An infinite-alleles version of the simple branching process

Abstract

Individuals in a population which grows according to the rules defining the simple branching process can mutate to novel allelic forms. We obtain limit theorems for the number of alleles present in any generation, the total number of alleles ever seen and the number of the generation containing the last mutation event.In addition we define a notion of frequency spectrum for each generation as the expected number of alleles having a given number of representatives. As the generation number increases we prove the existence of a limiting notion of the frequency spectrum and discuss its upper tail behaviour. Our results here are incomplete and we make some conjectures which are supported by informal argument and specific examples.