Gaussian limits associated with the Poisson-Dirichlet distribution and the Ewens sampling formula. (English) Zbl 1010.62101

Joyce, Paul; Krone, Stephen M.; Kurtz, Thomas G.

Ann. Appl. Probab. 12, No. 1, 101-124 (2002).

Summary: We consider large \(\theta\) approximations for the stationary distribution of the neutral infinite alleles model as described by the Poisson-Dirichlet distribution with parameter \(\theta\). We prove a variety of Gaussian limit theorems for functions of the population frequencies as the mutation rate \(\theta\) goes to infinity.
In particular, we show that if a sample of size \(n\) is drawn from a population described by the Poisson-Dirichlet distribution, then the conditional probability of a particular sample configuration is asymptotically normal with mean and variance determined by the Ewens sampling formula [see W.J. Ewens, Mathematical population genetics. (1979; Zbl 0422.92011)]. The asymptotic normality of the conditional sampling distribution is somewhat surprising since it is a fairly complicated function of the population frequencies. Along the way, we also prove an invariance principle giving weak convergence at the process level for powers of the size-biased allele frequencies.

MSC:

62P10	Applications of statistics to biology and medical sciences; meta analysis
92D15	Problems related to evolution
60F05	Central limit and other weak theorems
62E20	Asymptotic distribution theory in statistics

Citations:

Zbl 0422.92011

References:

[1]	BILLINGSLEY, P. (1968). Convergence of Probability Measures. Wiley, New York. · Zbl 0172.21201
[2]	ETHIER, S. N. and KURTZ, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York. · Zbl 0592.60049
[3]	ETHIER, S. N. and KURTZ, T. G. (1994). Convergence to Fleming-Viot processes in the weak atomic topology. Stochastic Process. Appl. 54 1-27. · Zbl 0817.60029 · doi:10.1016/0304-4149(94)00006-9
[4]	EWENS, W. J. (1979). Mathematical Population Genetics. Springer, Berlin. · Zbl 0422.92011
[5]	DONNELLY, P. and JOYCE, P. (1989). Continuity and weak convergence of ranked and sizebiased permutations on the infinite simplex. Stochastic Process. Appl. 31 89-103. · Zbl 0694.60009 · doi:10.1016/0304-4149(89)90104-X
[6]	GILLESPIE, J. H. (1999). The role of population size in molecular evolution. Theoret. Population Biol. 55 145-156. · Zbl 0920.92017 · doi:10.1006/tpbi.1998.1391
[7]	GRIFFITHS, R. C. (1979). On the distribution of allele frequencies in a diffusion model. Theoret. Population Biol. 15 140-158. · Zbl 0422.92015 · doi:10.1016/0040-5809(79)90031-5
[8]	JOYCE, P. (1998). Partition structures and sufficient statistics. J. Appl. Probab. 35 622-632. · Zbl 0952.62005 · doi:10.1239/jap/1032265210
[9]	JOYCE, P., KRONE, S. and KURTZ, T. G. (2001). When can one detect overdominant selection in the infinite alleles model? Unpublished manuscript.
[10]	KARATZAS, I. and SHREVE, S. (1988). Brownian Motion and Stochastic Calculus. Springer, New York. · Zbl 0638.60065
[11]	KINGMAN, J. F. C. (1977). The population structure associated with the Ewens sampling formula. Theoret. Population Biol. 11 274-283. · Zbl 0421.92011 · doi:10.1016/0040-5809(77)90029-6
[12]	KURTZ, T. G. and PROTTER, P. (1991). Weak limit theorems for stochastic integrals and stochastic differential equations. Ann. Probab. 19 1035-1070. · Zbl 0742.60053 · doi:10.1214/aop/1176990334
[13]	WATTERSON, G. A. and GUESS, H. A. (1977). Is the most frequent allele the oldest? Theoret. Population Biol. 11 141-160. · Zbl 0361.92017 · doi:10.1016/0040-5809(77)90023-5
[14]	MOSCOW, IDAHO 83844-1103 E-MAIL: [e-mail] [e-mail] T. G. KURTZ DEPARTMENTS OF MATHEMATICS AND STATISTICS UNIVERSITY OF WISCONSIN MADISON, WISCONSIN 53706-1388 E-MAIL: [e-mail]