In this paper, we study random matrix models which are obtained as a noncommutative polynomial in random matrix variables of two kinds: (a) a first kind which have a discrete spectrum in the limit, (b) a second kind which have a joint limiting distribution in Voiculescu's sense and are globally rotationally invariant. We assume that each monomial constituting this polynomial contains at least one variable of type (a), and show that this random matrix model has a set of eigenvalues that almost surely converges to a deterministic set of numbers that is either finite or accumulating to only zero in the large dimension limit. For this purpose we define a framework (cyclic monotone independence) for analyzing discrete spectra and develop the moment method for the eigenvalues of compact (and in particular Schatten class) operators. We give several explicit calculations of discrete eigenvalues of our model.
Let I * and I ⊞ be the classes of all classical infinitely divisible distributions and free infinitely divisible distributions, respectively, and let Λ be the Bercovici-Pata bijection between I * and I ⊞ . The class type W of symmetric distributions in I ⊞ that can be represented as free multiplicative convolutions of the Wigner distribution is studied. A characterization of this class under the condition that the mixing distribution is 2-divisible with respect to free multiplicative convolution is given. A correspondence between symmetric distributions in I ⊞ and the free counterpart under Λ of the positive distributions in I * is established. It is shown that the class type W does not include all symmetric distributions in I ⊞ and that it does not coincide with the image under Λ of the mixtures of the Gaussian distribution in I * . Similar results for free multiplicative convolutions with the symmetric arcsine measure are obtained. Several well-known and new concrete examples are presented.
The so-called Bercovici-Pata bijection maps the set of classical infinitely divisible laws to the set of free infinitely divisible laws. The purpose of this work is to study the free infinitely divisible laws corresponding to the classical Generalized Gamma Convolutions (GGC). Characterizations of their free cumulant transforms are derived as well as free integral representations with respect to the free Gamma process. A random matrix model for free GGC is built consisting of matrix random integrals with respect to a classical matrix Gamma process. Nested subclasses of free GGC are shown to converge to the free stable class of distributions.
Abstract. Let ⊞, ⊠, and ⊎ be the free additive, free multiplicative, and boolean additive convolutions, respectively. For a probability measure µ on [0, ∞) with finite second moment, we find a scaling limit of (µ ⊠N ) ⊞Nas N goes to infinity. The R-transform of its limit distribution can be represented by the Lambert's W function. From this, we prove that the limiting distribution is freely infinitely divisible as well as the lognormal distribution in classical sense. We also show a similar limit theorem by replacing the free additive convolution with the boolean convolution. IntroductionIn probability theory, limit theorems and infinite divisibility are considered in various situations. The classical references are the books by Gnedenko and Kolmogorov [11] and Petrov [17]. One of the most famous limit theorems is the Central Limit Theorem (for short CLT) that is the scaling limit of the sum of independent, identically distributed (i.i.d.) random variables. Suppose that a random variable Z has the standard normal distribution. Let {X k } ∞ k=1 be a sequence of i.i.d. random variables with finite second moment. Then a scaling (1.1)When we consider the product of i.i.d. random variables, we have also a CLT type limit theorem. The simplest case is as follows: for a sequence of i.i.d. random variables {X k } ∞ k=1 with finite second moment, we consider a scalingBy the CLT, this scaling converges to e Z in distribution as N goes to infinity.
In this paper, we firstly characterize the class of free self-decomposable distributions as a class of limiting distributions of suitably normalized partial sums of free independent random variables. Secondly, we introduce nested classes between the class of free self-decomposable distributions and that of free stable distributions, characterize them and show that the limit of the nested classes coincides with the closure of the class of free stable distributions. All results here are analogues of the results known in classical probability theory.
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