Free exponential families as kernel families

Bryc, Włodzimierz

doi:10.1515/dema-2013-0184

Cited by 6 publications

(12 citation statements)

References 12 publications

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“…Barndorff-Nielsen and Thorbjørnsen (2006, p. 101). In particular, Bryc (2008) has developed the idea of free exponential families with quadratic variance functions.…”

Section: Discussionmentioning

confidence: 99%

Dispersion models for extremes

2009

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We propose extreme value analogues of natural exponential families and exponential dispersion models, and introduce the slope function as an analogue of the variance function. A class of extreme generalized linear regression models for analysis of extremes and lifetime data is introduced. The set of quadratic and power slope functions characterize well-known families such as the Rayleigh, Gumbel, power, Pareto, logistic, negative exponential, Weibull and Fréchet. We show a convergence theorem for slope functions, by which we may express the classical extreme value convergence results in terms of asymptotics for extreme dispersion models. The key idea is to explore the parallels between location families and natural exponential families, and between the convolution and minimum operations.

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“…Barndorff-Nielsen and Thorbjørnsen (2006, p. 101). In particular, Bryc (2008) has developed the idea of free exponential families with quadratic variance functions.…”

Section: Discussionmentioning

confidence: 99%

Dispersion models for extremes

2009

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“…As pointed out in the introduction, reference [1] describe the class of quadratic CSK families; that is the class of CSK families such that the corresponding variance function is a polynomial function in the mean of degree at most 2. Up to affine transformation and powers of free convolution, this class consists of, the Wigner's semi-circle (free gaussian), the Marchenko Pastur (free Poisson), the free Pascal (free negative binomial), the free Gamma, the free analog of hyperbolic type law and the free binomial families.…”

Section: Characterizations Of the Quadratic Csk Familiesmentioning

confidence: 99%

“…In particular they have studied the case when q = 0 which is related to free probability theory by using the Cauchy-Stieltjes kernel 1/(1 − θx) instead of the exponential kernel exp(θx). In [1], Bryc continue the study of Cauchy-Stieltjes Kernel (CSK) families for compactly supported probability measures ν. He has shown that such families can be parameterized by the mean and under this parametrization, the family (and measure ν) is uniquely determined by the variance function V (m) and the mean m 0 of ν.…”

Section: Introductionmentioning

confidence: 99%

Characterization of quadratic Cauchy–Stieltjes kernel families based on the orthogonality of polynomials

Fakhfakh

2018

Journal of Mathematical Analysis and Applications

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“…It follows from (15) by approximation that the vector on the right hand side of (17) belongs to X (n+1) . Furthermore,…”

Section: Proofsmentioning

confidence: 99%

Meixner class of orthogonal polynomials of a non-commutative monotone Lévy noise

Lytvynov

Rodionova

2018

Infin. Dimens. Anal. Quantum. Probab. Relat. Top.

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Let (X t ) t≥0 denote a non-commutative monotone Lévy process. Let ω = (ω(t)) t≥0 denote the corresponding monotone Lévy noise, i.e., formallyis defined as the orthogonal projection of the monomial ω ⊗n , f (n) onto the subspace of L 2 (τ ) that is orthogonal to all continuous polynomials of ω of order ≤ n − 1. We denote by OCP the linear span of the orthogonal polynomials. Each orthogonal polynomial P (n) (ω), f (n) depends only on the restriction of the function f (n) to the set {(t 1 , . . . , t n ) ∈ R n + | t 1 ≥ t 2 ≥ · · · ≥ t n }. The orthogonal polynomials allow us to construct a unitary operator J : L 2 (τ ) → F, where F is an extended monotone Fock space. Thus, we may think of the monotone noise ω as a distribution of linear operators acting in F. We say that the orthogonal polynomials belong to the Meixner class if CP = OCP. We prove that each system of orthogonal polynomials from the Meixner class is characterized by two parameters: λ ∈ R and η ≥ 0. In this case, the monotone Lévy noise has the representation ωHere, ∂ † t and ∂ t are the (formal) creation and annihilation operators at t ∈ R + acting in F.

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Free exponential families as kernel families

Cited by 6 publications

References 12 publications

Dispersion models for extremes

Dispersion models for extremes

Characterization of quadratic Cauchy–Stieltjes kernel families based on the orthogonality of polynomials

Meixner class of orthogonal polynomials of a non-commutative monotone Lévy noise

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