D.V. Voiculescu, K.J. Dykema, and A. Nica, Free Random Variables,American Mathematical Society, CRM Monograph Series, Volume 1, Providence, Rhode Island, USA, 1992. 157  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of Z converges almost surely, as m 1, to a non random limit that is uniquely determined by the S transform SZ (z) S XX H(z) S Y (z) 7) Moreover, S XX H(z) 1 z) 1 . The proof is placed in Appendix A. Note that in addition to the results on multiplicative free convolution in [10], Theorem 1 states almost sure convergence and it is not restricted to Gaussian, diagonal, or unitary random matrices. The asymptotic limits for m 1 may serve as good estimates for the eigenvalues in the non asymptotic case. This has been verified for code division multiple access systems in ....

D. V. Voiculescu, K. J. Dykema, and A. Nica. Free Random Variables. American Mathematical Society, Providence, RI, 1992.

....years the question has drawn attention, what distributions are obtained in this limit, if one replaces the classical commutative notion of independence by some other type. Anti commutative independence, as occuring in Fermi noise [Mey93] and free independence as occurs in large random matrices [Maa92, Spe90, VDN92] have now been studied. In the former case the Gauss distribution is replaced by the measure (# 1 # 1 ) 2 and in the latter by Wigner s semicircle distribution, which on its support [ 2, 2] has the density x , # 4 x 2 2# . It is clear, however, that these two cases exhaust the ....

D.V. Voiculescu, K.J. Dykema, and A. Nica, Free random variables, CRM Monograph Series, vol. 1, American Mathematical Society, Providence, Rhode Island USA, 1992. 18

....we analyze a simpler method. Let the polynomial detector ignore the unbalanced powers of the interfering users. This means that we address the detector MwL 4 = L X i=0 w i R i : 26) For its analysis, the following lemma which follows form the de nition of free random variables in [15] will be crucial: Lemma 2 Let X and Y be two independent 2 K K random matrices with entries having nite moments. Then, with probability one, we have lim K 1 1 K tr(XY ) lim K 1 1 K tr(X) lim K 1 1 K tr(Y ) 27) The total received power and the useful signal power for the ....

.... 1 K tr(Y ) 27) The total received power and the useful signal power for the detector de ned in (26) are obviously given by P = tr R M 2 wL RA 2 N 0 tr M 2 wL R (28) S = tr diag 2 MwL RA ; 29) 2 It would be sucient to assume freeness (see [15] for de nition) respectively. Note the discrepancies to (7) and (8) due to the lack of scaling with A in (26) With the help of Lemma 2 and the normalization of A in (2) we get S P lim K 1 S P (30) the same asymptotic signal to total power ratio as found for equal powers in (16) ....

D. V. Voiculescu, K. J. Dykema, A. Nica. Free Random Variables. American Mathematical Society, Providence, RI, 1992.

....limits. The Stieltjes transforms corresponding to the asymptotic PDF of the factors in the product are available from (15) A machinery that yields the Stieltjes transform corresponding to a matrix product given the Stieltjes transforms corresponding to the factors is found in Section 3. 6 of [28] and called multiplicative free convolution. It can be applied if the factors of the matrix product form a free family in the large matrix limit. This condition is shown to hold in the proof of Theorem 1. Its key tool is the S transform S(z) which is defined in terms of Stieltjes transforms as ....

....corresponding to the factors in (20) can be shown with (21) and (22) by straightforward algebra to be S H =S (z) 1 z and S H =S (z) 1 z 1 ; 24) 11 respectively. Since the S transform of a product is the product of the factors S transforms, see Theorem 3.6. 3 in [28], we get S C= P ) z) 1 (z ) z 1 ) 25) Definitions (21) and (22) yield C= P ) s) s C= P ) s) 1 C= P ) s) C= P ) s) 1 : 26) and s 2 G 3 C= P ) s) s 1 2 G 2 C= P ) s) s ( 1) 1 ....

