Distributional properties of means of random probability measures

Lijoi, Antonio; Prünster, Igor

doi:10.1214/09-ss041

“…(Y j ) j ≥1 with distribution β 1,a and by taking W j = Y j j −1 k=1 (1 − Y k ). The survey by Lijoi and Prunster [10] contains a wealth of information on the Dirichlet process P ∼ D(α) and on the distributions of the functionals f (w)P (dw) with numerous references. The survey by Lijoi and Prunster [10] contains a wealth of information on the Dirichlet process P ∼ D(α) and on the distributions of the functionals f (w)P (dw) with numerous references.…”

Section: Quasi-bernoulli and Dirichlet Processesmentioning

confidence: 99%

Dirichlet and Quasi-Bernoulli Laws for Perpetuities

Hitczenko

¹

,

Letac

²

2014

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Let X, B, and Y be the Dirichlet, Bernoulli, and beta-independent random variables such that X ∼ D(a 0 , . . . , a d ), Pr(B = (0, . . . , 0, 1, 0, . . . , 0)) = a i /a with a = d i=0 a i , and Y ∼ β(1, a). Then, as proved by Sethuraman (1994)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 B k (a 0 , . . . , a d ) with k an integer such that the above result holds when B follows B k (a 0 , . . . , a d ) 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 a 0 = · · · = a d = 1.

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“…Thus, another way of stating the observation from [1] is that an arcsine random variable on (−1/2, 1/2) is a perpetuity generated by the distribution of (M, Q) ∼ (1 − Y, W Y /2). This property of the arcsine law is actually an instance of a much more general result due to Sethuraman (see [10] or Theorem 1.1 below) on Dirichlet distribution. To recall Sethuraman's result we will need the following notation.…”

Section: Introductionmentioning

confidence: 65%

Dirichlet and Quasi-Bernoulli Laws for Perpetuities

Hitczenko¹,

Letac²

2014

J. Appl. Probab.

0

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Let X, B, and Y be the Dirichlet, Bernoulli, and beta-independent random variables such that X ∼ D(a 0 , . . . , a d ), Pr(B = (0, . . . , 0, 1, 0, . . . , 0)) = a i /a with a = d i=0 a i , and Y ∼ β(1, a). Then, as proved by Sethuraman (1994)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 B k (a 0 , . . . , a d ) with k an integer such that the above result holds when B follows B k (a 0 , . . . , a d ) 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 a 0 = · · · = a d = 1.

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“…but μ(tα 1 ) for t = 1 is notoriously complicated, as it can be seen in [17]. Explicit calculations about this problem appear in [16], in the comments following Theorem 16.…”

Section: Examples Of Dirichlet Curvesmentioning

confidence: 98%

“…One says also that α is the governing probability of P t and that t is its intensity. Of course, (P t ) t≥0 has a venerable story and the papers by [4,6,9] and [17] are among the important papers to read on the subject. Some simple considerations about {D(tα), t > 0} are in order.…”

Section: Introductionmentioning

confidence: 99%

Dirichlet curves, convex order and Cauchy distribution

Letac¹,

Piccioni²

2018

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If α is a probability on R d and t > 0, the Dirichlet random probability P t ∼ D(tα) is such that for any measurable partitionThe Dirichlet curve associated to the probability α is the map t → μ(tα). It has simple properties like lim t 0 μ(tα) = α and lim t→∞ μ(tα) = δ m when m = R d xα(dx) exists. The present paper shows that if m exists and if ψ is a convex function on R d then t → R d ψ(x)μ(tα)(dx) is a decreasing function, which means that t → μ(tα) is decreasing according to the Strassen convex order of probabilities. The second aim of the paper is to prove a group of results around the following question: if μ(tα) = μ(sα) for some 0 ≤ s < t, can we claim that μ is Cauchy distributed in R d ?

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Distributional properties of means of random probability measures

Cited by 21 publications

References 83 publications

Dirichlet and Quasi-Bernoulli Laws for Perpetuities

Dirichlet and Quasi-Bernoulli Laws for Perpetuities

Dirichlet and Quasi-Bernoulli Laws for Perpetuities

Dirichlet curves, convex order and Cauchy distribution

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