Source: Adv. in Appl. Probab.
Volume 41, Number 1
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.
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