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D. V. Voiculescu, K. J. Dykema, and A. Nica. Free Random Variables. American Mathematical Society, Providence, RI, 1992.

....2 m 2 )w 1 : mD 2 2 mD 1 )wD m 2 = m 3 2 m 2 )w 0 (m 4 2 m 3 )w 1 : mD 3 2 mD 2 )wD . mD 1 = mD 2 2 mD 1 )w 0 (mD 3 2 mD 2 )w 1 : m 2D 2 2 m 2D 1 )wD (15) Random matrix theory [15] shows that the eigenvalue distributions of a large class of random matrices converge in probability to some deterministic limits if the sizes of the matrices grow over all bounds. The limiting statistics of H have been calculated in [10, 11] Since the eigenvalues converge to a deterministic ....

D. V. Voiculescu, K. J. Dykema, and A. Nica. Free Random Variables. American Mathematical Society, Providence, RI, 1992.

....K n 1 K n ; 1 n N; 4) and assume that all K n tend to infinity, but the ratios n remain constant. Then, the distribution of the eigenvalues of CN 4 = HNH H N = N Y n=1 M n N Y n=1 M n H (5) indeed converges to a non random limit described by the distribution function [17] F CN (x) 4 = lim Kn 1 1 KN f i : i xg : 6) where ( 4 = lim KN 1 lim K1 1 lim K0 1 ( 7) denotes the large size limit of the covariance matrix C . The distribution of the eigenvalues is conveniently represented in terms of its Stieltjes transform 1 ....

D. V. Voiculescu, K. J. Dykema, and A. Nica. Free Random Variables. American Mathematical Society, Providence, RI, 1992.

....to utilize this advantage in various ways while the physical nature of such multiple input multiple output (MIMO) channels is still not fully understood. Recently, a new model to describe the conditions for communication on such channels has been derived from random matrix and free probability [5] theory in [1] Since it reduces the description of the propagation conditions to the distribution of the eigenvalues of the channel s covariance matrices, it provides a direct link from the physical environment, which it comprehensively parameterizes by the number of signi cantly scattering ....

D. V. Voiculescu, K. J. Dykema, and A. Nica. Free Random Variables. American Mathematical Society, Providence, RI, 1992.

....G s 1 s 1 (10) where 1 (s) s. The S transforms corresponding to the factors in (9) can be shown with (10) to be S H =S (z) 1 z and S H =S (z) 1 z ; respectively. Since the S transform of a free product is the product of the factors S transforms [16], we get S C (z) 1 (z ) z ) Definitions (10) yield s 2 G 3 C s 2 G 2 C s ( 1) 1) G C = 1: Note that f C (x) f C (x= which reads in the Stieltjes domain as G C (s) GC (s= Together with (8) this yields s 2 G 3 ....

D. V. Voiculescu, K. J. Dykema, and A. Nica, Free Random Variables, American Mathematical Society, Providence, RI, 1992.

....matrices converge in probability to non random limits as the sizes of the matrices grow over all bounds. Asymptotic results for N = 1 are found in [2] For N 1, the entries within HN become dependent. Asymptotic results are reported in [3] for N = 2 making use of multiplicative free convolution [4]. The present paper will give results for arbitrary N as well as the limit N 1. Theorem 1 Let al..l the entries of all M n be zero mean independent Gaussian 1 random variables with variance 1=Kn . De ne the ratios n=Kn=KN and assume that all Kn tend to in nity, but the ratios n remain ....

....by SCN (z) N Y n=1 n z n 1 : 2) with SCN (z) z 1 z 1 CN (z) CN (s) GCN s 1 s 1; 3) 1 The assumption of Gaussianity is rather technical. 1 ) z) z and GCN (s) Z 1 x s dFCN (x) 4) The proof is based on multiplicative free convolution [4] and omitted due to space limitations. The Stieltjes transform (4) is a key tool to derive information and communication theoretic performance measures for random vector channels, see e.g. 3] It ful lls GCN (s) N Y n=1 sGCN (s) 1 n 1 n sGCN (s) 1: 5) It is interesting to ....

