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A convergence theorem for symmetric functionals of random partitions

Part of: Limit theorems Combinatorial probability

Published online by Cambridge University Press:  14 July 2016

Paul Joyce  and
Simon Tavaré
Paul Joyce*
Affiliation:
University of Idaho
Simon Tavaré*
Affiliation:
University of Southern California
*
∗ Postal address: Department of Mathematics and Statistics, University of Idaho, Moscow, ID 83843, USA.
∗∗ Postal address: Department of Mathematics, University of Southern California, Los Angeles, CA 90089–1113, USA.
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Abstract

This paper gives general conditions under which symmetric functionals of random partitions of the integer m converge in distribution as m → ∞. The main result is used to settle a conjecture of Donnelly et al. (1991) to the effect that the mean of the sum of the square roots of the relative sizes of the components of a random mapping of m integers converges to π/2 as m → ∞.

Keywords

RANDOM MAPPINGSRANDOM PERMUTATIONSPOISSON–DIRICHLET DISTRIBUTION

MSC classification

Primary: 60C05: Combinatorial probability
Secondary: 60F05: Central limit and other weak theorems 92D10: Genetics

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 29 , Issue 2 , June 1992 , pp. 280 - 290
Copyright
Copyright © Applied Probability Trust 1992 

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Footnotes

The authors were supported in part by NSF grant DMS90-05833.

References

Abramowitz, M. and Stegun, I. A. (1965) Handbook of Mathematical Functions. Dover, New York.Google Scholar
Aldous, D. J. (1985) Exchangeability and related topics. In école d'été de Probabilités de Saint-Flour XIII, 1983, ed. Hennequin, P. L., pp. 1–198. Lecture Notes in Mathematics 1117, Springer-Verlag, Berlin.Google Scholar
Donnelly, P. and Joyce, P. (1989) Continuity and weak convergence of ranked and size-biased permutations on the infinite simplex. Stoch. Proc. Appl. 31, 89–103.Google Scholar
Donnelly, P., Ewens, W. J. and Padmadisastra, S. (1991) Random functions: exact and asymptotic results. Adv. Appl. Prob. 23, 437–455.Google Scholar
Ethier, S. N. and Kurtz, T. G. (1986) Markov Processes: Characterization and Convergence. Wiley, New York.Google Scholar
Ewens, W. J. (1972) The sampling theory of selectively neutral alleles. Theoret Popn. Biol. 3, 87–112.Google Scholar
Joyce, P. and Tavaré, S. (1987) Cycles, permutations and the structure of the Yule process with immigration. Stoch. Proc. Appl. 25, 309–314.Google Scholar
Kingman, J. F. C. (1977) The population structure associated with the Ewens sampling formula. Theoret Popn. Biol. 11, 274–283.Google Scholar
Watterson, G. A. (1974) Models for logarithmic species abundance distributions. Theoret. Popn. Biol. 6, 217–250.Google Scholar