence is that all of the moments of the conditional time to extinction of a line, given ultimate extinction, are finite, regardless of the type of the ancestral individual.