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Dirichlet and Quasi-Bernoulli Laws for Perpetuities

Part of: Distribution theory - Probability Markov processes

Published online by Cambridge University Press:  19 February 2016

Paweł Hitczenko  and
Gérard Letac
Paweł Hitczenko*
Affiliation:
Drexel University
Gérard Letac*
Affiliation:
Université Paul Sabatier
*
∗ Postal address: Department of Mathematics, Drexel University, Philadelphia PA 19104, USA.
∗∗ Postal address: Laboratoire de Statistique et Probabilités, Université Paul Sabatier, 31062 Toulouse, France. Email address: gerard.letac@alsatis.net.
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Abstract

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Let X, B, and Y be the Dirichlet, Bernoulli, and beta-independent random variables such that X ~ D(a0, …, ad), Pr(B = (0, …, 0, 1, 0, …, 0)) = ai / a with a = ∑i=0dai, and Y ~ β(1, a). Then, as proved by Sethuraman (1994), X ~ X(1 - Y) + BY. This gives the stationary distribution of a simple Markov chain on a tetrahedron. In this paper we introduce a new distribution on the tetrahedron called a quasi-Bernoulli distribution Bk(a0, …, ad) with k an integer such that the above result holds when B follows Bk(a0, …, ad) and when Y ~ β(k, a). We extend it even more generally to the case where X and B are random probabilities such that X is Dirichlet and B is quasi-Bernoulli. Finally, the case where the integer k is replaced by a positive number c is considered when a0 = · · · = ad = 1.

Keywords

PerpetuitiesDirichlet processEwens' distributionquasi-Bernoulli lawprobabilities on a tetrahedronTc transformstationary distribution

MSC classification

Primary: 60J05: Discrete-time Markov processes on general state spaces 60E99: None of the above, but in this section
Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 2 , June 2014 , pp. 400 - 416
Copyright
© Applied Probability Trust 

Footnotes

Partially supported by the Simons Foundation (grant number #208766).

This author thanks the Fields Institute for its hospitality during the preparation of this paper.

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