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Two-parameter non-commutative central limit theorem. (English. French summary) Zbl 1314.60073
Summary: In [Math. Z. 209, No. 1, 55–66 (1992; Zbl 0724.60023)], R. 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 Bo$$\dot{z}$$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 $$\ast$$-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 first and the second moments 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.

##### MSC:
 60F05 Central limit and other weak theorems 60B20 Random matrices (probabilistic aspects) 46L54 Free probability and free operator algebras 81S05 Commutation relations and statistics as related to quantum mechanics (general)
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