Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The law of large numbers for the free multiplicative convolution   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 \ln x is additive with respect to the free multiplicative
convolution while the variance of \ln x is not in general additive. Furthermore
we study the two parameter family (\mu_{\alpha,\beta})_{\alpha,\beta \ge 0} of
measures on (0,\infty) for which the S-transform is given by
S_{\mu_{\alpha,\beta}}(z) = (-z)^\beta (1+z)^{-\alpha}, 0 < z < 1.

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The categories of this paper are: supr-con
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The categories of this paper are: astro-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: comp-gas
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The categories of this paper are: econ
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The categories of this paper are: acc-phys
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The categories of this paper are: hep-ph
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The categories of this paper are: cs
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The categories of this paper are: hep-lat
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The categories of this paper are: nucl-th
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The categories of this paper are: dg-ga
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The categories of this paper are: q-alg
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The categories of this paper are: q-bio
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The categories of this paper are: mtrl-th
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The categories of this paper are: adap-org
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The categories of this paper are: hep-ex