Abstract
In classical probability the law of large numbers for the multiplicative convolution follows directly from the law for the additive convolution. In free probability this is not the case. The free additive law was proved by D. Voiculescu in 1986 for probability measures with bounded support and extended to all probability measures with first moment by J.M. Lindsay and V. Pata in 1997, while the free multiplicative law was proved only recently by G. Tucci in 2010. In this paper we extend Tucci’s result to measures with unbounded support while at the same time giving a more elementary proof for the case of bounded support. In contrast to the classical multiplicative convolution case, the limit measure for the free multiplicative law of large numbers is not a Dirac measure, unless the original measure is a Dirac measure. We also show that the mean value of lnx is additive with respect to the free multiplicative convolution while the variance of lnx is not in general additive. Furthermore we study the two parameter family (μ α, β ) α, β ≥ 0 of measures on (0, ∞) for which the S-transform is given by \(S_{\mu _{\alpha,\beta }}(z) = {(-z)}^{\beta }{(1 + z)}^{-\alpha }\), 0 < z < 1.
Mathematics Subject Classification (2010): 46L54, 60F05.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Arizmendi, O., Hasebe, T.: Classical and free infinite divisibility for Boolean stable laws (2012), arXiv:1205.1575
Banica, T., Belinschi, S.T., Capitaine, M., Collins, B.: Free Bessel laws. Canad. J. Math. 63(1), 3–37 (2011). DOI 10.4153/CJM-2010-060-6. URL http://dx.doi.org/10.4153/CJM-2010-060-6
Bercovici, H., Pata, V.: Stable laws and domains of attraction in free probability theory. Ann. of Math. (2) 149(3), 1023–1060 (1999). DOI 10.2307/121080. URL http://dx.doi.org/10.2307/121080. With an appendix by Philippe Biane
Bercovici, H., Voiculescu, D.: Free convolution of measures with unbounded support. Indiana Univ. Math. J. 42(3), 733–773 (1993). DOI 10.1512/iumj.1993.42.42033. URL http://dx.doi.org/10.1512/iumj.1993.42.42033
Biane, P.: Processes with free increments. Math. Z. 227(1), 143–174 (1998). DOI 10.1007/ PL00004363. URL http://dx.doi.org/10.1007/PL00004363
Haagerup, U., Larsen, F.: Brown’s spectral distribution measure for R-diagonal elements in finite von Neumann algebras. J. Funct. Anal. 176(2), 331–367 (2000). DOI 10.1006/ jfan.2000.3610. URL http://dx.doi.org/10.1006/jfan.2000.3610
Haagerup, U., Schultz, H.: Brown measures of unbounded operators affiliated with a finite von Neumann algebra. Math. Scand. 100(2), 209–263 (2007)
Haagerup, U., Schultz, H.: Invariant subspaces for operators in a general II1-factor. Publ. Math. Inst. Hautes Études Sci. (109), 19–111 (2009). DOI 10.1007/s10240-009-0018-7. URL http://dx.doi.org/10.1007/s10240-009-0018-7
Johnson, N.L., Kotz, S., Balakrishnan, N.: Continuous univariate distributions. Vol. 1, second edn. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics. John Wiley & Sons Inc., New York (1994). A Wiley-Interscience Publication
Larsen, F.: Powers of R-diagonal elements. J. Operator Theory 47(1), 197–212 (2002)
Lindsay, J.M., Pata, V.: Some weak laws of large numbers in noncommutative probability. Math. Z. 226(4), 533–543 (1997). DOI 10.1007/PL00004356. URL http://dx.doi.org/10.1007/PL00004356
Mlotkowski, W., Penson, K.A., Zyczkowski, K.: Densities of the Raney distributions (2012), arXiv:1211.7259
Penson, K.A., Zyczkowski, K.: Product of Ginibre matrices: Fuss-Catalan and Raney distributions. Phys. Rev. E 83, 061,118 (2011), arXiv:1103.3453
Rosenthal, J.S.: A first look at rigorous probability theory, second edn. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2006)
Tucci, G.H.: Limits laws for geometric means of free random variables. Indiana Univ. Math. J. 59(1), 1–13 (2010). DOI 10.1512/iumj.2010.59.3775. URL http://dx.doi.org/10.1512/iumj.2010.59.3775
Voiculescu, D.: Addition of certain noncommuting random variables. J. Funct. Anal. 66(3), 323–346 (1986). DOI 10.1016/0022-1236(86)90062-5. URL http://dx.doi.org/10.1016/0022-1236(86)90062-5
Voiculescu, D.: Multiplication of certain noncommuting random variables. J. Operator Theory 18(2), 223–235 (1987)
Voiculescu, D.V., Dykema, K.J., Nica, A.: Free random variables, CRM Monograph Series, vol. 1. American Mathematical Society, Providence, RI (1992). A noncommutative probability approach to free products with applications to random matrices, operator algebras and harmonic analysis on free groups
Acknowledgements
The first author is supported by ERC Advanced Grant No. OAFPG 27731 and the Danish National Research Foundation through the Center for Symmetry and Deformation (DNRF92).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Haagerup, U., Möller, S. (2013). The Law of Large Numbers for the Free Multiplicative Convolution. In: Carlsen, T., Eilers, S., Restorff, G., Silvestrov, S. (eds) Operator Algebra and Dynamics. Springer Proceedings in Mathematics & Statistics, vol 58. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39459-1_8
Download citation
DOI: https://doi.org/10.1007/978-3-642-39459-1_8
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-39458-4
Online ISBN: 978-3-642-39459-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)