Identification of the theory of orthogonal polynomials in d-indeterminates with the theory of 3-diagonal symmetric interacting Fock spaces on ℂd

The identification mentioned in the title allows a formulation of the multidimensional Favard lemma different from the ones currently used in the literature and which parallels the original 1-dimensional formulation in the sense that the positive Jacobi sequence is replaced by a sequence of positive Hermitean (square) matrices and the real Jacobi sequence by a sequence of positive definite kernels. The above result opens the way to the program of a purely algebraic classification of probability measures on [Formula: see text] with moments of any order and more generally of states on the polynomial algebra on [Formula: see text]. The quantum decomposition of classical real-valued random variables with all moments is one of the main ingredients in the proof.

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INTERPOLATION OF CHEBYSHEV POLYNOMIALS AND INTERACTING FOCK SPACES

Infinite Dimensional Analysis Quantum Probability and Related Topics ◽

10.1142/s0219025706002421 ◽

2006 ◽

Vol 09 (03) ◽

pp. 361-371 ◽

Cited By ~ 7

Author(s):

IZUMI KUBO ◽

HUI-HSIUNG KUO ◽

SUAT NAMLI

Keyword(s):

Orthogonal Polynomials ◽

Anderson Model ◽

Chebyshev Polynomials ◽

Field Operator ◽

Parameter Family ◽

Probability Measures ◽

Fock Spaces ◽

Arcsine Distribution ◽

Multiplicative Renormalization ◽

Renormalization Method

We discover a family of probability measures μa, 0 < a ≤ 1, [Formula: see text] which contains the arcsine distribution (a = 1) and semi-circle distribution (a = 1/2). We show that the multiplicative renormalization method can be used to produce orthogonal polynomials, called Chebyshev polynomials with one parameter a, which reduce to Chebyshev polynomials of the first and second kinds when a = 1 and 1/2 respectively. Moreover, we derive the associated Jacobi–Szegö parameters. This one-parameter family of probability measures coincides with the vacuum distribution of the field operator of the interacting Fock spaces related to the Anderson model.

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BESSIS–MOUSSA–VILLANI CONJECTURE AND GENERALIZED GAUSSIAN RANDOM VARIABLES

Infinite Dimensional Analysis Quantum Probability and Related Topics ◽

10.1142/s0219025708003129 ◽

2008 ◽

Vol 11 (03) ◽

pp. 313-321 ◽

Cited By ~ 2

Author(s):

MAREK BOŻEJKO

Keyword(s):

Hilbert Space ◽

Real Line ◽

Real Hilbert Space ◽

Random Variables ◽

Positive Definite Function ◽

Positive Definite ◽

Definite Function ◽

Gaussian Random Variables ◽

The Real ◽

Bmv Conjecture

In this paper we give the solution of Bessis–Moussa–Villani (BMV) conjecture for the generalized Gaussian random variables [Formula: see text] where f is in the real Hilbert space [Formula: see text]. The main examples of generalized Gaussian random variables are q-Gaussian random variables, (-1 ≤ q ≤ 1), related to q-CCR relation and other commutation relations. We will prove that BMV conjecture is true for all operators A = G(f), B = G(g); i.e. we will show that the function [Formula: see text] is positive-definite function on the real line. The case q = 0, i.e. when G(f) are the free Gaussian (Wigner) random variables and the operators A and B are free with respect to the vacuum trace was proved by Fannes and Petz.23

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Random Variables and Positive Definite Kernels Associated with the Schrödinger Algebra

10.1063/1.3460158 ◽

2010 ◽

Author(s):

Luigi Accardi ◽

Andreas Boukas ◽

Vladimir Dobrev

Keyword(s):

Random Variables ◽

Positive Definite ◽

Positive Definite Kernels

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Characterization of Product Probability Measures on ℝd in Terms of Their Orthogonal Polynomials

Open Systems & Information Dynamics ◽

10.1142/s1230161216500220 ◽

2016 ◽

Vol 23 (04) ◽

pp. 1650022

Author(s):

Luigi Accardi ◽

Abdallah Dhahri ◽

Ameur Dhahri

Keyword(s):

Orthogonal Polynomial ◽

Orthogonal Polynomials ◽

Positive Definite ◽

Linear Operators ◽

Polynomial Basis ◽

Probability Measures ◽

Positive Definite Kernel ◽

Product Measures

In paper [1] the d-dimensional analogue of the Jacobi parameters has been individuated in a pair of sequences ((a.|n0),(Ω∼n)), where (a.|n0) is a sequence of Hermitean matrices and Ω∼n(n ∈ ℕ) a positive definite kernel with values in the linear operators on the n-th space of the orthogonal gradation. In this paper we prove that product measures on ℝd are characterized by the property that the (a.|n0) are diagonal and the (Ω∼n) quasidiagonal (see Definition 2 below) in the orthogonal polynomial basis.

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M-convolutions of products and ratios, statistical distributions and fractional calculus

Analysis ◽

10.1515/anly-2015-5002 ◽

2016 ◽

Vol 36 (1) ◽

Cited By ~ 2

Author(s):

Arakaparampil M. Mathai

Keyword(s):

Fractional Calculus ◽

Random Variables ◽

Positive Definite Matrix ◽

Positive Definite ◽

Positive Definite Matrices ◽

The Real ◽

Real Scalar ◽

Scalar Variable ◽

The Matrix

AbstractIt is shown that Mellin convolutions of products and ratios in the real scalar variable case can be considered as densities of products and ratios of two independently distributed real scalar positive random variables. It is also shown that these are also connected to Krätzel integrals and to the Krätzel transform in applied analysis, to reaction-rate probability integrals in astrophysics and to other related aspects when the random variables have gamma or generalized gamma densities, and to fractional calculus when one of the variables has a type-1 beta density and the other variable has an arbitrary density. Matrix-variate analogues are also discussed. In the matrix-variate case, the M-convolutions introduced by the author are shown to be directly connected to densities of products and ratios of statistically independently distributed positive definite matrix random variables in the real case and to Hermitian positive definite matrices in the complex domain. These M-convolutions reduce to Mellin convolutions in the scalar variable case.

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Central limit theorem for linear processes generated by IID random variables under the sub-linear expectation

Applied Mathematics-A Journal of Chinese Universities ◽

10.1007/s11766-021-3882-7 ◽

2021 ◽

Vol 36 (2) ◽

pp. 243-255

Author(s):

Wei Liu ◽

Yong Zhang

Keyword(s):

Central Limit Theorem ◽

Limit Theorem ◽

Central Limit ◽

Invariance Principle ◽

Random Variables ◽

Probability Measures ◽

Linear Processes ◽

The Central Limit Theorem

AbstractIn this paper, we investigate the central limit theorem and the invariance principle for linear processes generated by a new notion of independently and identically distributed (IID) random variables for sub-linear expectations initiated by Peng [19]. It turns out that these theorems are natural and fairly neat extensions of the classical Kolmogorov’s central limit theorem and invariance principle to the case where probability measures are no longer additive.

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