Annals of Probability

Local limit theorems in free probability theory

Jiun-Chau Wang

Full-text: Open access


In this paper, we study the superconvergence phenomenon in the free central limit theorem for identically distributed, unbounded summands. We prove not only the uniform convergence of the densities to the semicircular density but also their Lp-convergence to the same limit for p > 1/2. Moreover, an entropic central limit theorem is obtained as a consequence of the above results.

Article information

Ann. Probab., Volume 38, Number 4 (2010), 1492-1506.

First available in Project Euclid: 8 July 2010

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46L54: Free probability and free operator algebras
Secondary: 60F05: Central limit and other weak theorems 60F25: $L^p$-limit theorems

Free central limit theorem superconvergence local limit theorems free entropy


Wang, Jiun-Chau. Local limit theorems in free probability theory. Ann. Probab. 38 (2010), no. 4, 1492--1506. doi:10.1214/09-AOP505.

Export citation


  • [1] Artstein, S., Ball, K. M., Barthe, F. and Naor, A. (2004). Solution of Shannon’s problem on the monotonicity of entropy. J. Amer. Math. Soc. 17 975–982.
  • [2] Barron, A. R. (1986). Entropy and the central limit theorem. Ann. Probab. 14 336–342.
  • [3] Belinschi, S. T. and Bercovici, H. (2004). Atoms and regularity for measures in a partially defined free convolution semigroup. Math. Z. 248 665–674.
  • [4] Belinschi, S. T. and Nica, A. (2008). On a remarkable semigroup of homomorphisms with respect to free multiplicative convolution. Indiana Univ. Math. J. 57 1679–1713.
  • [5] Bercovici, H. and Voiculescu, D. (1993). Free convolution of measures with unbounded support. Indiana Univ. Math. J. 42 733–773.
  • [6] Bercovici, H. and Voiculescu, D. (1995). Superconvergence to the central limit and failure of the Cramér theorem for free random variables. Probab. Theory Related Fields 103 215–222.
  • [7] Biane, P. (1997). On the free convolution with a semi-circular distribution. Indiana Univ. Math. J. 46 705–718.
  • [8] Biane, P. (2003). Free probability for probabilists. In Quantum Probability Communications, Vol. XI (Grenoble, 1998) 55–71. World Scientific, River Edge, NJ.
  • [9] Biane, P. (2003). Logarithmic Sobolev inequalities, matrix models and free entropy. Acta Math. Sin. (Engl. Ser.) 19 497–506.
  • [10] Donoghue, W. F. Jr. (1974). Monotone Matrix Functions and Analytic Continuation. Springer, New York.
  • [11] Gnedenko, B. V. and Kolmogorov, A. N. (1968). Limit Distributions for Sums of Independent Random Variables. Addison-Wesley, Reading, MA.
  • [12] Hiai, F. and Petz, D. (1998). Maximizing free entropy. Acta Math. Hungar. 80 335–356.
  • [13] Johnson, O. (2004). Information Theory and the Central Limit Theorem. Imperial College Press, London.
  • [14] Kargin, V. (2007). On superconvergence of sums of free random variables. Ann. Probab. 35 1931–1949.
  • [15] Maassen, H. (1992). Addition of freely independent random variables. J. Funct. Anal. 106 409–438.
  • [16] Pastur, L. and Vasilchuk, V. (2000). On the law of addition of random matrices. Comm. Math. Phys. 214 249–286.
  • [17] Pata, V. (1996). The central limit theorem for free additive convolution. J. Funct. Anal. 140 359–380.
  • [18] Schultz, H. (2008). Semicircularity, Gaussianity and monotonicity of entropy. J. Operator Theory 60 125–136.
  • [19] Segal, I. E. (1953). A non-commutative extension of abstract integration. Ann. of Math. (2) 57 401–457.
  • [20] Shlyakhtenko, D. (2007). A free analogue of Shannon’s problem on monotonicity of entropy. Adv. Math. 208 824–833.
  • [21] Voiculescu, D. (1985). Symmetries of some reduced free product C*-algebras. In Operator Algebras and Their Connections with Topology and Ergodic Theory (BuŞteni, 1983). Lecture Notes in Math. 1132 556–588. Springer, Berlin.
  • [22] Voiculescu, D. (1986). Addition of certain noncommuting random variables. J. Funct. Anal. 66 323–346.
  • [23] Voiculescu, D. (1991). Limit laws for random matrices and free products. Invent. Math. 104 201–220.
  • [24] Voiculescu, D. (1993). The analogues of entropy and of Fisher’s information measure in free probability theory. I. Comm. Math. Phys. 155 71–92.
  • [25] Voiculescu, D. (1997). The analogues of entropy and of Fisher’s information measure in free probability theory. IV. Maximum entropy and freeness. In Free Probability Theory (Waterloo, ON, 1995). Fields Institute Communications 12 293–302. Amer. Math. Soc., Providence, RI.
  • [26] Voiculescu, D. (1998). The analogues of entropy and of Fisher’s information measure in free probability theory. V. Noncommutative Hilbert transforms. Invent. Math. 132 189–227.
  • [27] Voiculescu, D. (2002). Free entropy. Bull. Lond. Math. Soc. 34 257–278.
  • [28] Voiculescu, D. V., Dykema, K. J. and Nica, A. (1992). Free Random Variables. CRM Monograph Series 1. Amer. Math. Soc., Providence, RI.