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Limit theorems associated with the Pitman–Yor process

Part of: Limit theorems

Published online by Cambridge University Press:  26 June 2017

Shui Feng ,
Fuqing Gao  and
Youzhou Zhou
Shui Feng*
Affiliation:
McMaster University
Fuqing Gao*
Affiliation:
Wuhan University
Youzhou Zhou*
Affiliation:
Zhongnan University of Economics and Law
*
* Postal address: Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, L8S 4K1, Canada. Email address: shuifeng@mcmaster.ca
** Postal address: School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China. Email address: fqgao@whu.edu.cn
*** Postal address: School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan, 430073, China. Email address: youzhouzhou2014@zuel.edu.cn
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Abstract

The Pitman–Yor process is a random discrete measure. The random weights or masses follow the two-parameter Poisson–Dirichlet distribution with parameters 0 < α < 1, θ > -α. The parameters α and θ correspond to the stable and gamma components, respectively. The distribution of atoms is given by a probability η. In this paper we consider the limit theorems for the Pitman–Yor process and the two-parameter Poisson–Dirichlet distribution. These include the law of large numbers, fluctuations, and moderate or large deviation principles. The limiting procedures involve either α tending to 0 or 1. They arise naturally in genetics and physics such as the asymptotic coalescence time for explosive branching process and the approximation to the generalized random energy model for disordered systems.

Keywords

Pitman–Yor processexplosive branchinglarge deviationphase transitionPoisson–Dirichlet distributionrandom energy model

MSC classification

Primary: 60F10: Large deviations
Secondary: 92D10: Genetics
Type
Research Article
Information
Advances in Applied Probability , Volume 49 , Issue 2 , June 2017 , pp. 581 - 602
Copyright
Copyright © Applied Probability Trust 2017 

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