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Dirichlet Random Walks

Part of: Geometric probability and stochastic geometry Limit theorems

Published online by Cambridge University Press:  30 January 2018

Gérard Letac  and
Mauro Piccioni
Gérard Letac*
Affiliation:
Université Paul Sabatier
Mauro Piccioni*
Affiliation:
Sapienza Università di Roma
*
∗ Postal address: Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse, France. Email address: letac@math.univ-toulouse.fr
∗∗ Postal address: Dipartimento di Matematica, Sapienza Università di Roma, 00185 Rome, Italia.
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Abstract

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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.

Keywords

Dirichlet processStieltjes transformrandom flightdistributions in a spherehyperuniformityinfinite divisibility in the sense of Dirichlet

MSC classification

Primary: 60D99: None of the above, but in this section 60F99: None of the above, but in this section
Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 4 , December 2014 , pp. 1081 - 1099
Copyright
© Applied Probability Trust 

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