Large deviations for homozygosity

For any m ≥ 2, the homozygosity of order m of a population is the probability that a sample of size m from the population consists of the same type individuals. Assume that the type proportions follow Kingman’s Poisson-Dirichlet distribution with parameter θ. In this paper we establish the large deviation principle for the naturally scaled homozygosity as θ tends to infinity. The key step in the proof is a new representation of the homozygosity. This settles an open problem raised in [1]. The result is then generalized to the two-parameter Poisson-Dirichlet distribution