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Two-Parameter Poisson–Dirichlet Measures and Reversible Exchangeable Fragmentation–Coalescence Processes

Published online by Cambridge University Press:  01 May 2008

JEAN BERTOIN*
Affiliation:
Laboratoire de Probabilités, Université Pierre et Marie Curie and DMA, Ecole Normale Supérieure, 45, rue d'Ulm, F-75005 Paris, France (e-mail: [email protected])

Abstract

We show that for 0<α<1 and θ>−α, the Poisson–Dirichlet distribution with parameter (α, θ) is the unique reversible distribution of a rather natural fragmentation–coalescence process. This completes earlier results in the literature for certain split-and-merge transformations and the parameter α = 0.

Type
Paper
Copyright
Copyright © Cambridge University Press 2007

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References

[1]Arratia, R., Barbour, A. D., and Tavaré, S. (2003) Logarithmic Combinatorial Structures: A Probabilistic Approach, Monographs in Mathematics, European Mathematical Society, Zürich.CrossRefGoogle Scholar
[2]Basdevant, A.-L. (2006) Ruelle's probability cascades seen as a fragmentation process. Markov Process. Rel. Fields 12 447474.Google Scholar
[3]Berestycki, J. (2004) Exchangeable fragmentation–coalescence processes and their equilibrium measures. Electron. J. Probab. 9 770–824. Available via: http://www.math.washington.edu/~ejpecp/viewarticle.php?id=1477&layout=abstract.CrossRefGoogle Scholar
[4]Bertoin, J. (2001) Homogeneous fragmentation processes. Probab. Theory Rel. Fields 121 301318.CrossRefGoogle Scholar
[5]Bertoin, J. (2006) Random Fragmentation and Coagulation Processes, Cambridge University Press, Cambridge.CrossRefGoogle Scholar
[6]Diaconis, P., Mayer-Wolf, E., Zeitouni, O., and Zerner, M. P. W. (2004) The Poisson–Dirichlet law is the unique invariant distribution for uniform split-merge transformations. Ann. Probab. 32 915938.CrossRefGoogle Scholar
[7]Durrett, R., Granovsky, B. L. and Gueron, S. (1999) The equilibrium behavior of reversible coagulation–fragmentation processes. J. Theoret. Probab. 12 447474.CrossRefGoogle Scholar
[8]Erlihson, M. M. and Granovsky, B. L. (2004) Reversible coagulation–fragmentation processes and random combinatorial structures: Asymptotics for the number of groups. Random Struct. Alg. 25 227245.CrossRefGoogle Scholar
[9]Gnedin, A. and Kerov, S. (2001) A characterization of GEM distributions. Combin. Probab. Comput. 10 213217.CrossRefGoogle Scholar
[10]Kelly, F. P. (1979) Reversibility and Stochastic Networks, Wiley Series in Probability and Mathematical Statistics, Chichester.Google Scholar
[11]Mayer-Wolf, E., Zeitouni, O. and Zerner, M. P. W. (2002) Asymptotics of certain coagulation–fragmentation processes and invariant Poisson–Dirichlet measures. Electron. J. Probab. 7. Available via: http://www.math.washington.edu/~ejpecp/viewarticle.php?id=1309&layout=abstract.CrossRefGoogle Scholar
[12]Möhle, M. and Sagitov, S. (2001) A classification of coalescent processes for haploid exchangeable population models. Ann. Probab. 29 15471562.CrossRefGoogle Scholar
[13]Perman, M., Pitman, J. and Yor, M. (1992) Size-biased sampling of Poisson point processes and excursions. Probab. Theory Rel. Fields 92 2139.CrossRefGoogle Scholar
[14]Pitman, J. (1999) Coalescents with multiple collisions. Ann. Probab. 27 18701902.CrossRefGoogle Scholar
[15]Pitman, J. (2002) Poisson–Dirichlet and GEM invariant distributions for split-and-merge transformation of an interval-partition. Combin. Probab. Comput. 11 501514.CrossRefGoogle Scholar
[16]Pitman, J. and Yor, M. (1997) The two-parameter Poisson–Dirichlet distribution derived from a stable subordinator. Ann. Probab. 25 855900.CrossRefGoogle Scholar
[17]Sagitov, S. (1999) The general coalescent with asynchronous mergers of ancestor lines. J. Appl. Probab. 36 11161125.CrossRefGoogle Scholar
[18]Schramm, O. (2005) Composition of random transpositions. Israel J. Math. 147 221243.CrossRefGoogle Scholar
[19]Schweinsberg, J. (2000) Coalescents with simultaneous multiple collisions. Electron. J. Probab. 5–12 1–50. Available via: http://www.math.washington.edu/~ejpecp/viewarticle.php?id=1270&layout=abstractCrossRefGoogle Scholar
[20]Tsilevich, N. V. (2000) Stationary random partitions of positive integers. Theor. Probab. Appl. 44 6074.CrossRefGoogle Scholar
[21]Whittle, P. (1986) Systems in Stochastic Equilibrium, Chichester, Wiley.Google Scholar