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Limit theorems in free probability theory. I
 I ARXIV:MATH. OA/0602219 V
, 2006
"... 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 nonidentically distributed random variables in classical Probability Theory. ..."
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Cited by 21 (5 self)
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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 nonidentically distributed random variables in classical Probability Theory.
DECOMPOSITIONS OF THE FREE ADDITIVE CONVOLUTION
, 2006
"... We introduce and study a new type of convolution of probability measures called the orthogonal convolution, which is related to the monotone convolution. Using this convolution, we derive alternating decompositions of the free additive convolution µ�ν of compactly supported probability measures in f ..."
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Cited by 15 (5 self)
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We introduce and study a new type of convolution of probability measures called the orthogonal convolution, which is related to the monotone convolution. Using this convolution, we derive alternating decompositions of the free additive convolution µ�ν of compactly supported probability measures in free probability. These decompositions are directly related to alternating decompositions of the associated subordination functions. In particular, they allow us to compute free additive convolutions of compactly supported measures without using free cumulants or Rtransforms. In simple cases, representations of Cauchy transforms Gµ�ν z as continued fractions are obtained in a natural way. Moreover, this approach establishes a clear connection between convolutions and products associated with the main notions of independence (free, monotone and boolean) in noncommutative probability. Finally, our result leads to natural decompositions of the free product of rooted graphs.
The Lebesgue decomposition of the free additive convolution of two probability distributions
, 2006
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OPERATORS RELATED TO SUBORDINATION FOR FREE MULTIPLICATIVE CONVOLUTIONS
, 2007
"... Abstract. It has been shown by Voiculescu and Biane that the analytic subordination property holds for free additive and multiplicative convolutions. In this paper, we present an operatorial approach to subordination for free multiplicative convolutions. This study is based on the concepts of ‘freen ..."
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Abstract. It has been shown by Voiculescu and Biane that the analytic subordination property holds for free additive and multiplicative convolutions. In this paper, we present an operatorial approach to subordination for free multiplicative convolutions. This study is based on the concepts of ‘freeness with subordination’, or ‘sfree independence’, and ‘orthogonal independence’, introduced recently in the context of free additive convolutions. In particular, we introduce and study the associated multiplicative convolutions and construct related operators, called ‘subordination operators ’ and ‘subordination branches’. Using orthogonal independence, we derive decompositions of subordination branches and related decompositions of sfree and free multiplicative convolutions. The operatorial methods lead to several new types of graph products, called ‘loop products’, associated with different notions of independence (monotone, boolean, orthogonal, sfree). We also prove that the enumeration of rooted ‘alternating double return walks ’ on the loop products of graphs and on the free product of graphs gives the moments of the corresponding multiplicative convolutions. 1.
Eigenvalue distributions of sums and products of . ..
, 2005
"... This paper uses an incremental matrix expansion approach to derive asymptotic eigenvalue distributions (a.e.d.’s) of sums and products of large random matrices. We show that the result can be derived directly as a consequence of two common assumptions, and matches the results obtained from using R ..."
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This paper uses an incremental matrix expansion approach to derive asymptotic eigenvalue distributions (a.e.d.’s) of sums and products of large random matrices. We show that the result can be derived directly as a consequence of two common assumptions, and matches the results obtained from using R and Stransforms in free probability theory. We also give a direct derivation of the a.e.d. of the sum of certain random matrices which are not free. This is used to determine the asymptotic signaltointerferenceratio of a multiuser CDMA system with a minimum meansquare error linear receiver.
Limit theorems in free probability theory. II
, 2007
"... Based on an analytical approach to the definition of multiplicative free convolution on probability measures on the nonnegative line R+ and on the unit circle T we prove analogs of limit theorems for nonidentically distributed random variables in classical Probability Theory. ..."
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Based on an analytical approach to the definition of multiplicative free convolution on probability measures on the nonnegative line R+ and on the unit circle T we prove analogs of limit theorems for nonidentically distributed random variables in classical Probability Theory.
3 LOCAL DEFORMED SEMICIRCLE LAW AND COMPLETE DELOCALIZATION FOR WIGNER MATRICES WITH RANDOM POTENTIAL
, 2013
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4 Extremal Eigenvalues and Eigenvectors of Deformed Wigner Matrices
, 2014
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