The Law of Large Numbers for the Free Multiplicative Convolution
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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. red. / Toke M. Clausen; Søren Eilers; Gunnar Restorff; Sergei Silvestrov. Springer, 2013. s. 157-186 (Springer Proceedings in Mathematics & Statistics , Bind 58).Publikation: Bidrag til bog/antologi/rapport › Konferencebidrag i proceedings › Forskning › fagfællebedømt
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TY - GEN
T1 - The Law of Large Numbers for the Free Multiplicative Convolution
AU - Haagerup, Uffe
AU - Møller, Søren
PY - 2013
Y1 - 2013
N2 - 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.
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 (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.
U2 - 10.1007/978-3-642-39459-1_8
DO - 10.1007/978-3-642-39459-1_8
M3 - Article in proceedings
SN - 9783642394584
T3 - Springer Proceedings in Mathematics & Statistics
SP - 157
EP - 186
BT - Operator Algebra and Dynamics
A2 - Clausen, Toke M.
A2 - Eilers, Søren
A2 - Restorff, Gunnar
A2 - Silvestrov, Sergei
PB - Springer
ER -
ID: 97158884