The Annals of Probability

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

Jim Pitman and Marc Yor

Full-text: Open access


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

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

First available in Project Euclid: 18 June 2002

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Digital Object Identifier

Zentralblatt MATH identifier

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

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


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.

Export citation


  • [1] Aldous, D. (1985). Exchangeability and related topics. ´Ecole d' ´Et´e de Probabilit´es de SaintFlour XII. Lecture Notes in Math. 1117. Springer, Berlin.
  • [2] Aldous, D. J. and Eagleson, G. K. (1978). On mixing and stability of limit theorems. Ann. Probab. 6 325-331.
  • [3] Aldous, D. and Pitman, J. (1994). Brownian bridge asymptotics for random mappings. Random Structures Algorithms 5 487-512.
  • [4] Arov, D. and Bobrov, A. (1960). The extreme terms of a sample and their role in the sum of independent variables. Theory Probab. Appl. 5 377-396.
  • [5] Barlow, M., Pitman, J. and Yor, M. (1989). Une extension multidimensionnelle de la loi de l'arc sinus. In S´eminaire de Probabilit´es XXIII. Lecture Notes in Math. 1372 294-314. Springer, Berlin.
  • [6] Billingsley, P. (1972). On the distribution of large prime factors. Period. Math. Hungar. 2 283-289.
  • [7] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Encylopedia of Mathematics and Its Applications: Regular Variation. Cambridge Univ. Press.
  • [8] Blackwell, D. and MacQueen, J. (1973). Ferguson distributions via P´olya urn schemes. Ann. Statist. 1 353-355.
  • [9] Brockwell, P. J. and Brown, B. M. (1978). Expansion for the positive stable laws. Z. Wahrsch. Verw. Gebiete 45 213-224.
  • [10] Chung, K. and Erd ¨os, P. (1952). On the application of the Borel-Cantelli Lemma. Trans. Amer. Math. Soc. 72 179-186.
  • [11] Cs´aki, E., Erd ¨os, P. and Revesz, P. (1985). On the length of the longest excursion. Z. Wahrsch. Verw. Gebiete 68 365-382.
  • [12] Darling, D. (1952). The influence of the maximum term in the addition of independent random variables. Trans. Amer. Math. Soc. 73 95-107.
  • [13] Dickman, K. (1930). On the frequency of numbers containing prime factors of a certain relative magnitude. Arkiv. for Matematik Astronomi och Fysik 22 1-14.
  • [14] Donnelly, P. (1986). Partition structures, P´olya urns, the Ewens sampling formula, and the ages of alleles. Theoret. Population Biol. 30 271-288.
  • [15] Donnelly, P. and Grimmett, G. (1993). On the asymptotic distribution of large prime factors. J. London Math. Soc. 2 47 395-404.
  • [16] Donnelly, P. and Joyce, P. (1989). Continuity and weak convergence of ranked and sizebiased permutations on the infinite simplex. Stochastic Process. Appl. 31 89-103.
  • [17] Engen, S. (1978). Stochastic Abundance Models with Emphasis on Biological Communities and Species Diversity. Chapman and Hall, London.
  • [18] Evans, S. and Pitman, J. (1996). Construction of Markovian coalescents. Technical Report 465, Dept. Statistics, Univ. California, Berkeley.
  • [19] Ewens, W. (1972). The sampling theory of selectively neutral alleles. Theoret. Population Biol. 3 87-112.
  • [20] Ewens, W. (1988). Population genetics theory-the past and the future. In Mathematical and Statistical Problems in Evolution (S. Lessard, ed.). Univ. Montreal Press.
  • [21] Ewens, W. and Tavar´e, S. (1995). The Ewens sampling formula. In Multivariate Discrete Distributions (N. S. Johnson, S. Kotz and N. Balakrishnan, eds.). Wiley, New York. To appear.
  • [22] Ferguson, T. (1973). A Bayesian analysis of some nonparametric problems. Ann. Statist. 1 209-230.
