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1.
J. Armando Domínguez-Molina Alfonso Rocha-Arteaga 《Periodica Mathematica Hungarica》2012,64(2):145-160
The Bercovici-Pata bijection maps the set of classical infinitely divisible distributions to the set of free infinitely divisible distributions. The purpose of this work is to study random matrix models for free infinitely divisible distributions under this bijection. First, we find a specific form of the polar decomposition for the Lévy measures of the random matrix models considered in Benaych-Georges [6] who introduced the models through their laws. Second, random matrix models for free infinitely divisible distributions are built consisting of infinitely divisible matrix stochastic integrals whenever their corresponding classical infinitely divisible distributions admit stochastic integral representations. These random matrix models are realizations of random matrices given by stochastic integrals with respect to matrix-valued Lévy processes. Examples of these random matrix models for several classes of free infinitely divisible distributions are given. In particular, it is shown that any free selfdecomposable infinitely divisible distribution has a random matrix model of Ornstein-Uhlenbeck type ?? 0 ?? e ?1 d?? t d , d ?? 1, where ?? t d is a d × d matrix-valued Lévy process satisfying an I log condition. 相似文献
2.
Wei Liu 《应用数学学报(英文版)》2011,27(4):713-720
In this paper,based on the recent results of Gozlan and Léonard we give optimal transportationentropy inequalities for several usual distributions on R,such as Bernoulli,Binomial,Poisson,Gamma distributions and infinitely divisible distributions with positive or negative jumps. 相似文献
3.
Kumi Yasuda 《Journal of Theoretical Probability》2000,13(3):635-657
Our aim in this paper is to characterize some classes of infinitely divisible distributions on locally compact abelian groups. Firstly infinitely divisible distributions with no idempotent factor on locally compact abelian groups are characterized by means of limit distributions of sums of independent random variables. We introduce semi-selfdecomposable distributions on topological fields, and in case of totally disconnected fields we give a limit theorem for them. We also give a characterization of semistable laws on p-adic field and show that semistable processes are constructed as scaling limits of sums of i.i.d. 相似文献
4.
Florent Benaych-Georges 《Probability Theory and Related Fields》2007,139(1-2):143-189
In a previous paper (Benaych-Georges in Related Convolution 2006), we defined the rectangular free convolution ?λ. Here, we investigate the related notion of infinite divisibility, which happens to be closely related the classical infinite divisibility: there exists a bijection between the set of classical symmetric infinitely divisible distributions and the set of ?λ -infinitely divisible distributions, which preserves limit theorems. We give an interpretation of this correspondence in terms of random matrices: we construct distributions on sets of complex rectangular matrices which give rise to random matrices with singular laws going from the symmetric classical infinitely divisible distributions to their ?λ-infinitely divisible correspondents when the dimensions go from one to infinity in a ratio λ. 相似文献
5.
Shigeru Mase 《Annals of the Institute of Statistical Mathematics》1977,29(1):275-286
Summary The problem of characterizing the infinitely divisible characteristic functions which have only infinitely divisible factors
is considered. Under the assumption that both the absolutely continuous and the singular (or the discrete) components exist
in Poisson spectral measures, several necessary conditions for this problem are obtained. These conditions admit partial converses
and new examples of infinitely divisible characteristic functions which have only infinitely divisible factors are given. 相似文献
6.
F.W. Steutel 《Stochastic Processes and their Applications》1973,1(2):125-143
In this paper, a survey is given of some recent developments in infinite divisibility. There are three main topics: (i) the occurrence of infinitely divisible distributions in applied stochastic processes such as queueing processes and birth-death processes, (ii) the construction of infinitely divisible distributions, mainly by mixing, and (iii) conditions for infinite divisibility in terms of distribution functions and densities. 相似文献
7.
Serban T. Belinschi Marek Bo?ejko Franz Lehner Roland Speicher 《Advances in Mathematics》2011,(4):3677
We prove that the classical normal distribution is infinitely divisible with respect to the free additive convolution. We study the Voiculescu transform first by giving a survey of its combinatorial implications and then analytically, including a proof of free infinite divisibility. In fact we prove that a sub-family of Askey–Wimp–Kerov distributions are freely infinitely divisible, of which the normal distribution is a special case. At the time of this writing this is only the third example known to us of a nontrivial distribution that is infinitely divisible with respect to both classical and free convolution, the others being the Cauchy distribution and the free 1/2-stable distribution. 相似文献
8.
Shigeru Mase 《Annals of the Institute of Statistical Mathematics》1975,27(1):289-298
We shall consider the problem of characterizing infinitely divisible characteristic functions which have only infinitely divisible factors. Infinitely divisible characteristic functions treated in this paper are those which have absolutely continuous Poisson spectral measures and have no Gaussian component in their Lévy canonical representations. It will be shown that Ostrovskii's sufficient condition is also necessary in this case. 相似文献
9.
We investigate infinitely divisible distributions on cones in Fréchet spaces. We show that every infinitely divisible distribution
concentrated on a normal cone has the regular Lévy–Khintchine representation if and only if the cone is regular. These results
are relevant to the study of multidimensional subordination.
Research of J. Rosiński supported by a grant from the National Science Foundation. 相似文献
10.
Hari Bercovici Vittorino Pata 《Proceedings of the American Mathematical Society》2000,128(4):1011-1015
Hincin characterized the class of infinitely divisible distributions on the line as the class of all distributional limits of sums of infinitesimal independent random variables. We show that an analogue of this characterization is true in the addition theory of free random variables introduced by Voiculescu.
11.
