The law of large numbers for the free multiplicative convolution

TL;DR

This paper investigates the free multiplicative convolution's law of large numbers, extending previous results to measures with unbounded support and analyzing the behavior of the limit measure.

Contribution

It extends Tucci's free multiplicative law of large numbers to unbounded measures and provides a more elementary proof for bounded cases.

Findings

Limit measure is not a Dirac unless original measure is Dirac.

Mean of ln x is additive under free multiplicative convolution.

Variance of ln x is generally not additive.

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 \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.