  • [23] Getoor, R. (1963). The asymptotic distribution of the number of zero free intervals of a stable process. Trans. Amer. Math. Soc. 106 127-138.
  • [24] Gon charov, V. (1962). On the field of combinatory analysis. Amer. Math. Soc. Transl. 19 1-46.
  • [25] Griffiths, R. C. (1979). Exact sampling distributions from the infinite neutral alleles models. Adv. in Appl. Probab. 11 326-354.
  • [26] Griffiths, R. C. (1988). On the distribution of points in a Poisson Dirichlet process. J. Appl. Probab. 25 336-345.
  • [27] Hansen, J. (1994). Order statistics for decomposable combinatorial structures. Random Structures Algorithms 5 517-533.
  • [28] Hoppe, F. M. (1984). P´olya-like urns and the Ewens sampling formula. J. Math. Biol. 20 91-94.
  • [29] Hoppe, F. M. (1986). Size-biased filtering of Poisson-Dirichlet samples with an application to partition structures in genetics. J. Appl. Probab. 23 1008-1012.
  • [30] Hoppe, F. M. (1987). The sampling theory of neutral alleles and an urn model in population genetics. J. Math. Biol. 25 123-159.
  • [31] Horowitz, J. (1971). A note on the arc-sine law and Markov random sets. Ann. Math. Statist. 42 1068-1074.
  • [32] Horowitz, J. (1972). Semilinear Markov processes, subordinators and renewal theory. Z. Wahrsch. Verw. Gebiete 24 167-193.
  • [33] Hu, Y. and Shi, Z. (1995). Extreme lengths in Brownian and Bessel excursions. Preprint, Laboratoire de Probabilit´es, Univ. Paris VI.
  • [34] Ignatov, T. (1982). On a constant arising in the theory of symmetric groups and on Poisson- Dirichlet measures. Theory Probab. Appl. 27 136-147.
  • [35] Kallenberg, O. (1973). Canonical representations and convergence criteria for processes with interchangeable increments. Z. Wahrsch. Verw. Gebiete 27 23-36.
  • [36] Kerov, S. (1995). Coherent random allocations and the Ewens-Pitman formula. PDMI preprint, Steklov Math. Institute, St. Petersburg.
  • [37] Kingman, J. (1993). Poisson Processes. Clarendon, Oxford.
  • [38] Kingman, J. F. C. (1975). Random discrete distributions. J. Roy. Statist. Soc. Ser. B 37 1-22.
  • [39] Knight, F. (1985). On the duration of the longest excursion. In Seminar on Stochastic Processes (E. Çinlar, K. Chung and R. Getoor, eds.) 117-148. Birkh¨auser, Basel.
  • [40] K ¨uchler, U. and Lauritzen, S. L. (1989). Exponential families, extreme point models and minimal space-time invariant functions for stochastic processes with stationary and independent increments. Scand. J. Statist. 16 237-261.
  • [41] Lamperti, J. (1958). An occupation time theorem for a class of stochastic processes. Trans. Amer. Math. Soc. 88 380-387.
  • [42] Lamperti, J. (1961). A contribution to renewal theory. Proc. Amer. Math. Soc. 12 724-731.
  • [43] Lamperti, J. (1962). An invariance principle in renewal theory. Ann. Math. Statist. 33 685- 696.
  • [44] Lamperti, J. (1972). Semi-stable Markov processes. I. Z. Wahrsch. Verw. Gebiete 22 205-225.
  • [45] LePage, R., Woodroofe, M. and Zinn, J. (1981). Convergence to a stable distribution via order statistics. Ann. Probab. 9 624-632.
  • [46] L´evy, P. (1939). Sur certains processus stochastiques homog enes. Compositio Math. 7 283- 339.
  • [47] Lukacs, E. (1955). A characterization of the gamma distribution. Ann. Math. Statist. 26 319-324.
  • [48] McCloskey, J. W. (1965). A model for the distribution of individuals by species in an environment. Ph.D. thesis, Michigan State Univ.
  • [49] Molchanov, S. A. and Ostrovski, E. (1969). Symmetric stable processes as traces of degenerate diffusion processes. Theory Probab. Appl. 14 128-131.
  • [50] Perman, M. (1993). Order statistics for jumps of normalized subordinators. Stochastic Process Appl. 46 267-281.
