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The structure of the allelic partition of the total population for Galton-Watson processes with neutral mutations. (English) Zbl 1180.92063

Consider a critical or sub-critical Markov branching process with an infinite set of types (= alleles) in discrete time. Let the dynamics of the process be type-independent. Call children of the same type as its parent clones and children of a different type as its parent mutants. Decompose the total population into clusters of individuals of the same type.

The author specifies the law of this partition in terms of the distribution of the number of clone children and the number of mutant children of a typical individual, and proves limit theorems related to the distribution of the partition. The essential tool is an extension of the classical Harris’ representation of Bienaymé-Galton-Watson processes and a version of the ballot theorem.

MSC:
92D15Problems related to evolution
92D10Genetics
60J80Branching processes
60J85Applications of branching processes
60J20Applications of Markov chains and discrete-time Markov processes on general state spaces