Mathematics – Probability
Scientific paper
2007-10-18
Annals of Applied Probability 2008, Vol. 18, No. 5, 1794-1824
Mathematics
Probability
Published in at http://dx.doi.org/10.1214/07-AAP501 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Inst
Scientific paper
10.1214/07-AAP501
The Poisson--Dirichlet distribution arises in many different areas. The parameter $\theta$ in the distribution is the scaled mutation rate of a population in the context of population genetics. The limiting case of $\theta$ approaching infinity is practically motivated and has led to new, interesting mathematical structures. Laws of large numbers, fluctuation theorems and large-deviation results have been established. In this paper, moderate-deviation principles are established for the Poisson--Dirichlet distribution, the GEM distribution, the homozygosity, and the Dirichlet process when the parameter $\theta$ approaches infinity. These results, combined with earlier work, not only provide a relatively complete picture of the asymptotic behavior of the Poisson--Dirichlet distribution for large $\theta$, but also lead to a better understanding of the large deviation problem associated with the scaled homozygosity. They also reveal some new structures that are not observed in existing large-deviation results.
Feng Shui
Gao Fuqing
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