Journal of Applied Probability

Dirichlet random walks

Gérard Letac and Mauro Piccioni

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

This paper provides tools for the study of the Dirichlet random walk in Rd. We compute explicitly, for a number of cases, the distribution of the random variable W using a form of Stieltjes transform of W instead of the Laplace transform, replacing the Bessel functions with hypergeometric functions. This enables us to simplify some existing results, in particular, some of the proofs by Le Caër (2010), (2011). We extend our results to the study of the limits of the Dirichlet random walk when the number of added terms goes to ∞, interpreting the results in terms of an integral by a Dirichlet process. We introduce the ideas of Dirichlet semigroups and Dirichlet infinite divisibility and characterize these infinite divisible distributions in the sense of Dirichlet when they are concentrated on the unit sphere of Rd.

Article information

Source
J. Appl. Probab., Volume 51, Number 4 (2014), 1081-1099.

Dates
First available in Project Euclid: 20 January 2015

Permanent link to this document
https://projecteuclid.org/euclid.jap/1421763329

Mathematical Reviews number (MathSciNet)
MR3301290

Zentralblatt MATH identifier
1320.60108

Subjects
Primary: 60D99: None of the above, but in this section 60F99: None of the above, but in this section

Keywords
Dirichlet process Stieltjes transform random flight distributions in a sphere hyperuniformity infinite divisibility in the sense of Dirichlet

Citation

Letac, Gérard; Piccioni, Mauro. Dirichlet random walks. J. Appl. Probab. 51 (2014), no. 4, 1081--1099. https://projecteuclid.org/euclid.jap/1421763329


Export citation

References

  • Beghin, L. and Orsingher, E. (2010). Moving randomly amid scattered obstacles. Stochastics 82, 201–229.
  • Borwein, J. M., Straub, A., Wan, J. and Zudilin, W. (2012). Densities of short uniform random walks. Canad. J. Math. 64, 961–990.
  • Chamayou, J.-F. and Letac, G. (1994). A transient random walk on stochastic matrices with Dirichlet distributions. Ann. Prob. 22, 424–430.
  • Cifarelli, D. M. and Regazzini, E. (1979). A general approach to Bayesian analysis of nonparametric problems. The associative mean values within the framework of the Dirichlet process. II. Riv. Mat. Sci. Econom. Social. 2, 95–111 (in Italian).
  • Cifarelli, D. M. and Regazzini, E. (1990). Distribution functions of means of a Dirichlet process. Ann. Statist. 18, 429–442. (Correction: 22 (1994), 1633–1634.)
  • Hjort, N. L. and Ongaro, A. (2005). Exact inference for random Dirichlet means. Statist. Inference Stoch. Process. 8, 227–254.
  • Hjort, N. L. and Ongaro, A. (2006). On the distribution of random Dirichlet jumps. Metron 64, 61–92.
  • Kolesnik, A. D. (2009). The explicit probability distribution of a six-dimensional random flight. Theory Stoch. Process. 15, 33–39.
  • Le Caër, G. (2010). A Pearson random walk with steps of uniform orientation and Dirichlet distributed lengths. J. Statist. Phys. 140, 728–751.
  • Le Caër, G. (2011). A new family of solvable Pearson–Dirichlet random walks. J. Statist. Phys. 144, 23–45.
  • Lijoi, A. and Prünster, I. (2009). Distributional properties of means of random probability measures. Statist. Surveys 3, 47–95.
  • Sethuraman, J. (1994). A constructive definition of Dirichlet priors. Statistica Sinica 4, 639–650. \endharvreferences