The Semicircle Law, Free Random Variables and Entropy

Hiai, Fumio; Petz, Dénes

doi:10.1090/surv/077

Cited by 314 publications

(624 citation statements)

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“…The expectation of an element X in this algebra is defined as E (X) = E N −1 tr (X) , where E is the expectation with respect to underlying randomness and N is the dimension of the random matrix. For more details about these examples the reader may consult Hiai and Petz (2000).…”

Section: Resultsmentioning

confidence: 99%

The norm of products of free random variables

Kargin

2006

Probab. Theory Relat. Fields

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Let Xi denote free identically-distributed random variables. This paper investigates how the norm of products Πn = X1X2...Xn behaves as n approaches infinity. In addition, for positive Xi it studies the asymptotic behavior of the norm of Yn = X1 • X2 • ...• Xn, where • denotes the symmetric product of two positive operators:It is proved that if the expectation of Xi is 1, then the norm of the symmetric product Yn is between c1n 1/2 and c2n for certain constant c1 and c2. That is, the growth in the norm is at most linear.For the norm of the usual product P in, it is proved that the limit of n −1 log N orm(P in) exists and equals log p E (X * i Xi). In other words, the growth in the norm of the product is exponential and the rate equals the logarithm of the Hilbert-Schmidt norm of operator X.Finally, if π is a cyclic representation of the algebra generated by Xi, and if ξ is a cyclic vector, then n −1 log N orm(π (Πn) ξ) = log p E (X * i Xi) for all n. In other words, the growth in the length of the cyclic vector is exponential and the rate coincides with the rate in the growth of the norm of the product.These results are significantly different from analogous results for commuting random variables and generalize results for random matrices derived by Kesten and Furstenberg. IntroductionSuppose X 1 , X 2 , ... , X n are identically-distributed free random variables. These variables are infinite-dimensional linear operators but the reader may find it convenient to think of them as very large random matrices. The first question we will address in this paper is how the norm of Π n = X 1 X 2 ...X n behaves. If X i are all positive, then it is natural to look also at the symmetric product operation • defined as follows:The benefit is that unlike the usual operator product, this operation maps the set of positive variables to itself. For this operation we can ask how the norm of symmetric products Y n = X 1 • X 2 • ... • X n behaves. 1 * Courant Institute of Mathematical Sciences; 109-20 71st Road, Apt. 4A, Forest Hills NY 11375; kargin@cims.nyu.edu1 The operation • is neither commutative, nor associative. By convention we multiply starting on the right, so, for example, X 1 • X 2 • X 3 • X 4 = X 1 • (X 2 • (X 3 • X 4 )) . However,

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Section: Resultsmentioning

confidence: 99%

The norm of products of free random variables

Kargin

2006

Probab. Theory Relat. Fields

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“…x r )|| a.s. (5.15) follows, as in Lemma 7.2 in [9], from the a.s. asymptotic freeness of the (X (i) n ) i=1,...,r and sup n X (i) n < ∞ a.s.. The first point was proved by Hiai and Petz (see [10], [11]) and the second point follows from (5.14).…”

Section: The Main Theoremmentioning

confidence: 85%

Strong asymptotic freeness for Wigner and Wishart matrices

Capitaine¹,

Donati-Martin²

2007

Indiana Univ. Math. J.

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“…When more than one matrix is considered, the concept of asymptotic freeness [38] leaves us to compute the eigenvalue distribution of sums and products of random matrices.…”

Section: Historical Perspectivementioning

confidence: 99%

“…Free probability theory [38] was introduced by Voiculescu in the 1980s in order to attack some problems related to operator algebras and it can be considered as a generalization of classical probability theory to noncommutative algebras. The analogy between the concept of freeness and the independence in classical probability leaves us to work with noncommutative operators like matrices that can be considered elements in what is called a noncommutative probability space.…”

Section: Free Probability Frameworkmentioning

confidence: 99%

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Non-commutative large entries for cognitive radio applications

Masucci

Debbah

2012

J Wireless Com Network

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Cognitive radio has been proposed as a solution for the problem of underutilization of the radio spectrum. Indeed, measurements have shown that large portions of frequency bands are not efficiently assigned since large pieces of bandwidth are unused or only partially used. In the last decade, studies in different areas, such as signal processing, random matrix theory, information theory, game theory, etc., have brought us to the current state of cognitive radio research. These theoretical advancements represents a solid base for practical applications and even further developments. However, still open questions need to be answered. In this study, free probability theory, through the free deconvolution technique, is used to attack the huge problem of retrieving useful information from the network with a finite number of observations. Free deconvolution, based on the moments method, has shown to be a helpful approach to this problem. After giving the general idea of free deconvolution for known models, we show how the moments method works in the case where scalar random variables are considered. Since, in general, we have a situation where more complex systems are involved, the parameters of interest are no longer scalar random variables but random vectors and random matrices. Random matrices are non-commutative operators with respect to the matrix product and they can be considered elements of what is called noncommutative probability space.Therefore, we focus on the case where random matrices are considered. Concepts from combinatorics, such as crossing and non-crossing partitions are useful tools to express the moments of Gaussian and Vandermonde matrices, respectively. Our analysis and simulation results show that free deconvolution framework can be used for studying relevant information in cognitive radio such as power detection, users detection, etc.

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The Semicircle Law, Free Random Variables and Entropy

Cited by 314 publications

References 0 publications

The norm of products of free random variables

The norm of products of free random variables

Strong asymptotic freeness for Wigner and Wishart matrices

Non-commutative large entries for cognitive radio applications

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