D. V. Voiculescu, K. J. Dykema, A. Nica. Free Random Variables. American Mathematical Society, Providence, RI, 1992.

....years the question has drawn attention, what distributions are obtained in this limit, if one replaces the classical commutative notion of independence by some other type. Anti commutative independence, as occuring in Fermi noise [Mey93] and free independence as occurs in large random matrices [Maa92, Spe90, VDN92] have now been studied. In the former case the Gauss distribution is replaced by the measure (ffi 1 ffi Gamma1 ) 2 and in the latter by Wigner s semicircle distribution, which on its support [ Gamma2; 2] has the density x 7 p 4 Gamma x 2 =2 . It is clear, however, that these two cases ....

D.V. Voiculescu, K.J. Dykema, and A. Nica, Free random variables, CRM Monograph Series, vol. 1, American Mathematical Society, Providence, Rhode Island USA, 1992.

....some time considered a good candidate for a q deformed notion of the concept of independence itself. Indeed for q = 1 the random variables X(f) and X(g) with f g are independent Gaussian random variables in the classical sense, for q = 0 they are freely independent in the sense of Voiculescu [11]. In both cases a convolution law holds for sums of functions of X(f) and X(g) For q = 1 the convolution is ordinary convolution whereas for q = 0 the convolution is found to be an interesting operation involving Gauchy transforms and inverted functions [8, 11] In this paper we show, by a simple ....

.... independent in the sense of Voiculescu [11] In both cases a convolution law holds for sums of functions of X(f) and X(g) For q = 1 the convolution is ordinary convolution whereas for q = 0 the convolution is found to be an interesting operation involving Gauchy transforms and inverted functions [8, 11]. In this paper we show, by a simple example, that for q 2 (0; 1) no such convolution law holds since the distributions of functions of X(f) and X(g) do not determine the distribution of their sum. The construction of the Fock representation for (1) is described in [1, 3] but for completeness we ....

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D.V. Voiculescu, K.J. Dykema, and A. Nica, Free random variables, CRM Monograph Series, vol. 1, American Mathematical Society, Providence, Rhode Island USA, 1992.

.... general reference, see [12, 15] for the philosophy, see [6] Perhaps the most spectacular vindication of this is the work of Connes on non commutative geometry [4] but the influence can also be seen in C algebraic K theory (see Wegge Olsen [29] for example) and Voiculescu s free probability [26]. The theory of operator spaces, can, in a similar way, be thought of as a non commutative version of Banach space theory. More precisely, Banach spaces correspond to norm closed subspaces of 1 spaces; operator spaces are norm closed subspaces of B(H) the bounded operators on a Hilbert space ....

D. Voiculescu, K. Dykema, and A. Nica, Free random variables, American Mathematical Society, 1992.

.... E X 1 2 EXACTNESS, 10:38 o clock, 18 February 1999 In [8] Voiculescu introduced the noncommutative probabilistic theory of freeness, which has turned out to be instrumental to the study of C algebras and von Neumann algebras associated to free products of groups, see for example the book [9]) His amalgamated or B valued version of freeness is as follows: if A is a unital C algebra with a unital C subalgebra B and a conditional expectation (i.e. a projection of norm 1) OE : A B and if B A A are intermediate C subalgebras, 2 I) then the family (A ) ....

D. Voiculescu, K.J. Dykema, A. Nica, Free Random Variables, CRM Monograph Series vol. 1, American Mathematical Society, 1992.

....with respect to the Lebesgue measure, with density ae(t) 2(r 2 ) Gamma1 p r 2 Gamma t 2 on [ Gammar; r] A fundamental concept used throughout the paper is the one of freeness for a family of subsets of A. For the definition and basic properties of freeness, we refer the reader to [9], Chapter 2. The free analogues of entropy and of Fisher information for random variables were introduced and studied in a series of papers of D. Voiculescu ( 4] 8] in connection to the isomorphism problem for the von Neumann algebras associated to free groups. Free analogues for some ....

