The Matsumoto--Yor Property and Its Converse on Symmetric Cones

The Matsumoto--Yor (MY) property of the generalized inverse Gaussian and gamma distributions has many generalizations. As it was observed in (Letac and Weso{\l}owski in Ann Probab 28:1371--1383, 2000) the natural framework for the multivariate MY property is symmetric cones; however they prove their results for the cone of symmetric positive definite real matrices only. In this paper, we prove the converse to the symmetric cone-variate MY property, which extends some earlier results. The smoothness assumption for the densities of respective variables is reduced to the continuity only. This enhancement was possible due to the new solution of a related functional equation for real functions defined on symmetric cones.Comment: 11 pages. arXiv admin note: text overlap with arXiv:1206.609

The bi-Poisson process: a quadratic harness

This paper is a continuation of our previous research on quadratic harnesses, that is, processes with linear regressions and quadratic conditional variances. Our main result is a construction of a Markov process from given orthogonal and martingale polynomials. The construction uses a two-parameter extension of the Al-Salam--Chihara polynomials and a relation between these polynomials for different values of parameters.Comment: Published in at http://dx.doi.org/10.1214/009117907000000268 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

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.

q−q-Wiener (α,q)−\alpha,q)- Ornstein-Uhlenbeck processes. A generalization of known processes

We collect, scattered through literature, as well as we prove some new properties of two Markov processes that in many ways resemble Wiener and Ornstein--Uhlenbeck processes. Although processes considered in this paper were defined either in non-commutative probability context or through quadratic harnesses we define them once more as so to say 'continuous time ' generalization of a simple, symmetric, discrete time process satisfying simple conditions imposed on the form of its first two conditional moments. The finite dimensional distributions of the first one (say X=(X_{t})_{t\geq0} called q-Wiener) depends on one parameter q\in(-1,1] and of the second one (say Y=(Y_{t})_{t\inR} called ({\alpha},q)- Ornstein--Uhlenbeck) on two parameters ({\alpha},q)\in(0,\infty)\times(-1,1]. The first one resembles Wiener process in the sense that for q=1 it is Wiener process but also that for |q|<1 and \foralln\geq1: t^{n/2}H_{n}(X_{t}/\surdt|q), where (H_{n})_{n\geq0} are the so called q-Hermite polynomials, are martingales. It does not have however neither independent increments not allows continuous sample path modification. The second one resembles Ornstein--Uhlenbeck process. For q=1 it is a classical OU process. For |q|<1 it is also stationary with correlation function equal to exp(-{\alpha}|t-s|) and has many properties resembling those of its classical version. We think that these process are fascinating objects to study posing many interesting, open questions.Comment: 25 page