Two-parameter non-commutative Central Limit Theorem
Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 4, p. 1456-1473

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 q-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)-Gaussian statistics, which were recently shown to have rich connections to operator algebras, special functions, orthogonal polynomials, mathematical physics, and random matrix theory.

En 1992, Speicher a montré que les mesures de probabilités jouant le rôle des lois gaussiennes dans les différentes théories des probabilités non-commutatives (probabilités fermioniques, probabilités libres à la Voiculescu, probabilités q-déformées à la Bożejko et Speicher) apparaissent toutes comme limites d’un Théorème de la limite centrale généralisé. Ceci concerne des suites de variables aléatoires non-commutatives (éléments d’une *-algèbre munie d’un état) choisies dans un ensemble d’éléments qui commutent ou anti-commutent deux-à-deux, avec les distributions limites respectives déterminées par la valeur moyenne des coefficients de commutation. Dans ce papier, nous dérivons une forme plus générale du Théorème de la limite centrale où les coefficients de commutation deux-à-deux sont des nombres réels arbitraires. Les statistiques gaussiennes classiques dépendent maintenant d’un second paramètre comme résultat du contrôle du premier et du second moment des coefficients de commutation. Une application donne le modèle de matrices aléatoires pour les statistiques (q,t)-gaussiennes, pour lesquelles il a été montré récemment qu’elles ont des profondes connexions avec les algèbres d’opérateurs, les fonctions spéciales, les polynômes orthogonaux, la physique mathématique et la théorie des matrices aléatoires.

DOI : https://doi.org/10.1214/13-AIHP550
Classification:  60F05,  46L50,  60B20,  81S05
Keywords: central limit theorem, free probability, random matrices, q-gaussians
@article{AIHPB_2014__50_4_1456_0,
     author = {Blitvi\'c, Natasha},
     title = {Two-parameter non-commutative Central Limit Theorem},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {50},
     number = {4},
     year = {2014},
     pages = {1456-1473},
     doi = {10.1214/13-AIHP550},
     zbl = {06377561},
     mrnumber = {3270001},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2014__50_4_1456_0}
}
Blitvić, Natasha. Two-parameter non-commutative Central Limit Theorem. Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 4, pp. 1456-1473. doi : 10.1214/13-AIHP550. http://www.numdam.org/item/AIHPB_2014__50_4_1456_0/

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