69 research outputs found
Random nested tetrahedra
No full textInternational audienceIn a real n-1 dimensional affine space E, consider a tetrahedron T0, i.e. the convex hull of n points α1, α2, ..., αn of E. Choose n independent points β1, β2, ..., βn randomly and uniformly in T0, thus obtaining a new tetrahedron T1 contained in T0. Repeat the operation with T1 instead of T0, obtaining T2, and so on. The sequence of the Tk shrinks to a point Y of T0 and this note computes the distribution of the barycentric coordinates of Y with respect to (α1, α2, ..., αn) (Corollary 2.3). We also obtain the explicit distribution of Y in more general cases. The technique used is to reduce the problem to the study of a random walk on the semigroup of stochastic (n,n) matrices, and this note is a geometrical application of a former result of Chamayou and Letac (1994)
Dirichlet random walks
No full textThis paper provides tools for the study of the d-dimensional Dirichlet random walk. We compute explicitly, in a number of cases, the distribution using a form of Stieltjes transform. This allows us to simplify some existing results, due to Le Caer. We extend our results to the study of limit of the Dirichlet random walk when the number of added terms goes to infinity
Beta-hypergeometric distributions and random continued fractions
No full textIn this paper an enlargement of the beta family of distributions on (0, 1) is presented. Distributions in this class are characterized as being the laws of certain random continued fractions associated with products of independent random matrices of order 2 whose entries are either constant or beta distributed. The result can be proved by a famous 1879 Thomae formula on generalized hypergeometric functions 3F2
Seul le groupe des similitudes-inversions préserve le type de la loi de Cauchy-conforme de Rn pour n > 1
No full textRésuméL'article introduit le type de la loi de Cauchy-conforme comme l'ensemble des probabilités sur l'espace euclidien Rn défini par μp,a(dx = 2n − 1π−(n + 1)2Γ(n + 12)(pp2 + ∥a − x∥2)n dx1 … dxn où (p, a) décrit R+n + 1 = ]0, + ∞[× Rn et démontre que si F: Rn → Rn est une fonction mesurable, alors F préserve le type de la loi Cauchy-conforme si et seulement si F coïncide presque partout avec une similitude ou une similitude-inversion de Rn, dans le cas où n ⩾ 2. Ce résultat prolonge l'étude faite auparavant (Letac, Proc. Amer. Math. Soc. 67 (1977), 277–286) du même problème pour n = 1.AbstractThis paper introduces the conformal-Cauchy type as the set of probabilities on the Euclidean space Rn defined by μp,a(dx = 2n − 1π−(n + 1)2Γ(n + 12)(pp2 + ∥a − x∥2)n dx1 … dxn where (p, a) runs on R+n + 1 = ]0, + ∞[× Rn. It shows that if n ⩾ 2 and if F: Rn → Rn is measurable, then F preserves the conformal-Cauchy type if and only if there exists either a similitude or an inversion-similitude g of Rn such that F coincides with g almost everywhere. This result completes a study (Letac, Proc. Amer. Math. Soc. 67 (1977), 277–286), made before in the case n = 1
Exponential families of mixed Poisson distributions
No full textIf I=(I1,…,Id) is a random variable on [0,∞)d with distribution μ(dλ1,…,dλd), the mixed Poisson distribution MP(μ) on View the MathML source is the distribution of (N1(I1),…,Nd(Id)) where N1,…,Nd are ordinary independent Poisson processes which are also independent of I. The paper proves that if F is a natural exponential family on [0,∞)d then MP(F) is also a natural exponential family if and only if a generating probability of F is the distribution of v0+v1Y1+cdots, three dots, centered+vqYq for some qless-than-or-equals, slantd, for some vectors v0,…,vq of [0,∞)d with disjoint supports and for independent standard real gamma random variables Y1,…,Yq
Characterization of the simple cubic multivariate exponential families
No full textAbstractThe Letac–Mora class of real cubic natural exponential families has been characterized by a property of 2-orthogonality of an associated sequence of polynomials (see [G. Letac, M. Mora, Natural real exponential families with cubic variance functions, Ann. Statist. 18 (1990) 1–37; A. Hassairi, M. Zarai, Characterization of the cubic exponential families by orthogonality of polynomials, Ann. Probab. 32 (2004) 2463–2476]). The present paper introduces a notion of transorthogonality for a sequence of polynomial on Rd to extend the characterization to the multivariate version of the Letac–Mora class of real natural exponential families
Associated natural exponential families and elliptic functions
No full textInternational audienceThis paper studies the variance functions of the natural exponential families (NEF) on the real line of the form (Am 4 + Bm 2 + C) 1/2 where m denoting the mean. Surprisingly enough, most of them are discrete families concentrated on λ Z for some constant λ and the Laplace transform of their elements are expressed by elliptic functions. The concept of association of two NEF is an auxilliary tool for their study: two families F and G are associated if they are generated by symmetric probabilities and if the analytic continuations of their variance functions satisfy V F (m) = V G (m √ −1). We give some properties of the association before its application to these elliptic NEF. The paper is completed by the study of NEF with variance functions m(Cm 4 + Bm 2 + A) 1/2. They are easier to study and they are concentrated on a N
Haight's distribution and busy periods
No full textHaight's distribution as introduced by Letac and Seshadri is related to busy period distributions and generalised to a semigroup of infinitely divisible distributions.Haight's distribution busy period distribution
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