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The bi  Poisson process: a quadratic harness
 Annals of Probability
"... 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 twoparameter ..."
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Cited by 6 (3 self)
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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 twoparameter extension of the AlSalam–Chihara polynomials and a relation between these polynomials for different values of parameters. 1. Introduction. The
FREE EXPONENTIAL FAMILIES AS KERNEL FAMILIES
, 2008
"... Free exponential families have been previously introduced as a special case of the qexponential family. We show that free exponential families arise also from a procedure analogous to the definition of exponential families by using the CauchyStieltjes kernel (1−θx) −1 instead of the exponential k ..."
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Cited by 5 (0 self)
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Free exponential families have been previously introduced as a special case of the qexponential family. We show that free exponential families arise also from a procedure analogous to the definition of exponential families by using the CauchyStieltjes kernel (1−θx) −1 instead of the exponential kernel e θx. We use this approach to rederive several known results and to study further similarities with exponential families and reproductive exponential models.
BIPOISSON PROCESS
, 2004
"... 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 ..."
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Cited by 3 (3 self)
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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.
AskeyWilson polynomials, quadratic harnesses and martingales
 sumbitted) arxiv.org/abs/0812.0657, 2008. □ W̷LODEK BRYC AND JACEK WESO̷LOWSKI
"... Abstract. We use orthogonality measures of AskeyWilson polynomials to construct Markov processes with linear regressions and quadratic conditional variances. AskeyWilson polynomials are orthogonal martingale polynomials for these processes. ..."
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Cited by 2 (2 self)
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Abstract. We use orthogonality measures of AskeyWilson polynomials to construct Markov processes with linear regressions and quadratic conditional variances. AskeyWilson polynomials are orthogonal martingale polynomials for these processes.
CauchyStieltjès kernel families
, 2006
"... 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 CauchyStieltjes kernel (1 − θx) −1. This family exhibits similarities to exponential families and reproductive exponential models [12 ..."
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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 CauchyStieltjes 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 qexponential 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 qexponential 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 CauchyStieltjes 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
CONDITIONING OF QUADRATIC HARNESSES
, 903
"... 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 ..."
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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 conditionallyindependent quadratic harnesses and we show that the only qMeixner 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.
Appell Polynomials and Their Relatives
"... Let µ be a probability measure on the real line, all of whose moments ..."
Projection formulas for orthogonal polynomials
, 2006
"... We prove a new projection formula for the fourparameter family of orthogonal polynomials outside of the AskeyWilson class. By carefully analyzing the recurrence relations we manage to overcome the lack of explicit expression for the orthogonality measure. ∗Research partially supported by NSF grant ..."
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We prove a new projection formula for the fourparameter family of orthogonal polynomials outside of the AskeyWilson class. By carefully analyzing the recurrence relations we manage to overcome the lack of explicit expression for the orthogonality measure. ∗Research partially supported by NSF grants #INT0332062, #DMS0504198, and by the C.P. Taft Memorial Fund.
Multidimensional qNormal and related distributions Markov case
"... E l e c t r o n J o u r n a l ..."