The distribution of the frequencies of age-ordered alleles in a diffusion model

We prove that the frequencies of the oldest, second-oldest, third-oldest, . alleles in the stationary infinitely-many-neutral-alleles diffusion model are distributed as X1, (1 . X1)X2, (1 . X1)(1 . X2)X3, ., where X1, X2,X3, . are independent beta (1, .) random variables, . being twice the mutation intensity; that is, the frequencies of age-ordered alleles have the so-called Griffiths.Engen.McCloskey, or GEM, distribution. In fact, two proofs are given, the first involving reversibility and the size-biased Poisson.Dirichlet distribution, and the second relying on a result of Donnelly and Tavaré relating their age-ordered sampling formula to the GEM distribution.