Two-parameter non-commutative central limit theorem

Blitvic, Natasa (2014) Two-parameter non-commutative central limit theorem. Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, 50 (4). pp. 1456-1473. ISSN 0246-0203

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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 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.

Item Type:
Journal Article
Journal or Publication Title:
Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques
Uncontrolled Keywords:
/dk/atira/pure/subjectarea/asjc/2600/2613
Subjects:
ID Code:
83875
Deposited By:
Deposited On:
09 Jan 2017 09:10
Refereed?:
Yes
Published?:
Published
Last Modified:
05 Aug 2020 04:50