Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-24T21:00:19.533Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  01 May 2008

JEAN BERTOIN
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])
Get access Rights & Permissions [Opens in a new window]

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
Information
Combinatorics, Probability and Computing , Volume 17 , Issue 3 , May 2008 , pp. 329 - 337
Copyright
Copyright © Cambridge University Press 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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 12 447–474.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 121 301–318.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. 32 915–938.CrossRefGoogle Scholar
[7]Durrett, R., Granovsky, B. L. and Gueron, S. (1999) The equilibrium behavior of reversible coagulation–fragmentation processes. J. Theoret. Probab. 12 447–474.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. 25 227–245.CrossRefGoogle Scholar
[9]Gnedin, A. and Kerov, S. (2001) A characterization of GEM distributions. Combin. Probab. Comput. 10 213–217.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 1547–1562.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 21–39.CrossRefGoogle Scholar
[14]Pitman, J. (1999) Coalescents with multiple collisions. Ann. Probab. 27 1870–1902.CrossRefGoogle Scholar
[15]Pitman, J. (2002) Poisson–Dirichlet and GEM invariant distributions for split-and-merge transformation of an interval-partition. Combin. Probab. Comput. 11 501–514.CrossRefGoogle Scholar
[16]Pitman, J. and Yor, M. (1997) The two-parameter Poisson–Dirichlet distribution derived from a stable subordinator. Ann. Probab. 25 855–900.CrossRefGoogle Scholar
[17]Sagitov, S. (1999) The general coalescent with asynchronous mergers of ancestor lines. J. Appl. Probab. 36 1116–1125.CrossRefGoogle Scholar
[18]Schramm, O. (2005) Composition of random transpositions. Israel J. Math. 147 221–243.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 60–74.CrossRefGoogle Scholar
[21]Whittle, P. (1986) Systems in Stochastic Equilibrium, Chichester, Wiley.Google Scholar