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Large deviations for homozygosity. (English) Zbl 1352.60037
Summary: 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 [D. A. Dawson and S. Feng, Ann. Appl. Probab. 16, No. 2, 562–582 (2006; Zbl 1119.92046)]. The result is then generalized to the two-parameter Poisson-Dirichlet distribution.
MSC:
60F10 Large deviations
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