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**Two-parameter non-commutative central limit theorem.** / Blitvic, Natasa.

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Blitvic, N 2014, 'Two-parameter non-commutative central limit theorem', *Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques*, vol. 50, no. 4, pp. 1456-1473. https://doi.org/10.1214/13-AIHP550

Blitvic, N. (2014). Two-parameter non-commutative central limit theorem. *Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques*, *50*(4), 1456-1473. https://doi.org/10.1214/13-AIHP550

Blitvic N. Two-parameter non-commutative central limit theorem. Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques. 2014 Oct 17;50(4):1456-1473. doi: 10.1214/13-AIHP550

@article{8363100efe3d4d4d9e1f1dfae3ddbc29,

title = "Two-parameter non-commutative central limit theorem",

abstract = "In 1992, Speicher showed the fundamental fact that the probability measures playing the role of the classical Gaussian in the various non-commutative probability theories (viz. fermionic probability, Voiculescu{\textquoteright}s free probability, and qq-deformed probability of Bo{\.z}ejko and Speicher) all arise as the limits in a generalized Central Limit Theorem. The latter concerns sequences of non-commutative random variables (elements of a ∗∗-algebra equipped with a state) drawn from an ensemble of pair-wise commuting or anti-commuting elements, with the respective limiting distributions determined by the average value of the commutation coefficients.In this paper, we derive a more general form of the Central Limit Theorem in which the pair-wise commutation coefficients are arbitrary real numbers. The classical Gaussian statistics now undergo a second-parameter refinement as a result of controlling for the first and the second moments of the commutation coefficients. An application yields the random matrix models for the (q,t)(q,t)-Gaussian statistics, which were recently shown to have rich connections to operator algebras, special functions, orthogonal polynomials, mathematical physics, and random matrix theory.",

keywords = "Central Limit Theorem, Free probability, Random matrices, qq-Gaussians",

author = "Natasa Blitvic",

year = "2014",

month = oct,

day = "17",

doi = "10.1214/13-AIHP550",

language = "English",

volume = "50",

pages = "1456--1473",

journal = "Annales de l'Institut Henri Poincar{\'e} (B) Probabilit{\'e}s et Statistiques",

issn = "0246-0203",

publisher = "Institute of Mathematical Statistics",

number = "4",

}

TY - JOUR

T1 - Two-parameter non-commutative central limit theorem

AU - Blitvic, Natasa

PY - 2014/10/17

Y1 - 2014/10/17

N2 - In 1992, Speicher showed the fundamental fact that the probability measures playing the role of the classical Gaussian in the various non-commutative probability theories (viz. fermionic probability, Voiculescu’s free probability, and qq-deformed probability of Bożejko and Speicher) all arise as the limits in a generalized Central Limit Theorem. The latter concerns sequences of non-commutative random variables (elements of a ∗∗-algebra equipped with a state) drawn from an ensemble of pair-wise commuting or anti-commuting elements, with the respective limiting distributions determined by the average value of the commutation coefficients.In this paper, we derive a more general form of the Central Limit Theorem in which the pair-wise commutation coefficients are arbitrary real numbers. The classical Gaussian statistics now undergo a second-parameter refinement as a result of controlling for the first and the second moments of the commutation coefficients. An application yields the random matrix models for the (q,t)(q,t)-Gaussian statistics, which were recently shown to have rich connections to operator algebras, special functions, orthogonal polynomials, mathematical physics, and random matrix theory.

AB - In 1992, Speicher showed the fundamental fact that the probability measures playing the role of the classical Gaussian in the various non-commutative probability theories (viz. fermionic probability, Voiculescu’s free probability, and qq-deformed probability of Bożejko and Speicher) all arise as the limits in a generalized Central Limit Theorem. The latter concerns sequences of non-commutative random variables (elements of a ∗∗-algebra equipped with a state) drawn from an ensemble of pair-wise commuting or anti-commuting elements, with the respective limiting distributions determined by the average value of the commutation coefficients.In this paper, we derive a more general form of the Central Limit Theorem in which the pair-wise commutation coefficients are arbitrary real numbers. The classical Gaussian statistics now undergo a second-parameter refinement as a result of controlling for the first and the second moments of the commutation coefficients. An application yields the random matrix models for the (q,t)(q,t)-Gaussian statistics, which were recently shown to have rich connections to operator algebras, special functions, orthogonal polynomials, mathematical physics, and random matrix theory.

KW - Central Limit Theorem

KW - Free probability

KW - Random matrices

KW - qq-Gaussians

U2 - 10.1214/13-AIHP550

DO - 10.1214/13-AIHP550

M3 - Journal article

VL - 50

SP - 1456

EP - 1473

JO - Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques

JF - Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques

SN - 0246-0203

IS - 4

ER -