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.