The Law of Large Numbers for the Free Multiplicative Convolution

Publikation: Bidrag til bog/antologi/rapportKonferencebidrag i proceedingsForskningfagfællebedømt

  • Uffe Haagerup
  • Søren Møller
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.
OriginalsprogEngelsk
TitelOperator Algebra and Dynamics : Nordforsk Network Closing Conference, Faroe Islands, May 2012
RedaktørerToke M. Clausen, Søren Eilers, Gunnar Restorff, Sergei Silvestrov
ForlagSpringer
Publikationsdato2013
Sider157-186
ISBN (Trykt)9783642394584
ISBN (Elektronisk)9783642394591
DOI
StatusUdgivet - 2013
NavnSpringer Proceedings in Mathematics & Statistics
Vol/bind58

ID: 97158884