# The Law of Large Numbers for the Free Multiplicative Convolution

Uffe Haagerup, Sören Möller

Publikation: Kapitel i bog/rapport/konference-proceedingKonferencebidrag i proceedingsForskningpeer review

## Abstrakt

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 (μα, β)α, β ≥ 0 of measures on (0, ∞) for which the S-transform is given by S μ α,β (z)=(−z) β (1+z) −α , 0 < z < 1.
Originalsprog Engelsk Operator Algebra and Dynamics : Nordforsk Network Closing Conference, Faroe Islands, May 2012 Toke M. Carlsen, Søren Eilers, Gunnar Restorff, Sergei Silvestrov Springer 2013 157-186 978-3-642-39458-4 https://doi.org/10.1007/978-3-642-39459-1_8 Udgivet - 2013
Navn Springer Proceedings in Mathematics & Statistics 58
• ### A law of large numbers for the free multiplicative convolution - distributions and observations

Sören Möller (Foredragsholder)

27. feb. 2013

Aktivitet: Foredrag og mundtlige bidragForedrag og præsentationer i privat eller offentlig virksomhed

• ### A law of large numbers for the free multiplicative convolution

Sören Möller (Foredragsholder)

19. maj 2012

Aktivitet: Foredrag og mundtlige bidragForedrag og præsentationer i privat eller offentlig virksomhed

• ### A law of large numbers for the free multiplicative convolution

Sören Möller (Foredragsholder)

25. jan. 2012

Aktivitet: Foredrag og mundtlige bidragForedrag og præsentationer i privat eller offentlig virksomhed