## Annals of Probability

- Ann. Probab.
- Volume 25, Number 2 (1997), 855-900.

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

Jim Pitman and Marc Yor

#### Abstract

The two-parameter Poisson-Dirichlet distribution, denoted $\mathsf{PD}(\alpha, \theta)$ 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 Kingman, is $\mathsf{PD}(0, \theta)$. Known properties of $\mathsf{PD}(0, \theta)$, including the Markov chain description due to Vershik, Shmidt and Ignatov, are generalized to the two-parameter case. The size-biased random permutation of $\mathsf{PD}(\alpha, \theta)$ 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 $0 < \alpha < 1, \mathsf{PD}(\alpha, 0)$ 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 $\mathsf{PD}(1/2, 0)$, and the corresponding distribution for a Brownian bredge is $\mathsf{PD}(1/2, 1/2)$. The $\mathsf{PD}(\alpha, 0)$ and $\mathsf{PD}(\alpha, \alpha)$ 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 $\alpha$.

#### Article information

**Source**

Ann. Probab., Volume 25, Number 2 (1997), 855-900.

**Dates**

First available in Project Euclid: 18 June 2002

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1024404422

**Digital Object Identifier**

doi:10.1214/aop/1024404422

**Mathematical Reviews number (MathSciNet)**

MR1434129

**Zentralblatt MATH identifier**

0880.60076

**Subjects**

Primary: 60J30 60G57: Random measures 60E99: None of the above, but in this section

**Keywords**

Zero set semistable Markov process local time ranked lengths of excursions Poisson point process

#### Citation

Pitman, Jim; Yor, Marc. The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator. Ann. Probab. 25 (1997), no. 2, 855--900. doi:10.1214/aop/1024404422. https://projecteuclid.org/euclid.aop/1024404422