Free Infinitely Divisible Approximations of n-Fold Free Convolutions
Based on the method of subordinating functions we prove a free analog of error bounds in classical Probability Theory for the approximation of n-fold convolutions of probability measures by infinitely divisible distributions.
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Abstract Based on the method of subordinating functions we prove a free analog of error bounds in classical Probability Theory for the approximation of n-fold convolutions of probability measures by infinitely divisible distributions. Keywords Additive free convolution • Cauchy’s transform • Free infinitely divisible probability measures • n-fold additive free convolutions of probability measures
Mathematics Subject Classification (2010): 46L53, 46L54, 60E07
1 Introduction In recent years a number of papers are investigating limit theorems for the free convolution of probability measures defined by D. Voiculescu. The key concept of this definition is the notion of freeness, which can be interpreted as a kind of independence for noncommutative random variables. As in the classical probability where the concept of independence gives rise to the classical convolution, the concept of freeness leads to a binary operation on the probability measures on the real line, the free convolution. Classical results for the convolution of probability G. Chistyakov () Fakult¨at f¨ur Mathematik, Universit¨at Bielefeld, Postfach 100131, 33501, Bielefeld, Germany Institute for Low Temperature Physics and Engineering, Kharkov, Ukraine e-mail: [email protected] F. G¨otze Fakult¨at f¨ur Mathematik, Universit¨at Bielefeld, Postfach 100131, 33501, Bielefeld, Germany e-mail: [email protected] A.N. Shiryaev et al. (eds.), Prokhorov and Contemporary Probability Theory, Springer Proceedings in Mathematics & Statistics 33, DOI 10.1007/978-3-642-33549-5 12, © Springer-Verlag Berlin Heidelberg 2013
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measures have their counterpart in this new theory, such as the law of large numbers, the central limit theorem, the L´evy-Khintchin formula and others. We refer to Voiculescu, Dykema and Nica [26] and Hiai and Petz [17] for introduction to these topics. Bercovici and Pata [10] established the distributional behavior of sums of free identically distributed random variables and described explicitly the correspondence between limits laws for free and classical additive convolution. Chistyakov and G¨otze [14] generalized the results of Bercovici and Pata to the case of free non-identically distributed random variables. They showed that the parallelism found by Bercovici and Pata holds in the general case of free non-identically distributed random variables. Using the method of subordination functions they proved the semi-circle approximation theorem (an analog of the Berry-Esseen inequality). See Kargin’s paper [18] as well. In the classical probability Doeblin [15] showed that it is possible to construct independent identically distributed random variables X1 ; X2 ; : : : such that the distribution of the centered and normalized sum bn1 .X1 C C Xnk ank / does not k converge to any nondegenerate distribution, whatever the choice of the constants an and bn and of the sequence n1 < n2 < : : : . Kolmogorov [19] initiated the study of approximations of sequences fn g1 nD1 of convolu
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