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Two-parameter non-commutative central limit theorem

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Published
<mark>Journal publication date</mark>17/10/2014
<mark>Journal</mark>Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques
Issue number4
Volume50
Number of pages18
Pages (from-to)1456-1473
Publication StatusPublished
<mark>Original language</mark>English

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.