The Law of Large Numbers for the Free Multiplicative Convolution

Uffe Haagerup; Søren Møller

doi:10.1007/978-3-642-39459-1_8

The Law of Large Numbers for the Free Multiplicative Convolution

Uffe Haagerup, Søren Møller

Department of Mathematical Sciences

14 Citations (Scopus)

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_μαβ(z) = (-z)^β(1 + z)^-α, 0 < z < 1.

Original language	English
Title of host publication	Operator Algebra and Dynamics : Nordforsk Network Closing Conference, Faroe Islands, May 2012
Editors	Toke M. Clausen, Søren Eilers, Gunnar Restorff, Sergei Silvestrov
Publisher	Springer
Publication date	2013
Pages	157-186
ISBN (Print)	9783642394584
ISBN (Electronic)	9783642394591
DOIs	https://doi.org/10.1007/978-3-642-39459-1_8
Publication status	Published - 2013

Series	Springer Proceedings in Mathematics & Statistics
Volume	58

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Haagerup, U., & Møller, S. (2013). The Law of Large Numbers for the Free Multiplicative Convolution. In T. M. Clausen, S. Eilers, G. Restorff, & S. Silvestrov (Eds.), Operator Algebra and Dynamics: Nordforsk Network Closing Conference, Faroe Islands, May 2012 (pp. 157-186). Springer. Springer Proceedings in Mathematics & Statistics Vol. 58 https://doi.org/10.1007/978-3-642-39459-1_8

The Law of Large Numbers for the Free Multiplicative Convolution. / Haagerup, Uffe; Møller, Søren.

Operator Algebra and Dynamics: Nordforsk Network Closing Conference, Faroe Islands, May 2012. ed. / Toke M. Clausen; Søren Eilers; Gunnar Restorff; Sergei Silvestrov. Springer, 2013. p. 157-186 (Springer Proceedings in Mathematics & Statistics , Vol. 58).

Research output: Chapter in Book/Report/Conference proceeding › Article in proceedings › Research › peer-review

Haagerup, U & Møller, S 2013, The Law of Large Numbers for the Free Multiplicative Convolution. in TM Clausen, S Eilers, G Restorff & S Silvestrov (eds), Operator Algebra and Dynamics: Nordforsk Network Closing Conference, Faroe Islands, May 2012. Springer, Springer Proceedings in Mathematics & Statistics , vol. 58, pp. 157-186. https://doi.org/10.1007/978-3-642-39459-1_8

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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 Sμαβ(z) = (-z)β(1 + z)-α, 0 < z < 1.",

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AB - 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 Sμαβ(z) = (-z)β(1 + z)-α, 0 < z < 1.

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The Law of Large Numbers for the Free Multiplicative Convolution

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