Ramachandran (1969) [9, Theorem 8] has shown that for any univariate infinitely divisible distribution and any positive real number α, an absolute moment of order α relative to the distribution exists (as a finite number) if and only if this is so for a certain truncated version of the corresponding Lévy measure. A generalized version of this result in the case of multivariate infinitely divisible distributions, involving the concept of g-moments, was given by Sato (1999) [6, Theorem 25.3]. We extend Ramachandran’s theorem to the multivariate case, keeping in mind the immediate requirements under appropriate assumptions of cumulant studies of the distributions referred to; the format of Sato’s theorem just referred to obviously varies from ours and seems to have a different agenda. Also, appealing to a further criterion based on the Lévy measure, we identify in a certain class of multivariate infinitely divisible distributions the distributions that are self-decomposable; this throws new light on structural aspects of certain multivariate distributions such as the multivariate generalized hyperbolic distributions studied by Barndorff-Nielsen (1977) [12] and others. Various points relevant to the study are also addressed through specific examples. 相似文献
12.
本文考虑具有有限矩的1维无穷可分分布的正交多项式的母函数,通过“一步提升”原则得到的重正化核的显式表示,建立重正化核运算与Poisson随机积分之间的关系. 相似文献
13.
B. Grigelionis 《Lithuanian Mathematical Journal》2008,48(3):294-315
We define and characterize Thorin classes {ie294-01}, of infinitely divisible distributions on R
+. We investigate Poisson, Karlin, and Bessel transforms of Thorin classes and also consider extended Thorin classes {ie294-02}.
Canonical representation and self-decomposability properties of Thorin subordinated Gaussian Lévy processes are discussed.
As an example, a subordinated Cauchy process is considered in detail. 相似文献
14.
Víctor Pérez-Abreu 《Periodica Mathematica Hungarica》2006,52(1):47-65
Summary The Gaussian unitary ensemble is a random matrix model (RMM) for the Wigner law. While random matrices in this model are infinitely
divisible, the Wigner law is infinitely divisible not in the classical but in the free sense. We prove that any variance mixture
of Gaussian distributions -- whether infinitely divisible or not in the classical sense -- admits a RMM of non Gaussian infinitely
divisible random matrices. More generally, it is shown that any mixture of the Wigner law admits a RMM. A key role is played
by the fact that the Gaussian distribution is the mixture of Wigner law with the <InlineEquation ID=IE"1"><EquationSource
Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>2$-gamma
distribution. 相似文献
15.
Processes with independent increments are proven to be the unique solutions of duality formulas. This result is based on a simple characterization of infinitely divisible random vectors by a functional equation in which a difference operator appears. This operator is constructed by a variational method and compared to approaches involving chaos decompositions. We also obtain a related characterization of infinitely divisible random measures. 相似文献
16.
Henry A. Krieger 《Israel Journal of Mathematics》1984,47(1):32-64
In his 1972Periodica Mathematica Hungarica paper, H. Bergström stated a theorem on convergence in distribution for triangular arrays of dependent random variables satisfying, a ?-mixing condition. A gap in his proof of this theorem is explained and a more general version is proved under weakened hypotheses. The method used consists of comparisons between the given array and associated arrays which are parameterized by a truncation variable. In addition to the main theorem, this method yields a proof of equality of limiting finite-dimensional distributions for processes generated by the given associated arrays as well as the result that if a limit distribution for the centered row sums does exist, it must be infinitely divisible. Several corollaries to the main theorem specialize this result for convergence to distributions within certain subclasses of the infinitely divisible laws. 相似文献
17.
Peter F. Thall 《Stochastic Processes and their Applications》1984,16(2):205-210
If a regular infinitely divisible (Poisson cluster) point process is Coxian (doubly stochastic Poisson, subordinated Poisson), then the number of points per cluster either takes on each positive integer value with positive probability or is identically equal to one. In particular, a Gauss-Poisson process can not be Coxian. 相似文献
18.
Shigeru Mase 《Journal of multivariate analysis》1975,5(4):415-424
We shall consider the decomposition problem of multivariate infinitely divisible characteristic functions which have no Gaussian component and have absolutely continuous Poisson spectral measures. Under the condition that A = {x;f(x) > 0} is open, where f is the density of spectral measure, we shall show that a known sufficient condition for the membership of the class I0m (i.e., infinitely divisible characteristic functions having only infinitely divisible factors) is also necessary. 相似文献
19.
A. Yu. Zaitsev 《Journal of Mathematical Sciences》2005,128(1):2556-2563
It is shown that the results obtained earlier for the rate of approximation of convolutions of probability distributions by the accompanying infinitely divisible laws may be interpreted as estimates of the rate of approximation of a sample by a Poisson point process. The most interesting results are obtained for a scheme of rare events. Bibliography: 53 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 298, 2003, pp. 111–125. 相似文献
20.
A. F. Taraskin 《Journal of Mathematical Sciences》1997,84(3):1162-1174
Some results concerning the asymptotic behavior of the log-likelihood ratio (LLR) and also of certain other random variables
closely associated with the likelihood ratio are presented. More specifically, in the present paper we formulate the conditions
for the stable convergence in distribution of the LLR for two sequences of the probability measures to a mixture of infinitely
divisible distributions with finite variance. Moreover, the notion of a locally asymptotically mixed infinitely divisible
(LAMID) sequence of parametric families of the probability measures is introduced, and it is shown that when a certain kind
of differentiability-type regularity condition is satisfied, the given sequence of families satisfies the LAMID condition.
These results extend and supplement the previous investigations of the author concerning non-Gaussian asymptotic distributions
in statistics.
Proceedings of the XVII Seminar on Stability Problems for Stochastic Models, Kazan, Russia, 1995, Part III. 相似文献