Two-parameter non-commutative Central Limit Theorem

Natasha Blitvić

Annales de l'I.H.P. Probabilités et statistiques (2014)

  • Volume: 50, Issue: 4, page 1456-1473
  • ISSN: 0246-0203

Abstract

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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 q -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 firstand the secondmoments of the commutation coefficients. An application yields the random matrix models for the ( 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.

How to cite

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Blitvić, Natasha. "Two-parameter non-commutative Central Limit Theorem." Annales de l'I.H.P. Probabilités et statistiques 50.4 (2014): 1456-1473. <http://eudml.org/doc/272036>.

@article{Blitvić2014,
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’s free probability, and $q$-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 $\ast $-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 firstand the secondmoments of the commutation coefficients. An application yields the random matrix models for the $(q,t)$-Gaussian statistics, which were recently shown to have rich connections to operator algebras, special functions, orthogonal polynomials, mathematical physics, and random matrix theory.},
author = {Blitvić, Natasha},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {central limit theorem; free probability; random matrices; $q$-gaussians; central limit theorem; free probability; random matrices; -Gaussians},
language = {eng},
number = {4},
pages = {1456-1473},
publisher = {Gauthier-Villars},
title = {Two-parameter non-commutative Central Limit Theorem},
url = {http://eudml.org/doc/272036},
volume = {50},
year = {2014},
}

TY - JOUR
AU - Blitvić, Natasha
TI - Two-parameter non-commutative Central Limit Theorem
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2014
PB - Gauthier-Villars
VL - 50
IS - 4
SP - 1456
EP - 1473
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 $q$-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 $\ast $-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 firstand the secondmoments of the commutation coefficients. An application yields the random matrix models for the $(q,t)$-Gaussian statistics, which were recently shown to have rich connections to operator algebras, special functions, orthogonal polynomials, mathematical physics, and random matrix theory.
LA - eng
KW - central limit theorem; free probability; random matrices; $q$-gaussians; central limit theorem; free probability; random matrices; -Gaussians
UR - http://eudml.org/doc/272036
ER -

References

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