Multitype Bienaymé--Galton--Watson processes escaping extinction

Abstract

In the framework of a multitype Bienaymé--Galton--Watson (BGW) process, the event that the daughter's type differs from the mother's type can be viewed as a mutation event. Assuming that mutations are rare, we study a situation where all types except one produce on average less than one offspring. We establish a neat asymptotic structure for the BGW process escaping extinction due to a sequence of mutations toward the supercritical type. Our asymptotic analysis is performed by letting mutation probabilities tend to 0. The limit process, conditional on escaping extinction, is another BGW process with an enriched set of types, allowing us to delineate a stem lineage of particles that leads toward the escape event. The stem lineage can be described by a simple Markov chain on the set of particle types. The total time to escape becomes a sum of a random number of independent, geometrically distributed times spent at intermediate types.