....element; this is simply because, given as in (1.1) there does not exist in general a circular element a such that a a has distribution . In fact: if a is circular, then the distribution of a a can only be of the form 2(ff) Gamma1 p (ff Gamma t) tdt on [0; ff] for some ff 0 see [9], Section 5.1. A remarkable family of relatives of the circular element is provided by the so called R diagonal elements , introduced in [1] There are several possible descriptions for the fact that an element a 2 A is R diagonal. The one taken as starting point in [1] is that the 1 ....

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D.V. Voiculescu, K.J. Dykema, A. Nica. Free random variables. CRM Monograph Series, Vol. 1, American Mathematical Society, Providence, 1992. 31

....E X 1 2 DYKEMA AND RRDAM Introduction We construct three classes of examples of purely infinite, simple, unital C algebras, which may be of special interest. Some of these constructions use Voiculescu s theory freeness and his construction of reduced free products of operator algebras, see [14], see also [1] The first class of examples consists of separable, purely infinite, simple, unital C algebras which are not approximately divisible in the sense of [2] These are the first such examples, and they are constructed by applying a theorem of L. Barnett [4] concerning free ....

D. Voiculescu, K.J. Dykema, A. Nica, Free Random Variables, CRM Monograph Series vol. 1, American Mathematical Society, 1992.

....in (1.1) has normalized semicircular distribution with respect to #. The fact that an element a = a # # A has normalized semicircular distribution means by definition that #(a n ) 1 2# Z 2 2 t n p 4 t 2 dt, #n # 1. 1. 2) For the definition of freeness in (A, #) we refer to [11], Chapter 2. The definition of a circular system given above can be rephrased in a purely combinatorial way, by indicating the general formula of the joint moments of c 1 , c # 1 , c s , c # s , i.e. of the expressions #( c #(1) r 1 c #(n) rn ) n # 1, r 1 , r n # 1, ....

D.V. Voiculescu, K.J. Dykema, A. Nica. Free random variables. CRM Monograph Series, Vol. 1, American Mathematical Society, Providence, 1992. 32

.... on triangular operator algebras analogous to the algebra of analytic Toeplitz operators on the circle, and their ideal structures have been studied in [7] 13] and [21] see also [24] Moreover, the C # algebra E(H) is related to freeness in the sense of Voiculescu [31] see also the book [33]) For example, Speicher [29] has proved that if H =H 1 #H 2 then E(H) is isomorphic to the reduced amalgmated free product of C # algebras E(H 1 ) and E(H 2 ) amalgamating over B with respect to the canonical conditional expectations E(H i ) # B. The algebras E(H) are also the natural ....

D.V. Voiculescu, K.J. Dykema, A. Nica, Free Random Variables, CRM Monograph Series 1, American Mathematical Society, 1992.

....[20] and independently (in a more restricted way) by Avitzour [1] Thus A is a unital C algebra with canonical, injective, unital homomorphisms, A A, and is a state on A such that ffi = for all . It is the natural construction in Voiculescu s free probability theory (see [21]) and Voiculescu s theory has been vital to the study of these C algebras. In [12] for reduced free product C algebras A as in (1) when all the are faithful, we investigated projections in A and the related topic of positive elements in K 0 (A) The behaviour we discovered, under ....

D. Voiculescu, K.J. Dykema, A. Nica, Free Random Variables, CRM Monograph Series vol. 1, American Mathematical Society, 1992.

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D.Voiculescu, K.J.Dykema, A.Nica, Free Random Variables, CRM Monograph Series, vol. 1, American Mathematical Society, Providence, 1992.

