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Limit theorems associated with the Pitman–Yor process

Published online by Cambridge University Press:  26 June 2017

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: [email protected]
** Postal address: School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China. Email address: [email protected]
*** Postal address: School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan, 430073, China. Email address: [email protected]

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.

MSC classification

Type
Research Article
Copyright
Copyright © Applied Probability Trust 2017 

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References

[1] Athreya, K. B. (2012). Coalescence in the recent past in a rapidly growing population. Stoch. Process. Appl. 122, 37573766. CrossRefGoogle Scholar
[2] Bertoin, J. (1996). Lévy Processes (Camb. Tracts Math. 121). Cambridge University Press. Google Scholar
[3] Bertoin, J. (2006). Random Fragmentation and Coagulation Processes (Camb. Stud. Adv. Math. 102). Cambridge University Press. CrossRefGoogle Scholar
[4] Bolthausen, E. and Sznitman, A.-S. (1998). On Ruelle's probability cascades and an abstract cavity method. Commun. Math. Phys. 197, 247276. CrossRefGoogle Scholar
[5] Bovier, A. (2006). Statistical Mechanics of Disordered Systems (Camb. Ser. Statist. Prob. Math. 18). Cambridge University Press. CrossRefGoogle Scholar
[6] Dawson, D. A. and Feng, S. (2001). Large deviations for the Fleming–Viot process with neutral mutation and selection. II. Stoch. Process. Appl. 92, 131162. CrossRefGoogle Scholar
[7] Dawson, D. A. and Feng, S. (2006). Asymptotic behavior of Poisson–Dirichlet distribution for large mutation rate. Ann. Appl. Prob. 16, 562582. CrossRefGoogle Scholar
[8] Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications (Appl. Math. 38), 2nd edn. Springer, New York. CrossRefGoogle Scholar
[9] Derrida, B. (1980). Random-energy model: limit of a family of disordered models. Phys. Rev. Lett. 45, 7982. CrossRefGoogle Scholar
[10] Derrida, B. (1981). Random-energy model: an exactly solvable model of disordered systems. Phys. Rev. B 24, 26132626. CrossRefGoogle Scholar
[11] Derrida, B. (1985). A generalization of the random energy model which includes correlations between the energies. J. Phys. Lett. 46, 401407. CrossRefGoogle Scholar
[12] Derrida, B. and Gardner, E. (1986). Solution of the generalized random energy model. J. Phys. C 19, 22532274. CrossRefGoogle Scholar
[13] Eichelsbacher, P. and Ganesh, A. (2002). Moderate deviations for Bayes posteriors. Scand. J. Statist. 29, 153167. CrossRefGoogle Scholar
[14] Feng, S. (2007). Large deviations for Dirichlet processes and Poisson–Dirichlet distribution with two parameters. Electron. J. Prob. 12, 787807. CrossRefGoogle Scholar
[15] Feng, S. (2009). Poisson–Dirichlet distribution with small mutation rate. Stoch. Process. Appl. 119, 20822094. CrossRefGoogle Scholar
[16] Feng, S. (2010). The Poisson–Dirichlet Distribution and Related Topics. Springer, Heidelberg. CrossRefGoogle Scholar
[17] Feng, S. and Gao, F. (2008). Moderate deviations for Poisson–Dirichlet distribution. Ann. Appl. Prob. 18, 17941824. CrossRefGoogle Scholar
[18] Feng, S. and Gao, F. (2010). Asymptotic results for the two-parameter Poisson–Dirichlet distribution. Stoch. Process. Appl. 120, 11591177. CrossRefGoogle Scholar
[19] Feng, S. and Zhou, Y. (2015). Asymptotic behaviour of Poisson–Dirichlet distribution and random energy model. In XI Symp. on Probability and Stochastic Processes (Progress in Probability 69), eds R. H. Mena et al., Birkhäuser, Cham, pp. 141155. Google Scholar
[20] Ferguson, T. S. (1973). A Baysian analysis of some nonparametric problems. Ann. Statist. 1, 209230. CrossRefGoogle Scholar
[21] Ganesh, A. J. and O'Connell, N. (2000). A large-deviation principle for Dirichlet posteriors. Bernoulli 6, 10211034. CrossRefGoogle Scholar
[22] Griffiths, R. C. (1979). On the distribution of allele frequencies in a diffusion model. Theoret. Pop. Biol. 15, 140158. CrossRefGoogle Scholar
[23] Handa, K. (2009). The two-parameter Poisson–Dirichlet point process. Bernoulli 15, 10821116. CrossRefGoogle Scholar
[24] James, L. F. (2008). Large sample asymptotics for the two-parameter Poisson–Dirichlet process. In Pushing the Limits of Contemporary Statistics (Inst. Math. Statist. Collect. 3), eds B. Clarke and S. Ghosal, Institute of Mathematical Statistics, Beachwood, OH, pp. 187199. Google Scholar
[25] Joyce, P., Krone, S. M. and Kurtz, T. G. (2002). Gaussian limits associated with the Poisson–Dirichlet distribution and the Ewens sampling formula. Ann. Appl. Prob. 12, 101124. CrossRefGoogle Scholar
[26] Kanter, M. (1975). Stable densities under change of scale and total variation inequalities. Ann. Prob. 3, 697707. CrossRefGoogle Scholar
[27] Kingman, J. C. F. et al. (1975). Random discrete distribution. J. R. Statist. Soc. B 37, 122. Google Scholar
[28] Lynch, J. and Sethuraman, J. (1987). Large deviations for processes with independent increments. Ann. Prob. 15, 610627. CrossRefGoogle Scholar
[29] Perman, M., Pitman, J. and Yor, M. (1992). Size-biased sampling of Poisson point processes and excursions. Prob. Theory Relat. Fields 92, 2139. CrossRefGoogle Scholar
[30] Pitman, J. (1992). The two-parameter generalization of Ewens' random partition structure. Tech. Rep. 345, University of California, Berkeley. Google Scholar
[31] Pitman, J. (1996). Some developments of the Blackwell–MacQueen urn scheme. In Statistics, Probability and Game Theory, eds T. S. Ferguson, L. S. Shapley and J. B. MacQueen, Institute of Mathematical Statistics, Hayward, CA, pp. 245267. CrossRefGoogle Scholar
[32] Pitman, J. (2006). Combinatorial Stochastic Processes (Lecture Notes Math. 1875). Springer, Berlin. Google Scholar
[33] Pitman, J. and Yor, M. (1992). Arcsine laws and interval partitions derived from a stable subordinator. Proc. London Math. Soc. 65, 326356. CrossRefGoogle Scholar
[34] Pitman, J. and Yor, M. (1997). The two-parameter Poisson–Dirichlet distribution derived from a stable subordinator. Ann. Prob. 25, 855900. CrossRefGoogle Scholar
[35] Pollard, H. (1946). The representation of e-x λ as a Laplace integral. Bull. Amer. Math. Soc. 52, 908910. CrossRefGoogle Scholar
[36] Puhalskii, A. A. (1991). On functional principle of large deviations. In New Trends in Probability and Statistics Vol. 1, eds V. Sazonov and T. Shervashidze, VSP, Utrecht, pp. 198218. Google Scholar
[37] Talagrand, M. (2003). Spin Glasses: A Challenge for Mathematicians (Res. Math. Relat. Areas 3; Ser. Modern Surveys Math. 46). Springer, Berlin. Google Scholar