  • [51] Perman, M., Pitman, J. and Yor, M. (1992). Size-biased sampling of Poisson point processes and excursions. Probab. Theory Related Fields 92 21-39.
  • [52] Pitman, J. (1992). Partition structures derived from Brownian motion and stable subordinators. Bernoulli. To appear.
  • [53] Pitman, J. (1992). The two-parameter generalization of Ewens' random partition structure. Technical Report 345, Dept. Statistics, Univ. California, Berkeley.
  • [54] Pitman, J. (1995). Exchangeable and partially exchangeable random partitions. Probab. Theory Related Fields 102 145-158.
  • [55] Pitman, J. (1995). Species sampling models. Unpublished manuscript.
  • [56] Pitman, J. (1996). Random discrete distributions invariant under size-biased permutation. Adv. in Appl. Probab. 28 525-539.
  • [57] Pitman, J. (1996). Some developments of the Blackwell-MacQueen urn scheme. In Statistics, Probability and Game Theory. Papers in Honor of David Blackwell. IMS, Hayward, CA. To appear.
  • [58] Pitman, J. (1996). The additive coalescent, random trees, and Brownian excursions. Unpublished manuscript.
  • [59] Pitman, J. and Yor, M. (1992). Arcsine laws and interval partitions derived from a stable subordinator. Proc. London Math. Soc. 3 65 326-356.
  • [60] Pitman, J. and Yor, M. (1996). On the relative lengths of excursions derived from a stable subordinator. S´eminaire de Probabilit´es XXXI. To appear.
  • [61] Resnick, S. I. (1986). Point processes, regular variation and weak convergence. Adv. in Appl. Probab. 18 66-138.
  • [62] Scheffer, C. (1995). The rank of the present excursion. Stochastic Process Appl. 55 101- 118.
  • [63] Shepp, L. and Lloyd, S. (1966). Ordered cycle lengths in a random permutation. Trans. Amer. Math. Soc. 121 340-357.
  • [64] Stepanov, V. (1969). Limit distributions of certain characteristics of random mappings. Theory Probab. Appl. 14 612-626.
  • [65] Vershik, A. M. (1986). The asymptotic distribution of factorizations of natural numbers into prime divisors. Soviet Math. Dokl. 34 57-61.
  • [66] Vershik, A. and Shmidt, A. (1977). Limit measures arising in the theory of groups, I. Theory Probab. Appl. 22 79-85.
  • [67] Vershik, A. and Shmidt, A. (1978). Limit measures arising in the theory of symmetric groups, II. Theory Probab. Appl. 23 36-49.
  • [68] Vershik, A. and Yor, M. (1995). Multiplicativit´e du processus gamma et ´etude asymptotique des lois stables d'indice, lorsque tend vers 0. Technical Report 289, Laboratoire de Probabilit´es, Univ. Paris VI.
  • [69] Vervaat, W. (1972). Success Epochs in Bernoulli Trials. Math. Centre Tracts 42. Math. Centrum, Amsterdam.
  • [70] Vervaat, W. (1979). On a stochastic difference equation and a representation of nonnegative infinitely divisible random variables. Adv. in Appl. Probab. 11 750-783.
  • [71] Watanabe, S. (1995). Generalized arc-sine laws for one-dimensional diffusion processes and random walks. Proc. Sympos. Pure Math. 57 157-172.
  • [72] Watterson, G. A. (1976). The stationary distribution of the infinitely-many neutral alleles diffusion model. J. Appl. Probab. 13 639-651.
  • [73] Watterson, G. A. and Guess, H. (1977). Is the most frequent allele the oldest? Theoret. Population Biol. 11 141-160.
  • [74] Wendel, J. (1964). Zero-free intervals of semi-stable Markov processes. Math. Scand. 14 21-34.
  • [75] Zabell, S. (1996). The continuum of inductive methods revisited. In The Cosmos of Science (J. Earman and J. Norton, eds.). Univ. Pittsburgh Press/Univ. Konstanz. To appear.
  • [76] Derrida, B. (1994). Non-self-averaging effects in sums of random variables, spin glasses, random maps and random walks. In On Three Levels: Micro-, Meso-, and MacroApproaches in Physics (M. Fannes, C. Maes and A. Verbeure, eds.) NATO ASI Series 125-137. Plenum Press, New York.