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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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