Journal of Applied Probability

Dirichlet and quasi-Bernoulli laws for perpetuities

Paweł Hitczenko and Gérard Letac

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Abstract

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

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

Dates
First available in Project Euclid: 12 June 2014

Permanent link to this document
https://projecteuclid.org/euclid.jap/1402578633

Digital Object Identifier
doi:10.1239/jap/1402578633

Mathematical Reviews number (MathSciNet)
MR3217775

Zentralblatt MATH identifier
1296.60184

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

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

Citation

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. https://projecteuclid.org/euclid.jap/1402578633


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References

  • Ambrus, G., Kevei, P. and Vígh, V. (2012). The diminishing segment process. Statist. Prob. Lett. 82, 191–195.
  • Arizmendi, O. and Pérez-Abreu, V. (2010). On the non-classical infinite divisibility of power semicircle distributions. Commun. Stoch. Anal. 4, 161–178.
  • Chamayou, J.-F. and Letac, G. (1991). Explicit stationary distributions for compositions of random functions and products of random matrices. J. Theoret. Prob. 4, 3–36
  • Chamayou, J.-F. and Letac, G. (1994). A transient random walk on stochastic matrices with Dirichlet distributions. Ann. Prob. 22, 424–430.
  • Diaconis, P. and Kemperman, J. (1996). Some new tools for Dirichlet priors. In Bayesian Statistics 5, Oxford University Press, pp. 97–106.
  • Ferguson, T. S. (1973). A Bayesian analysis of some nonparametric problems. Ann. Statist. 1, 209–230.
  • Goldie, C. M. and Maller, R. A. (2000). Stability of perpetuities. Ann. Prob. 28, 1195–1218.
  • Kotz, S., Balakrishnan, N. and Johnson, N. L. (2000). Continuous Multivariate Distributions, Vol. 1, 2nd edn. John Wiley, New York.
  • Letac, G. and Massam, H. (1998). A formula on multivariate Dirichlet distribution. Statist. Prob. Lett. 38, 247–253.
  • 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.