The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator

• 成果类型: Article
• 署名作者: Pitman, J; Yor, M
• 署名单位: University of California System; University of California Berkeley
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 1997
• 页码: 855-900
• 关键词: random discrete-distributions; ewens sampling formula; neutral alleles; order-statistics; partition structures; WEAK-CONVERGENCE; Random mappings; point-processes; MODEL; stationary

摘要

The two-parameter Poisson-Dirichlet distribution, denoted PD(alpha, theta), is a probability distribution on the set of decreasing positive sequences with sum 1. The usual Poisson-Dirichlet distribution with a single parameter theta, introduced by King-man, is PD(0, theta). Known properties of PD(0, theta), 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(alpha, theta) 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 < alpha < 1, PD(alpha, 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 alpha. 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(alpha, 0) and PD(alpha, alpha) 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 alpha.