The Annals of Probability

An inequality for the distance between densities of free convolutions

V. Kargin

Full-text: Open access


This paper contributes to the study of the free additive convolution of probability measures. It shows that under some conditions, if measures $\mu_{i}$ and $\nu_{i}$, $i=1,2$, are close to each other in terms of the Lé vy metric and if the free convolution $\mu_{1}\boxplus\mu_{2}$ is sufficiently smooth, then $\nu_{1}\boxplus\nu_{2}$ is absolutely continuous, and the densities of measures $\nu_{1}\boxplus\nu_{2}$ and $\mu_{1}\boxplus\mu_{2}$ are close to each other. In particular, convergence in distribution $\mu_{1}^{(n)}\rightarrow\mu_{1}$, $\mu_{2}^{(n)}\rightarrow\mu_{2}$ implies that the density of $\mu_{1}^{(n)}\boxplus\mu_{2}^{(n)}$ is defined for all sufficiently large $n$ and converges to the density of $\mu_{1}\boxplus\mu_{2}$. Some applications are provided, including: (i) a new proof of the local version of the free central limit theorem, and (ii) new local limit theorems for sums of free projections, for sums of $\boxplus$-stable random variables and for eigenvalues of a sum of two $N$-by-$N$ random matrices.

Article information

Ann. Probab., Volume 41, Number 5 (2013), 3241-3260.

First available in Project Euclid: 12 September 2013

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46L54: Free probability and free operator algebras 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52)

Free probability free convolution convergence of measures


Kargin, V. An inequality for the distance between densities of free convolutions. Ann. Probab. 41 (2013), no. 5, 3241--3260. doi:10.1214/12-AOP756.

Export citation


  • [1] Belinschi, S. T. (2008). The Lebesgue decomposition of the free additive convolution of two probability distributions. Probab. Theory Related Fields 142 125–150.
  • [2] Belinschi, S. T. and Bercovici, H. (2007). A new approach to subordination results in free probability. J. Anal. Math. 101 357–365.
  • [3] Bercovici, H. and Pata, V. (1999). Stable laws and domains of attraction in free probability theory. Ann. of Math. (2) 149 1023–1060.
  • [4] Bercovici, H. and Voiculescu, D. (1993). Free convolution of measures with unbounded support. Indiana Univ. Math. J. 42 733–773.
  • [5] 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.
  • [6] Bercovici, H. and Voiculescu, D. (1998). Regularity questions for free convolution. In Nonselfadjoint Operator Algebras, Operator Theory, and Related Topics (H. Bercovici and C. Foias, eds.). Operator Theory: Advances and Applications 104 37–47. Birkhäuser, Basel.
  • [7] Biane, P. (1998). Processes with free increments. Math. Z. 227 143–174.
  • [8] Biane, P. (1998). Representations of symmetric groups and free probability. Adv. Math. 138 126–181.
  • [9] Feinberg, J. and Zee, A. (1997). Non-Gaussian non-Hermitian random matrix theory: Phase transition and addition formalism. Nuclear Phys. B 501 643–669.
  • [10] Guionnet, A., Krishnapur, M. and Zeitouni, O. (2011). The single ring theorem. Ann. of Math. (2) 174 1189–1217.
  • [11] Haagerup, U. and Thorbjørnsen, S. (2005). A new application of random matrices: $\mathrm{Ext}(C^{*}_{\mathrm{red}}(F_{2}))$ is not a group. Ann. of Math. (2) 162 711–775.
  • [12] Henrici, P. (1986). Applied and Computational Complex Analysis. Vol. 3. Wiley, New York.
  • [13] Kantorovič, L. V. (1948). Functional analysis and applied mathematics. Uspekhi Matem. Nauk (N.S.) 3 89–185.
  • [14] Kargin, V. (2012). A concentration inequality and a local law for the sum of two random matrices. Probab. Theory Related Fields. To appear. Available at
  • [15] Maassen, H. (1992). Addition of freely independent random variables. J. Funct. Anal. 106 409–438.
  • [16] Nica, A. and Speicher, R. (2006). Lectures on the Combinatorics of Free Probability. London Mathematical Society Lecture Note Series 335. Cambridge Univ. Press, Cambridge.
  • [17] Pastur, L. and Vasilchuk, V. (2000). On the law of addition of random matrices. Comm. Math. Phys. 214 249–286.
  • [18] Shiryaev, A. N. (1996). Probability, 2nd ed. Graduate Texts in Mathematics 95. Springer, New York.
  • [19] Speicher, R. (1993). Free convolution and the random sum of matrices. Publ. Res. Inst. Math. Sci. 29 731–744.
  • [20] 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.
  • [21] Voiculescu, D. (1986). Addition of certain noncommuting random variables. J. Funct. Anal. 66 323–346.
  • [22] Voiculescu, D. (1991). Limit laws for random matrices and free products. Invent. Math. 104 201–220.
  • [23] Voiculescu, D. (1993). The analogues of entropy and of Fisher’s information measure in free probability theory. I. Comm. Math. Phys. 155 71–92.
  • [24] Voiculescu, D. (1996). The analogues of entropy and of Fisher’s information measure in free probability theory. III. The absence of Cartan subalgebras. Geom. Funct. Anal. 6 172–199.
  • [25] Voiculescu, D. V., Dykema, K. J. and Nica, A. (1992). Free Random Variables: A Noncommutative Probability Approach to Free Products With Applications to Random Matrices, Operator Algebras and Harmonic Analysis on Free Groups. CRM Monograph Series 1. Amer. Math. Soc., Providence, RI.
  • [26] Wang, J.-C. (2010). Local limit theorems in free probability theory. Ann. Probab. 38 1492–1506.
  • [27] Zee, A. (1996). Law of addition in random matrix theory. Nuclear Phys. B 474 726–744.