Results 1  10
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30
Conditional moments of qMeixner processes
, 2004
"... Abstract. We show that stochastic processes with linear conditional expectations and quadratic conditional variances are Markov, and their transition probabilities are related to a threeparameter family of orthogonal polynomials which generalize the Meixner polynomials. Special cases of these proce ..."
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Cited by 9 (5 self)
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Abstract. We show that stochastic processes with linear conditional expectations and quadratic conditional variances are Markov, and their transition probabilities are related to a threeparameter family of orthogonal polynomials which generalize the Meixner polynomials. Special cases of these processes are known to arise from the noncommutative generalizations of the Lévy processes. 1.
Orthogonal polynomials with a resolventtype generating function
, 2004
"... Free Sheffer polynomials are a polynomial family in noncommuting variables with a resolventtype generating function. Among such families, we describe the ones that are orthogonal with respect to a state. Their free cumulant generating functions satisfy a quadratic condition. If this condition is ..."
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Cited by 8 (1 self)
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Free Sheffer polynomials are a polynomial family in noncommuting variables with a resolventtype generating function. Among such families, we describe the ones that are orthogonal with respect to a state. Their free cumulant generating functions satisfy a quadratic condition. If this condition is linear and the state is tracial, we show that the state is a rotation of a free product state. We also describe interesting examples of nontracial infinitely divisible states with the quadratic property.
Quadratic harnesses, qcommutations, and orthogonal martingale polynomials
 Trans. Amer. Math. Soc
, 2007
"... Abstract. We introduce the quadratic harness condition and show that integrable quadratic harnesses have orthogonal martingale polynomials with a three step recurrence that satisfies a qcommutation relation. This implies that quadratic harnesses are essentially determined uniquely by five numerical ..."
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Cited by 6 (6 self)
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Abstract. We introduce the quadratic harness condition and show that integrable quadratic harnesses have orthogonal martingale polynomials with a three step recurrence that satisfies a qcommutation relation. This implies that quadratic harnesses are essentially determined uniquely by five numerical constants. Explicit recurrences for the orthogonal martingale polynomials are derived in several cases of interest. 1.
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.
Asymptotic behavior of random determinants
 in the Laguerre, Gram and Jacobi ensembles, arXiv math.PR/0607767
, 2007
"... Abstract. We consider properties of determinants of some random symmetric matrices issued from multivariate statistics: Wishart/Laguerre ensemble (sample covariance matrices), Uniform Gram ensemble (sample correlation matrices) and Jacobi ensemble (MANOVA). If n is the size of the sample, r ≤ n the ..."
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Cited by 4 (4 self)
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Abstract. We consider properties of determinants of some random symmetric matrices issued from multivariate statistics: Wishart/Laguerre ensemble (sample covariance matrices), Uniform Gram ensemble (sample correlation matrices) and Jacobi ensemble (MANOVA). If n is the size of the sample, r ≤ n the number of variates and Xn,r such a matrix, a generalization of the Bartletttype theorems gives a decomposition of det Xn,r into a product of r independent gamma or beta random variables. For n fixed, we study the evolution as r grows, and then take the limit of large r and n with r/n = t ≤ 1. We derive limit theorems for the sequence of processes with independent increments {n −1 log detX n,⌊nt⌋, t ∈ [0, T]}n for T ≤ 1: convergence in probability, invariance principle, large deviations. Since the logarithm of the determinant is a linear statistic of the empirical spectral distribution, we connect the results for marginals (fixed t) with those obtained by the spectral method. Actually, all the results hold true for log gases or β models, if we define the determinant as the product of charges. The classical matrix models (real, complex, and quaternionic) correspond to the particular values β = 1, 2, 4 of the Dyson parameter. 1.
APPELL POLYNOMIALS AND THEIR RELATIVES II. BOOLEAN THEORY
"... ABSTRACT. The Appelltype polynomial family corresponding to the simplest noncommutative derivative operator turns out to be connected with the Boolean probability theory, the simplest of the three universal noncommutative probability theories (the other two being free and tensor/classical probabi ..."
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Cited by 3 (0 self)
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ABSTRACT. The Appelltype polynomial family corresponding to the simplest noncommutative derivative operator turns out to be connected with the Boolean probability theory, the simplest of the three universal noncommutative probability theories (the other two being free and tensor/classical probability). The basic properties of the Boolean Appell polynomials are described. In particular, their generating function turns out to have a resolventtype form, just like the generating function for the free Sheffer polynomials. It follows that the Meixner (that is, Sheffer plus orthogonal) polynomial classes, in the Boolean and free theory, coincide. This is true even in the multivariate case. A number of applications of this fact are described, to the BelinschiNica and BercoviciPata maps, conditional freeness, and the LahaLukacs type characterization. A number of properties which hold for the Meixner class in the free and classical cases turn out to hold in general in the Boolean theory. Examples include the behavior of the Jacobi parameters under convolution, the relationship between the Jacobi parameters and cumulants, and an operator model for cumulants. Along the way, we obtain a multivariate version of the Stieltjes continued fraction expansion for the moment generating function of an arbitrary state with monic orthogonal polynomials. 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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Cited by 1 (0 self)
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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
BOCHNERPEARSONTYPE CHARACTERIZATION OF THE FREE MEIXNER CLASS
, 909
"... ABSTRACT. The operator Lµ: f ↦ → ∫ f(x)−f(y) x−y dµ(y) is, for a compactly supported measure µ with an L 3 density, a closed, densely defined operator on L 2 (µ). We show that the operator Q = pL 2 µ−qLµ has polynomial eigenfunctions if and only if µ is a free Meixner distribution. The only time Q ..."
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Cited by 1 (0 self)
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ABSTRACT. The operator Lµ: f ↦ → ∫ f(x)−f(y) x−y dµ(y) is, for a compactly supported measure µ with an L 3 density, a closed, densely defined operator on L 2 (µ). We show that the operator Q = pL 2 µ−qLµ has polynomial eigenfunctions if and only if µ is a free Meixner distribution. The only time Q has orthogonal polynomial eigenfunctions is if µ is a semicircular distribution. More generally, the only time the operator p(LνLµ) − qLµ has orthogonal polynomial eigenfunctions is when µ and ν are related by a Jacobi shift. 1.