For any $m \geq 2$, 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 $\theta $. In this paper we establish the large deviation principle for the naturally scaled homozygosity as $\theta $ tends to infinity. The key step in the proof is a new representation of the homozygosity. This settles an open problem raised in . The result is then generalized to the two-parameter Poisson-Dirichlet distribution.
"Large deviations for homozygosity." Electron. Commun. Probab. 21 1 - 8, 2016. https://doi.org/10.1214/16-ECP34