Poisson-Dirichlet and GEM invariant distributions for split-and-merge transformations of an interval partition

June, 2001
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Jim Pitman

This paper introduces a split-and-merge transformation of interval partitions which combines some features of one model studied by Gnedin and Kerov and another studied by Tsilevich and by Mayer-Wolf, Zeitouni and Zerner. 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 Mayer-Wolf, Zeitouni and Zerner 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.

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