The normal distribution is ⊞\boxplus-infinitely divisible

We prove that the classical normal distribution is infinitely divisible with respect to the free additive convolution. We study the Voiculescu transform first by giving a survey of its combinatorial implications and then analytically, including a proof of free infinite divisibility. In fact we prove that a subfamily Askey-Wimp-Kerov distributions are freely infinitely divisible, of which the normal distribution is a special case. At the time of this writing this is only the third example known to us of a nontrivial distribution that is infinitely divisible with respect to both classical and free convolution, the others being the Cauchy distribution and the free 1/2-stable distribution.Comment: AMS LaTeX, 29 pages, using tikz and 3 eps figures; new proof including infinite divisibility of certain Askey-Wilson-Kerov distibution

Fock space associated to Coxeter group of type B

In this article we construct a generalized Gaussian process coming from Coxeter groups of type B. It is given by creation and annihilation operators on an $(\alpha,q)$-Fock space, which satisfy the commutation relation $$b_{\alpha,q}(x)b_{\alpha,q}^\ast(y)-qb_{\alpha,q}^\ast(y)b_{\alpha,q}(x)=\langle x, y\rangle I+\alpha\langle \overline{x}, y \rangle q^{2N},$$ where $x,y$ are elements of a complex Hilbert space with a self-adjoint involution $x\mapsto\bar{x}$ and $N$ is the number operator with respect to the grading on the $(\alpha,q)$-Fock space. We give an estimate of the norms of creation operators. We show that the distribution of the operators $b_{\alpha,q}(x)+b_{\alpha,q}^\ast(x)$ with respect to the vacuum expectation becomes a generalized Gaussian distribution, in the sense that all mixed moments can be calculated from the second moments with the help of a combinatorial formula related with set partitions. Our generalized Gaussian distribution associates the orthogonal polynomials called the $q$-Meixner-Pollaczek polynomials, yielding the $q$-Hermite polynomials when $\alpha=0$ and free Meixner polynomials when $q=0$.Comment: 22 pages, 6 figure

On a class of free Levy laws related to a regression problem

The free Meixner laws arise as the distributions of orthogonal polynomials with constant-coefficient recursions. We show that these are the laws of the free pairs of random variables which have linear regressions and quadratic conditional variances when conditioned with respect to their sum. We apply this result to describe free Levy processes with quadratic conditional variances, and to prove a converse implication related to asymptotic freeness of random Wishart matrices.Comment: LaTeX, v2: strengthened main theore

On summable, positive Poisson-Mehler kernels built of Al-Salam--Chihara and related polynomials

Using special technique of expanding ratio of densities in an infinite series of polynomials orthogonal with respect to one of the densities, we obtain simple, closed forms of certain kernels built of the so called Al-Salam-Chihara (ASC) polynomials. We consider also kernels built of some other families of polynomials such as the so called big continuous q-Hermite polynomials that are related to the ASC polynomials. The constructed kernels are symmetric and asymmetric. Being the ratios of the densities they are automatically positive. We expand also reciprocals of some of the kernels, getting nice identities built of the ASC polynomials involving 6 variables like e.g. formula (nice). These expansions lead to asymmetric, positive and summable kernels. The particular cases (referring to q=1 and q=0) lead to the kernels build of certain linear combinations of the ordinary Hermite and Chebyshev polynomials

Conditional moments of q-Meixner processes

We show that stochastic processes with linear conditional expectations and quadratic conditional variances are Markov, and their transition probabilities are related to a three-parameter family of orthogonal polynomials which generalize the Meixner polynomials. Special cases of these processes are known to arise from the non-commutative generalizations of the Levy processes.Comment: LaTeX, 24 pages. Corrections to published version affect formulas in Theorem 4.

Koszul Theorem for S-Lie coalgebras

For a symmetry braid S-Lie coalgebras, as a dual object to algebras introduced by Gurevich, are considered. For an Young antisymmetrizer an S-exterior algebra is introduced. From this differential point of view S-Lie coalgebras are investigated. The dual Koszul theorem in this case is proved.Comment: 8 pages, AMSLaTe

Semigroups of distributions with linear Jacobi parameters

We show that a convolution semigroup of measures has Jacobi parameters polynomial in the convolution parameter $t$ if and only if the measures come from the Meixner class. Moreover, we prove the parallel result, in a more explicit way, for the free convolution and the free Meixner class. We then construct the class of measures satisfying the same property for the two-state free convolution. This class of two-state free convolution semigroups has not been considered explicitly before. We show that it also has Meixner-type properties. Specifically, it contains the analogs of the normal, Poisson, and binomial distributions, has a Laha-Lukacs-type characterization, and is related to the $q=0$ case of quadratic harnesses.Comment: v3: the article is merged back together with arXiv:1003.4025. A significant revision following suggestions by the referee. 2 pdf figure

Hypercontractivity on the qq-Araki-Woods algebras

Extending a work of Carlen and Lieb, Biane has obtained the optimal hypercontractivity of the $q$-Ornstein-Uhlenbeck semigroup on the $q$-deformation of the free group algebra. In this note, we look for an extension of this result to the type III situation, that is for the $q$-Araki-Woods algebras. We show that hypercontractivity from $L^p$ to $L^2$ can occur if and only if the generator of the deformation is bounded.Comment: 17 page

Remarks on tt-transformations of measures and convolutions

ABSTRACT. – A family of transformations of probability measures is constructed, and used to define transformations of convolutions. The relations between moments and cumulants of a measure and its transformation are presented. For transformed classical and free convolutions the central limit measures and the Poisson type limit measures are computed. Families of non-commutative random variables are constructed, which are associated to these central limit measures. They provide examples of “position operators ” which act on the Interacting Fock Spaces. 2001 Éditions scientifiques et médicales Elsevier SAS AMS classification: 60B10; 60E05; 60F05 RÉSUMÉ. – Une famille de transformations de mesures de probabilités est construite et utilisée pour définir des transformations des convolutions. On présente les relations entre les moments et cumulants d’une mesure et de sa transformation. Pour les convolutions transformées classiques et libres les mesures limites du théorème limite central et les mesures limites du théorème limite Poisson sont calculées. On construit des familles de variables aléatoires non-commutatives qui sont associées à ces mesures limites. Elles sont une source d’exemples d’espaces de Fock interactifs. 2001 Éditions scientifiques et médicales Elsevier SAS 1