Two-parameter non-commutative Central Limit Theorem

Blitvić, Natasha

Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 4, pp. 1456-1473.

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.

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.

Classification : 60F05, 46L50, 60B20, 81S05
Mots clés : central limit theorem, free probability, random matrices, $q$-gaussians

[1] M. Arik and D. D. Coon. Hilbert spaces of analytic functions and generalized coherent states. J. Math. Phys. 17 (1976) 524-527. | MR | Zbl

[2] P. Biane. Free hypercontractivity. Comm. Math. Phys. 184 (1997) 457-474. | MR | Zbl

[3] P. Biane. Free probability for probabilists. In Quantum Probability Communications, Vol. XI (Grenoble, 1998), QP-PQ, XI 55-71. World Sci. Publ., River Edge, NJ, 2003. | MR | Zbl

[4] L. C. Biedenharn. The quantum group $\mathrm{SU}q\left(2\right)$ and a $q$-analogue of the boson operators. J. Phys. A 22 (1989) L873-L878. | MR | Zbl

[5] N. Blitvić. The $(q,t)$-Gaussian process. J. Funct. Anal. 10 (2012) 3270-3305. | MR | Zbl

[6] M. Bożejko, B. Kümmerer and R. Speicher. $q$-Gaussian processes: Non-commutative and classical aspects. Comm. Math. Phys. 185 (1997) 129-154. | MR | Zbl

[7] M. Bożejko and R. Speicher. An example of a generalized Brownian motion. Comm. Math. Phys. 137 (1991) 519-531. | MR | Zbl

[8] M. Bożejko and H. Yoshida. Generalized $q$-deformed Gaussian random variables. In Quantum Probability 127-140. Banach Center Publ. 73. Polish Acad. Sci. Inst. Math., Warsaw, 2006. | MR | Zbl

[9] E. A. Carlen and E. H. Lieb. Optimal hypercontractivity for Fermi fields and related noncommutative integration inequalities. Comm. Math. Phys. 155 (1993) 27-46. | MR | Zbl

[10] R. Chakrabarti and R. Jagannathan. A $(p,q)$-oscillator realization of two-parameter quantum algebras. J. Phys. A 24 (1991) L711-L718. | MR | Zbl

[11] W. Y. C. Chen, E. Y. P. Deng, R. R. X. Du, R. P. Stanley and C. H. Yan. Crossings and nestings of matchings and partitions. Trans. Amer. Math. Soc. 359 (2007) 1555-1575 (electronic). | MR | Zbl

[12] U. Frisch and R. Bourret. Parastochastics. J. Math. Phys. 11 (1970) 364-390. | MR | Zbl

[13] M. E. H. Ismail. Asymptotics of $q$-orthogonal polynomials and a $q$-Airy function. Int. Math. Res. Not. 2005 (2005) 1063-1088. | MR | Zbl

[14] G. Iwata. Transformation functions in the complex domain. Progr. Theoret. Phys. 6 (1951) 524-528. | MR

[15] A. Kasraoui and J. Zeng. Distribution of crossings, nestings and alignments of two edges in matchings and partitions. Electron. J. Combin. 13 (2006) Research Paper 33, 12 pp. (electronic). | MR | Zbl

[16] T. Kemp. Hypercontractivity in non-commutative holomorphic spaces. Comm. Math. Phys. 259 (2005) 615-637. | MR | Zbl

[17] A. Khorunzhy. Products of random matrices and $q$-catalan numbers. Preprint, 2001. Available at arXiv:math/0104074v2 [math.CO].

[18] M. Klazar. On identities concerning the numbers of crossings and nestings of two edges in matchings. SIAM J. Discrete Math. 20 (2006) 960-976. | MR | Zbl

[19] A. J. Macfarlane. On $q$-analogues of the quantum harmonic oscillator and the quantum group $\mathrm{SU}\left(2\right)q$. J. Phys. A 22 (1989) 4581. | MR | Zbl

[20] C. Mazza and D. Piau. Products of correlated symmetric matrices and $q$-Catalan numbers. Probab. Theory Related Fields 124 (2002) 574-594. | MR | Zbl

[21] A. Nica and R. Speicher. Lectures on the Combinatorics of Free Probability. London Mathematical Society Lecture Note Series 335. Cambridge Univ. Press, Cambridge, 2006. | MR | Zbl

[22] R. Speicher. A noncommutative central limit theorem. Math. Z. 209 (1992) 55-66. | EuDML | MR | Zbl

[23] D. Voiculescu. Addition of certain noncommuting random variables. J. Funct. Anal. 66 (1986) 323-346. | MR | Zbl

[24] D. V. Voiculescu, K. J. Dykema and A. Nica. Free Random Variables. CRM Monograph Series 1. American Mathematical Society, Providence, RI, 1992. | MR | Zbl