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. 1. Introduction. The

Abstract. We study a two parameter family of processes with linear regressions and linear conditional variances. We give conditions for the unique solution of this problem, and point out the connection between the resulting Markov processes and the generalized convolutions introduced by Bo˙zejko and Speicher [2]. 1.

Free exponential families have been previously introduced as a special case of the q-exponential family. We show that free exponential families arise also from a procedure analogous to the definition of exponential families by using the Cauchy-Stieltjes kernel (1−θx) −1 instead of the exponential kernel e θx. We use this approach to re-derive several known results and to study further similarities with exponential families and reproductive exponential models.

Abstract. We consider a family of probability measures obtained by a procedure analogous to the definition of exponential families replacing the kernel e θx with the Cauchy-Stieltjes kernel (1 − θx) −1. This family exhibits similarities to exponential families and reproductive exponential models [12]; in some results, free additive convolution of measures plays the role of the classical convolution. 1. Kernel families of measures An operator approach to exponential families in [8], lead the authors to the concept of the q-exponential family of measures, where q> −1 was a real parameter; the classical exponential families correspond to q = 1. The purpose of this note is to provide direct approach to the q-exponential families with q = 0. This case is technically simpler, exhibits remarkable connections to free probability, and shows additional similarities to exponential families. Our approach relies on the concept of the kernel family proposed by Weso̷lowski [21] as a generalization of the natural exponential family. According to Weso̷lowski, the kernel family generated by a kernel k(x, θ) consists of the probability measures {k(x, θ)/L(θ)ν(dx) : θ ∈ Θ}, where L(θ) = � k(x, θ)ν(dx) is the normalizing constant, and ν is the generating measure. Exponential families are based on the integral kernel k(x, θ) = e θx, and can also be based on the kernel k(x, θ) = e θ(x−m0) , where the auxiliary parameter m0 cancels out. In this paper, we are interested in the Cauchy-Stieltjes kernel with m0 chosen appropriately. k(x, θ) = 1, (1.1) 1 − θ(x − m0) �Definition 1.1. Suppose ν is a compactly supported probability measure with xdν = m0. Let 1 M(θ) = ν(dx). (1.2) 1 − θ(x − m0) The kernel family generated by (1.1) is the family of probability measures

Abstract. We describe quadratic harnesses that arise through the double sided conditioning of an already known quadratic harness and we characterize quadratic harnesses that arise by this construction from bridges of Lévy processes. We also analyze a construction that produces quadratic harnesses by ”gluing together ” two conditionally-independent quadratic harnesses and we show that the only q-Meixner processes that can be used in this construction are pairs of Poisson processes [4] or pairs of negative binomial processes [15]. Our main tool is a deterministic time and space transformation of quadratic harnesses.