.... Delta Delta Delta b sn ) for every n 1 and s 1 ; s n 2 f1; g: 1. 2) A fundamental concept used throughout the paper is the one of freeness for a family of subsets of A (where (A; is a probability space) For the definition and basic properties of freeness, we refer the reader to [13], Chapter 2. The concept of R diagonal element was introduced in [5] and was subsequently found to play an important role in several problems in free probability (see e.g. 6] 2] 3] Loosely speaking, the name R diagonal refers to elements which have a factorization of the form: a = up; ....

....of moments of a = up are implicitly contained in the equations describing the freeness of fu; u g from fp; p g. The main goal of this paper is to present a new approach to R diagonality, relying on freeness with amalgamation. For basic facts about freeness with amalgamation see e.g. [13] Section 3.8, or [10] the definition of the concept is also reviewed in Section 3 below) Several characterizations of R diagonality being now available, it is no longer obvious which of them is the most suitable to be used as the definition of this notion. Since we could not come to an agreement ....

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D.V. Voiculescu, K.J. Dykema, A. Nica. Free random variables. CRM Monograph Series, Vol. 1, American Mathematical Society, Providence, 1992.

....i ; i ) i 2 I, be a family of unital C algebras A i with faithful normalized traces i . To each such family one can associate the reduced free product C algebra (A; i2I (A i ; i ) where A is a unital C algebra and is a normalized faithful trace on A ( 15] see also [16]) By construction, A i is a sub C algebra of A, and extends i for each i 2 I. Elements in A of the form w = a 1 a 2 a 3 Delta Delta Delta a n ; where a j 2 A i(j) a j ) 0, and i(1) 6= i(2) i(2) 6= i(3) i(n Gamma 1) 6= i(n) are said to be reduced words of (block ) ....

D. Voiculescu, K.J. Dykema, A. Nica, Free Random Variables, CRM Monograph Series vol. 1, American Mathematical Society, 1992.

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D.V. Voiculescu, K.J. Dykema, and A. Nica, Free Random Variables,American Mathematical Society, CRM Monograph Series, Volume 1, Providence, Rhode Island, USA, 1992. 157

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D.V. Voiculescu, K.J. Dykema, and A. Nica, Free Random Variables, American Mathematical Society, CRM Monograph Series, Volume 1, Providence, Rhode Island, USA, 1992.

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D.V. Voiculescu, K.J. Dykema, and A. Nica. Free Random Variables. American Mathematical Society, CRM Monograph Series, Volume 1, Providence, Rhode Island, USA, 1992.

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D.V. Voiculescu, K.J. Dykema, and A. Nica, Free Random Variables, American Mathematical Society, CRM Monograph Series, Volume 1, Providence, Rhode Island, USA, 1992.

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D.V. Voiculescu, K.J. Dykema, and A. Nica, Free Random Variables, American Mathematical Society, CRM Monograph Series, Volume 1, Providence, Rhode Island, USA, 1992. 24

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D.V. Voiculescu, K.J. Dykema, and A. Nica, Free Random Variables, American Mathematical Society, CRM Monograph Series, Volume 1, Providence, Rhode Island, USA, 1992.

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D.V. Voiculescu, K.J. Dykema, and A. Nica, Free Random Variables, American Mathematical Society, CRM Monograph Series, Volume 1, Providence, Rhode Island, USA, 1992.

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D.V. Voiculescu, K.J. Dykema, and A. Nica. Free Random Variables. American Mathematical Society, CRM Monograph Series, Volume 1, Providence, Rhode Island, USA, 1992.

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D.V. Voiculescu, K.J. Dykema, and A. Nica, Free Random Variables, American Mathematical Society, CRM Monograph Series, Volume 1, Providence, Rhode Island, USA, 1992.

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D.V. Voiculescu, K.J. Dykema, and A. Nica, Free Random Variables, American Mathematical Society, CRM Monograph Series, Volume 1, Providence, Rhode Island, USA, 1992.

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D. V. Voiculescu, K. J. Dykema, and A. Nica, Free Random Variables, American Mathematical Society, Providence, RI, 1992.

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