Journal of Applied Probability

Dirichlet and quasi-Bernoulli laws for perpetuities

Paweł Hitczenko and Gérard Letac

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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.

Article information

J. Appl. Probab., Volume 51, Number 2 (2014), 400-416.

First available in Project Euclid: 12 June 2014

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Zentralblatt MATH identifier

Primary: 60J05: Discrete-time Markov processes on general state spaces 60E99: None of the above, but in this section

Perpetuities Dirichlet process Ewens' distribution quasi-Bernoulli law probabilities on a tetrahedron T_c transform stationary distribution


Hitczenko, Paweł; Letac, Gérard. Dirichlet and quasi-Bernoulli laws for perpetuities. J. Appl. Probab. 51 (2014), no. 2, 400--416. doi:10.1239/jap/1402578633.

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