An Integral Characterization of the Dirichlet Process

  • Günter LastEmail author


We give a new integral characterization of the Dirichlet process on a general phase space. To do so, we first prove a characterization of the nonsymmetric Beta distribution via size-biased sampling. Two applications are a new characterization of the Dirichlet distribution and a marked version of a classical characterization of the Poisson–Dirichlet distribution via invariance and independence properties.


Dirichlet process Dirichlet distribution Beta distribution Poisson process Mecke equation Poisson–Dirichlet distribution Size-biased sampling 

Mathematics Subject Classification (2010)

60G55 60G57 



I wish to thank Lorenzo Dello Schiavo for drawing my attention to the topic of the paper and the referee for making several helpful comments.


  1. 1.
    Dello Schiavo, L., Lytvynov, E.W.: A Mecke-type characterization of the Dirichlet–Ferguson measure. (2017). arXiv:1706.07602
  2. 2.
    Ferguson, T.S.: A Bayesian analysis of some nonparametric problems. Ann. Stat. 1, 209–230 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Hoppe, F.M.: Size-biased filtering of Poisson–Dirichlet samples with an application to partition structures in genetics. J. Appl. Probab. 23, 1008–1012 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Kallenberg, O.: Foundations of Modern Probability, 2nd edn. Springer, New York (2002)CrossRefzbMATHGoogle Scholar
  5. 5.
    Kingman, J.F.C.: Random discrete distributions. J. R. Stat. Soc. Ser. B 37, 1–22 (1975)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Kingman, J.F.C.: Poisson Processes. Oxford University Press, Oxford (1993)zbMATHGoogle Scholar
  7. 7.
    Last, G., Penrose, M.: Lectures on the Poisson Process. Cambridge University Press, Cambridge (2017)CrossRefzbMATHGoogle Scholar
  8. 8.
    McCloskey, J.W.: A Model for the Distribution of Individuals by Species in an Environment. Unpublished Ph.D. Dissertation Thesis, Michigan State University (1965)Google Scholar
  9. 9.
    Mecke, J.: Stationäre zufällige Maße auf lokalkompakten Abelschen Gruppen. Z. Wahrsch. verw. Gebiete 9, 36–58 (1967)CrossRefzbMATHGoogle Scholar
  10. 10.
    Pakes, A.G.: Characterization by invariance under length-biasing and random scaling. J. Stat. Plan. Inference 63, 285–310 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Pitman, J.: Some developments of the Blackwell–MacQueen urn scheme. In: Ferguson, T.S., Shapley, L.S., MacQueen, J.B. (eds.), Statistics, Probability and Game Theory. IMS Lecture Notes-Monograph Series, vol. 30, pp. 245–267 (1995)Google Scholar
  12. 12.
    Pitman, J.: Random discrete distributions invariant under size-biased permutation. Adv. Appl. Probab. 28, 525–539 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Pitman, J.: Combinatorial Stochastic Processes. Ecole de’Éte Probabilités de Saint Flour. Lecture Notes in Mathematics, vol. 1875. Springer, Berlin (2006)zbMATHGoogle Scholar
  14. 14.
    Seshadri, V., Wesolowski, J.: Constancy of regressions for beta distributions. Sankhya A 65, 284–291 (2003)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Sethuraman, J.: A constructive definition of Dirichlet priors. Stat. Sin. 4, 639–650 (1994)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute of StochasticsKarlsruhe Institute of TechnologyKarlsruheGermany

Personalised recommendations