Poisson–Dirichlet and GEM Invariant Distributions for Split-and-Merge Transformations of an Interval Partition

Authors:
Jim Pitman
Affiliations:
Department of Statistics, University of California, 367 Evans Hall # 3860, Berkeley, CA 94720-3860, USA (e-mail: pitman@stat.Berkeley.edu)
Venue:
Combinatorics, Probability and Computing
Year:
2002

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Abstract

This paper introduces a split-and-merge transformation of interval partitions which combines some features of one model studied by Gnedin and Kerov [12, 11] and another studied by Tsilevich [30, 31] and Mayer-Wolf, Zeitouni and Zerner [21]. The invariance under this split-and-merge transformation of the interval partition generated by a suitable Poisson process yields a simple proof of the recent result of [21] that a Poisson–Dirichlet distribution is invariant for a closely related fragmentation–coagulation process. Uniqueness and convergence to the invariant measure are established for the split-and-merge transformation of interval partitions, but the corresponding problems for the fragmentation–coagulation process remain open.