arXiv subjects
Publications and source records attributed to Abdelhamid Hassairi.
In this paper, we determine the effect of the free multiplicative convolution on the pseudo-variance function of a Cauchy-Stieltjes kernel family. We then use the machinery of variance functions to establish some limit theorems related to this type of convolution and involving the free additive convolution and the boolean additive convolution.
The Wishart probability distribution on symmetricmatrices has been initially defined by mean of the multivariateGaussian distribution as an of the chi-square distribution. A moregeneral definition is given using results for harmonic analysis.Recently a probability distribution on symmetric matrices called theRiesz distribution has been defined by its Laplace transform as ageneralization of the Wishart distribution. The aim of the presentpaper is to show that some Riesz probability distributions which arenot necessarily Wishart may also be presented by mean of theGaussian distribution using Gaussian samples with missing data.
In this paper, we give an explicitdescription of a class of positive measures on symmetric conesdefined by their Laplace transforms in the framework of the Rieszintegrals. This work is motivated by the importance of thesemeasures in probability theory and statistics sincethey represent a generalization of the measures generating the famous Wishart distributions.
In this paper, we use a notion of ratio based on a division algorithm, to extend to a symmetric cone the definition of a continued fraction in its more general form. We then give a criteria of convergence of a non ordinary random continued fraction that has arisen in the study of some probability distributions related to the beta distribution on the cone of positive definite symmetric matrices or on any symmetric cone.
Some remarkable properties of the beta distribution are based on relations involving independence between beta random variables such that a parameter of one among them is the sum of the parameters of an other (see (1.1) et (1.2) below). Asci, Letac and Piccioni \cite{6} have used the real beta-hypergeometric distribution on $ \reel$ to give a general version of these properties without the condition on the parameters. In the present paper, we extend the properties of the real beta to the beta distribution on symmetric matrices, we use on the positive definite matrices the division algorithm defined by the Cholesky decomposition to define a matrix-variate beta-hypergeometric distribution, and we extend to this distribution the proprieties established in the real case by Asci, Letac and Piccioni
For a natural exponential family (NEF), one can associate in a natural way two standard families of conjugate priors, one on the natural parameter and the other on the mean parameter. These families of conjugate priors have been used to establish some remarkable properties and characterization results of the quadratic NEF's. In the present paper, we show that for a NEF, we can associate a class of NEF's, and for each one of these NEF's, we define a family of conjugate priors on the natural parameter and a family of conjugate priors on the mean parameter which are different of the standard ones. These families are then used to extend to the Letac-Mora class of real cubic natural exponential families the properties and characterization results related to the Bayesian theory established for the quadratic natural exponential families.
We explore properties of Cauchy-Stieltjes families that have no counterpart in exponential families. We relate the variance function of the iterated Cauchy-Stieltjes family to the pseudo-variance function of the initial Cauchy-Stieltjes family. We also investigate when the domain of means can be extended beyond the "natural domain".
In this paper, we give three equivalent properties of the class of multivariate simple cubic natural exponential families (NEF's). The first property says that the cumulant function of any basis of the family is a solution of some Monge-Ampére equation, the second is that the variance function satisfies a differential equation, and the third is characterized by the equality between two families of prior distributions related to the NEF. These properties represent the extensions to this class of the properties stated in $\cite{Casalis(1996)}$ and satisfied by the Wishart and the simple quadratic NEF's. We also show that in the real case, each of these properties provides a new characterization of the Letac-Mora class of real cubic NEF's.
This paper continues the study of a kernel family which uses the Cauchy-Stieltjes kernel in place of the celebrated exponential kernel of the exponential families theory. We extend the theory to cover generating measures with support that is unbounded on one side. We illustrate the need for such an extension by showing that cubic pseudo-variance functions correspond to free-infinitely divisible laws without the first moment. We also determine the domain of means, advancing the understanding of Cauchy-Stieltjes kernel families also for compactly supported generating measures.
The Wishart distribution on an homogeneous cone is a generalization of the Riesz distribution on a symmetric cone which corresponds to a given graph. The paper extends to this distribution, the famous Olkin and Rubin characterization of the ordinary Wishart distribution on symmetric matrices.
In this paper, we introduce, for a multiplier $χ$, a notion of generalized power function $x\mapsto Δ_χ(x),$ defined on the homogeneous cone ${\mathcal{P}}$ of a Vinberg algebra ${\mathcal{A}}$. We then extend to ${\mathcal{A}}$ the famous Gindikin result, that is we determine the set of multipliers $χ$ such that the map $θ\mapsto Δ_χ(θ^{-1})$, defined on ${\mathcal{P}}^{\ast}$, is the Laplace transform of a positive measure $R_χ$. We also determine the set of $χ$ such that $R_χ$ generates an exponential family, and we calculate the variance function of this family
The aim of this paper is to study the mixture of the Riesz distribution on symmetric matrices with respect to the multivariate Poisson distribution. We show, in particular, that this distribution is related to the modified Bessel function of the first kind. We also study the generated natural exponential family. We determine the domain of the means and the variance function of this family.
Bobecka and Wesolowski (2002) have shown that, in the Olkin and Rubin characterization of the Wishart distribution (See Casalis and Letac (1996)), when we use the division algorithm defined by the quadratic representation and replace the property of invariance by the existence of twice differentiable densities, we still have a characterization of the Wishart distribution. In the present work, we show that, when we use the division algorithm defined by the Cholesky decomposition, we get a characterization of the Riesz distribution.
This paper introduces a notion of 2-orthogonality for a sequence of polynomials to give extended versions of the Meixner and Feinsilver characterization results based on orthogonal polynomials. These new versions subsume the Letac-Mora characterization of the real natural exponential families having cubic variance function.