Electronic Communications in Probability

Universality of asymptotically Ewens measures on partitions

James Zhao

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We give a criterion for functionals of partitions to converge to a universal limit under a class of measures that "behaves like" the Ewens measure. Various limit theorems for the Ewens measure, most notably the Poisson-Dirichlet limit for the longest parts, the functional central limit theorem for the number of parts, and the Erdos-Turan limit for the product of parts, extend to these asymptotically Ewens measures as easy corollaries. Our major contributions are: (1) extending the classes of measures for which these limit theorems hold; (2) characterising universality by an intuitive and easily-checked criterion; and (3) providing a new and much shorter proof of the limit theorems by taking advantage of the Feller coupling.

Article information

Electron. Commun. Probab., Volume 17 (2012), paper no. 16, 11 pp.

Accepted: 23 April 2012
First available in Project Euclid: 7 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Ewens sampling formula Feller coupling logarithmic combinatorial structures perturbation Poisson-Dirichlet limit central limit theorem Erdos-Turan theorem

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Zhao, James. Universality of asymptotically Ewens measures on partitions. Electron. Commun. Probab. 17 (2012), paper no. 16, 11 pp. doi:10.1214/ECP.v17-1956. https://projecteuclid.org/euclid.ecp/1465263149

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  • Arratia, Richard; Barbour, A. D.; Tavaré, Simon. Poisson process approximations for the Ewens sampling formula. Ann. Appl. Probab. 2 (1992), no. 3, 519–535.
  • Arratia, Richard; Barbour, A. D.; Tavaré, Simon. Random combinatorial structures and prime factorizations. Notices Amer. Math. Soc. 44 (1997), no. 8, 903–910.
  • Arratia, Richard; Barbour, A. D.; Tavaré, Simon. On Poisson-Dirichlet limits for random decomposable combinatorial structures. Combin. Probab. Comput. 8 (1999), no. 3, 193–208.
  • Arratia, R.; Barbour, A. D.; Tavaré, S. Limits of logarithmic combinatorial structures. Ann. Probab. 28 (2000), no. 4, 1620–1644.
  • Arratia, Richard; Barbour, A. D.; Tavaré, Simon. Logarithmic combinatorial structures: a probabilistic approach. EMS Monographs in Mathematics. European Mathematical Society (EMS), Zürich, 2003. xii+363 pp. ISBN: 3-03719-000-0
  • Barbour, A. D.; Tavaré, Simon. A rate for the ErdÅ‘s-Turán law. Combin. Probab. Comput. 3 (1994), no. 2, 167–176.
  • Betz, Volker; Ueltschi, Daniel. Spatial random permutations and infinite cycles. Comm. Math. Phys. 285 (2009), no. 2, 469–501.
  • Betz, Volker; Ueltschi, Daniel; Velenik, Yvan. Random permutations with cycle weights. Ann. Appl. Probab. 21 (2011), no. 1, 312–331.
  • Billingsley, Patrick. Convergence of probability measures. John Wiley & Sons, Inc., New York-London-Sydney 1968 xii+253 pp.
  • DeLaurentis, J. M.; Pittel, B. G. Random permutations and Brownian motion. Pacific J. Math. 119 (1985), no. 2, 287–301.
  • Diaconis, Persi; McGrath, Michael; Pitman, Jim. Riffle shuffles, cycles, and descents. Combinatorica 15 (1995), no. 1, 11–29.
  • Diaconis, P.; Ram, A. A probabilistic interpretation of the Macdonald polynomials. Preprint.
  • Ercolani, N. M.; Ueltschi, D. Cycle structure of random permutations with cycle weights. Preprint.
  • ErdÅ‘s, P.; Turán, P. On some problems of a statistical group-theory. I. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 4 1965 175–186 (1965).
  • ErdÅ‘s, P.; Turán, P. On some problems of a statistical group-theory. III. Acta Math. Acad. Sci. Hungar. 18 1967 309–320.
  • Ewens, W. J. The sampling theory of selectively neutral alleles. Theoret. Population Biology 3 (1972), 87–112; erratum, ibid. 3 (1972), 240; erratum, ibid. 3 (1972), 376.
  • Feller, W. The fundamental limit theorems in probability. Bull. Amer. Math. Soc. 51, (1945). 800–832.
  • Flajolet, Philippe; Soria, Michèle. Gaussian limiting distributions for the number of components in combinatorial structures. J. Combin. Theory Ser. A 53 (1990), no. 2, 165–182.
  • Fristedt, Bert. The structure of random partitions of large integers. Trans. Amer. Math. Soc. 337 (1993), no. 2, 703–735.
  • Golomb, S. W. Random Permutations. Bull. Amer. Math. Soc. 70 (1964), no. 6, 747.
  • Gontcharoff, W. Sur la distribution des cycles dans les permutations. C. R. (Doklady) Acad. Sci. URSS (N.S.) 35, (1942). 267–269.
  • Gončarov, V. On the field of combinatory analysis. Amer. Math. Soc. Transl. (2) 19 1962 1–46.
  • Griffiths, R. C. On the distribution of allele frequencies in a diffusion model. Theoret. Population Biol. 15 (1979), no. 1, 140–158.
  • Hansen, Jennie C. A functional central limit theorem for the Ewens sampling formula. J. Appl. Probab. 27 (1990), no. 1, 28–43.
  • Hansen, Jennie C. Order statistics for decomposable combinatorial structures. Random Structures Algorithms 5 (1994), no. 4, 517–533.
  • Hoppe, Fred M. Pólya-like urns and the Ewens' sampling formula. J. Math. Biol. 20 (1984), no. 1, 91–94.
  • Kingman, J. F. C. Random discrete distributions. J. Roy. Statist. Soc. Ser. B 37 (1975), 1–22.
  • Kingman, J. F. C. The population structure associated with the Ewens sampling formula. Theoret. Population Biology 11 (1977), no. 2, 274–283.
  • Knuth, Donald E.; Trabb Pardo, Luis. Analysis of a simple factorization algorithm. Theoret. Comput. Sci. 3 (1976/77), no. 3, 321–348.
  • Lugo, Michael. Profiles of permutations. Electron. J. Combin. 16 (2009), no. 1, Research Paper 99, 20 pp.
  • Okounkov, Andrei. Infinite wedge and random partitions. Selecta Math. (N.S.) 7 (2001), no. 1, 57–81.
  • Pitman, Jim. The two-parameter generalization of Ewens' random partition structure. Technical Report 345, Department of Statistics, UC Berkeley, 1992.
  • Pitman, Jim; Yor, Marc. The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator. Ann. Probab. 25 (1997), no. 2, 855–900.
  • Shepp, L. A.; Lloyd, S. P. Ordered cycle lengths in a random permutation. Trans. Amer. Math. Soc. 121 1966 340–357.
  • VerÅ¡ik, A. M.; Å midt, A. A. Limit measures that arise in the asymptotic theory of symmetric groups. I. (Russian) Teor. Verojatnost. i Primenen. 22 (1977), no. 1, 72–88.
  • VerÅ¡ik, A. M.; Å midt, A. A. Limit measures that arise in the asymptotic theory of symmetric groups. II. (Russian) Teor. Verojatnost. i Primenen. 23 (1978), no. 1, 42–54.
  • Watterson, G. A. The stationary distribution of the infinitely-many neutral alleles diffusion model. J. Appl. Probability 13 (1976), no. 4, 639–651.