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Teor. Veroyatnost. i Primenen., 1999, Volume 44, Issue 1, Pages 55–73 (Mi tvp597)  

This article is cited in 10 scientific papers (total in 10 papers)

Stationary random partitions of positive integers

N. V. Tsilevich

St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences

Abstract: This paper gives a description of stationary random partitions of positive integers (equivalently, stationary coherent sequences of random permutations) under the action of the infinite symmetric group. Equivalently, all stationary coherent sequences of random permutations are described. This result gives a new characterization of the Poisson–Dirichlet distribution PD(1) with the unit parameter, which turns out to be the unique invariant distribution for a family of Markovian operators on the infinite-dimensional simplex. This result also provides a new characterization of the Haar measure on the projective limit of finite symmetric groups.

Keywords: random partitions, random permutations, stationary distribution, Markovian operator, Poisson–Dirichlet distribution.

DOI: https://doi.org/10.4213/tvp597

Full text: PDF file (2865 kB)

English version:
Theory of Probability and its Applications, 2000, 44:1, 60–74

Bibliographic databases:

Received: 15.09.1998

Citation: N. V. Tsilevich, “Stationary random partitions of positive integers”, Teor. Veroyatnost. i Primenen., 44:1 (1999), 55–73; Theory Probab. Appl., 44:1 (2000), 60–74

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Pitman J., “Poisson–Dirichlet and GEM invariant distributions for split-and-merge transformations of an interval partition”, Combin. Probab. Comput., 11:5 (2002), 501–514  crossref  mathscinet  zmath  isi  scopus
    2. Brooks R., Makover E., “Random construction of Riemann surfaces”, J. Differential Geom., 68:1 (2004), 121–157  crossref  mathscinet  zmath  isi  scopus
    3. Diaconis P., Mayer-Wolf E., Zeitouni O., Zerner M.P.W., “The Poisson–Dirichlet law is the unique invariant distribution for uniform split–merge transformations”, Ann. Probab., 32:1B (2004), 915–938  crossref  mathscinet  zmath  isi
    4. Brooks R., “A statistical model of Riemann surfaces”, Complex analysis and dynamical systems, Contemp. Math., 364, Amer. Math. Soc., Providence, RI, 2004, 15–25  crossref  mathscinet  zmath  isi
    5. Gamburd A., “Poisson-Dirichlet distribution for random Belyi surfaces”, Ann. Probab., 34:5 (2006), 1827–1848  crossref  mathscinet  zmath  isi  elib  scopus
    6. A. M. Vershik, “Does There Exist a Lebesgue Measure in the Infinite-Dimensional Space?”, Proc. Steklov Inst. Math., 259 (2007), 248–272  mathnet  crossref  mathscinet  zmath  elib  elib
    7. Bertoin J., “Two-parameter Poisson-Dirichlet measures and reversible exchangeable fragmentation–coalescence processes”, Combin. Probab. Comput., 17:3 (2008), 329–337  crossref  mathscinet  zmath  isi  elib  scopus
    8. Goldschmidt Ch., Ueltschi D., Windridge P., “Quantum Heisenberg Models and their Probabilistic Representations”, Entropy and the Quantum II, Contemporary Mathematics, 552, eds. Sims R., Ueltschi D., Amer Mathematical Soc, 2011, 177–224  crossref  mathscinet  zmath  isi
    9. Grosskinsky S., Lovisolo A.A., Ueltschi D., “Lattice Permutations and Poisson-Dirichlet Distribution of Cycle Lengths”, J. Stat. Phys., 146:6 (2012), 1105–1121  crossref  mathscinet  zmath  adsnasa  isi  scopus
    10. Kammoun M.S., “Monotonous Subsequences and the Descent Process of Invariant Random Permutations”, Electron. J. Probab., 23 (2018), 118  crossref  mathscinet  zmath  isi
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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