Research output: Contribution to Journal/Magazine › Journal article › peer-review
Two-parameter non-commutative central limit theorem. / Blitvic, Natasa.
In: Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, Vol. 50, No. 4, 17.10.2014, p. 1456-1473.Research output: Contribution to Journal/Magazine › Journal article › peer-review
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TY - JOUR
T1 - Two-parameter non-commutative central limit theorem
AU - Blitvic, Natasa
PY - 2014/10/17
Y1 - 2014/10/17
N2 - 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.
AB - 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.
KW - Central Limit Theorem
KW - Free probability
KW - Random matrices
KW - qq-Gaussians
U2 - 10.1214/13-AIHP550
DO - 10.1214/13-AIHP550
M3 - Journal article
VL - 50
SP - 1456
EP - 1473
JO - Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques
JF - Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques
SN - 0246-0203
IS - 4
ER -