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Local limit theorems for multiplicative free convolutions  [PDF]
Michael Anshelevich,Jiun-Chau Wang,Ping Zhong
Mathematics , 2013,
Abstract: This paper describes the quality of convergence to an infinitely divisible law relative to free multiplicative convolution. We show that convergence in distribution for products of identically distributed and infinitesimal free random variables implies superconvergence of their probability densities to the density of the limit law. Superconvergence to the marginal law of free multiplicative Brownian motion at a specified time is also studied. In the unitary case, the superconvergence to free Brownian motion and that to the Haar measure are shown to be uniform over the entire unit circle, implying further a free entropic limit theorem and a universality result for unitary free L\'{e}vy processes. Finally, the method of proofs on the positive half-line gives rise to a new multiplicative Boolean to free Bercovici-Pata bijection.
Limit theorems for free multiplicative convolutions  [PDF]
Hari Bercovici,Jiun-Chau Wang
Mathematics , 2006,
Abstract: We determine the distributional behavior for products of free random variables in a general infinitesimal triangular array. In the case of positive variables, the main theorem extends a result proved earlier for arrays with identically distributed rows. The case of unitary variables is considered as well.
Convolution and Limit Theorems for Conditionally Free Random Variables  [PDF]
Marek Bozejko,Michael Leinert,Roland Speicher
Mathematics , 1994,
Abstract: We introduce the notion of a conditionally free product and conditionally free convolution. We describe this convolution both from a combinatorial point of view, by showing its connection with the lattice of non-crossing partitions, and from an analytic point of view, by presenting the basic formula for its $R$-transform. We calculate explicitly the distributions of the conditionally free Gaussian and conditionally free Poisson distribution.
Limit theorems for additive c-free convolution  [PDF]
Jiun-Chau Wang
Mathematics , 2008,
Abstract: In this paper we find necessary and sufficient conditions for the weak convergence of c-free convolution of pairs of measures, where the measures are assumed to be infinitesimal and their support may be unbounded. These results are obtained by complex analytic methods.
On multiplicative conditionally free convolution  [PDF]
Mihai Popa,Jiun-Chau Wang
Mathematics , 2008,
Abstract: Using the combinatorics of non-crossing partitions, we construct a conditionally free analogue of the Voiculescu's S-transform. The result is applied to analytical description of conditionally free multiplicative convolution and characterization of infinite divisibility.
Limit theorems for monotonic convolution and the Chernoff product formula  [PDF]
Michael Anshelevich,John D. Williams
Mathematics , 2012,
Abstract: Bercovici and Pata showed that the correspondence between classically, freely, and Boolean infinitely divisible distributions holds on the level of limit theorems. We extend this correspondence also to distributions infinitely divisible with respect to the additive monotone convolution. Because of non-commutativity of this convolution, we use a new technique based on the Chernoff product formula. In fact, the correspondence between the Boolean and monotone limit theorems extends from probability measures to positive measures of total weight at most one. Finally, we study this correspondence for multiplicative monotone convolution, where the Bercovici-Pata bijection no longer holds.
The law of large numbers for the free multiplicative convolution  [PDF]
Uffe Haagerup,S?ren M?ller
Mathematics , 2012, DOI: 10.1007/978-3-642-39459-1_8
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.
Matricial model for the free multiplicative convolution  [PDF]
Guillaume Cébron
Mathematics , 2014,
Abstract: This paper investigates homomorphisms \`a la Bercovici-Pata between additive and multiplicative convolutions. We also consider their matricial versions which are associated with measures on the space of Hermitian matrices and on the unitary group. The previous results combined with a matricial model of Benaych-Georges and Cabanal-Duvillard allows us to define and study the large N limit of a new matricial model on the unitary group for free multiplicative L\'evy processes.
Limit Theorems in Free Probability Theory I  [PDF]
G. P. Chistyakov,F. G?tze
Mathematics , 2006,
Abstract: Based on a new analytical approach to the definition of additive free convolution on probability measures on the real line we prove free analogs of limit theorems for sums for non-identically distributed random variables in classical Probability Theory.
On regularity for measures in multiplicative free convolution semigroups  [PDF]
Ping Zhong
Mathematics , 2011,
Abstract: Given a probability measure $\mu$ on the real line, there exists a semigroup $\mu_t$ with real parameter $t>1$ which interpolates the discrete semigroup of measures $\mu_n$ obtained by iterating its free convolution. It was shown in \cite{[BB2004]} that it is impossible that $\mu_t$ has no mass in an interval whose endpoints are atoms. We extend this result to semigroups related to multiplicative free convolution. The proofs use subordination results.
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