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Dirichlet Random Walks

Published online by Cambridge University Press:  30 January 2018

Gérard Letac
Affiliation:
Université Paul Sabatier
Mauro Piccioni
Affiliation:
Sapienza Università di Roma
Corresponding
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Abstract

This paper provides tools for the study of the Dirichlet random walk in R d . 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 R d .

Type
Research Article
Copyright
© Applied Probability Trust 

References

Beghin, L. and Orsingher, E. (2010). Moving randomly amid scattered obstacles. Stochastics 82, 201229.CrossRefGoogle Scholar
Borwein, J. M., Straub, A., Wan, J. and Zudilin, W. (2012). Densities of short uniform random walks. Canad. J. Math. 64, 961990.CrossRefGoogle Scholar
Chamayou, J.-F. and Letac, G. (1994). A transient random walk on stochastic matrices with Dirichlet distributions. Ann. Prob. 22, 424430.CrossRefGoogle Scholar
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, 95111 (in Italian).Google Scholar
Cifarelli, D. M. and Regazzini, E. (1990). Distribution functions of means of a Dirichlet process. Ann. Statist. 18, 429442. (Correction: 22 (1994), 1633–1634.)CrossRefGoogle Scholar
Hjort, N. L. and Ongaro, A. (2005). Exact inference for random Dirichlet means. Statist. Inference Stoch. Process. 8, 227254.CrossRefGoogle Scholar
Hjort, N. L. and Ongaro, A. (2006). On the distribution of random Dirichlet Jumps. Metron 64, 6192.Google Scholar
Kolesnik, A. D. (2009). The explicit probability distribution of a six-dimensional random flight. Theory Stoch. Process. 15, 3339.Google Scholar
Le Caër, G. (2010). A Pearson random walk with steps of uniform orientation and Dirichlet distributed lengths. J. Statist. Phys. 140, 728751.CrossRefGoogle Scholar
Le Caër, G. (2011). A new family of solvable Pearson–Dirichlet random walks. J. Statist. Phys. 144, 2345.CrossRefGoogle Scholar
Lijoi, A. and Prünster, I. (2009). Distributional properties of means of random probability measures. Statist. Surveys 3, 4795.CrossRefGoogle Scholar
Sethuraman, J. (1994). A constructive definition of Dirichlet priors. Statistica Sinica 4, 639650.Google Scholar

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