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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.

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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