3rd ed. p. cm. -(Grundlehren der mathematischen Wissenschaften; 293) Includes bibliographical references and index. ISBN 978-3-642-08400-3 ISBN 978-3-662-06400-9 (eBook)

The two-parameter Poisson-Dirichlet distribution, denoted PD α θ , is a probability distribution on the set of decreasing positive sequences with sum 1. The usual Poisson-Dirichlet distribution with a single parameter θ, introduced by Kingman, is PD 0 θ . Known properties of PD 0 θ , including the Markov chain description due to Vershik, Shmidt and Ignatov, are generalized to the two-parameter case. The size-biased random permutation of PD α θ is a simple residual allocation model proposed by Engen in the context of species diversity, and rediscovered by Perman and the authors in the study of excursions of Brownian motion and Bessel processes. For 0 < α < 1, PD α 0 is the asymptotic distribution of ranked lengths of excursions of a Markov chain away from a state whose recurrence time distribution is in the domain of attraction of a stable law of index α. Formulae in this case trace back to work of Darling, Lamperti and Wendel in the 1950s and 1960s. The distribution of ranked lengths of excursions of a one-dimensional Brownian motion is PD 1/2 0 , and the corresponding distribution for a Brownian bridge is PD 1/2 1/2 . The PD α 0 and PD α α distributions admit a similar interpretation in terms of the ranked lengths of excursions of a semistable Markov process whose zero set is the range of a stable subordinator of index α.

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Library ofCongress Cataloging-in-Publication Data Revuz, D. Continuous Martingales and Brownian motion 1 Daniel Revuz, Mare Y or. p. cm. -(Grundlehren der mathematischen Wissenschaften =A Series of comprehensive studies in mathematics; 293) lncludes bibliographical references (p.) and indexes.

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