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On the Law of Addition of Random Matrices
 COMMUNICATIONS IN MATHEMATICAL PHYSICS
, 2000
"... Normalized eigenvalue counting measure of the sum of two Hermitian (or real symmetric) matrices An and Bn rotated independently with respect to each other by the random unitary (or orthogonal) Haar distributed matrix Un (i.e. An + U ∗ n BnUn) is studied in the limit of large matrix order n. Converg ..."
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Cited by 6 (1 self)
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Normalized eigenvalue counting measure of the sum of two Hermitian (or real symmetric) matrices An and Bn rotated independently with respect to each other by the random unitary (or orthogonal) Haar distributed matrix Un (i.e. An + U ∗ n BnUn) is studied in the limit of large matrix order n. Convergence in probability to a limiting nonrandom measure is established. A functional equation for the Stieltjes transform of the limiting measure in terms of limiting eigenvalue measures of An and Bn is obtained and studied.
The Arithmetic of Distributions in Free Probability Theory
, 2005
"... We give a new approach to the definition of additive and multiplicative free convolutions which is based on the theory of Nevanlinna and of Schur functions. We consider the set of probability distributions as a semigroup M equipped with the operation of free convolution and prove a Khintchine type ..."
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Cited by 6 (2 self)
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We give a new approach to the definition of additive and multiplicative free convolutions which is based on the theory of Nevanlinna and of Schur functions. We consider the set of probability distributions as a semigroup M equipped with the operation of free convolution and prove a Khintchine type theorem for factorization of elements of this semigroup. Any element of M contains either indecomposable (“prime”) factors or it belongs to a class, say I0, of distributions without indecomposable factors. In contrast to the classical convolution semigroup in the free additive and multiplicative convolution semigroups the class I0 consists of units (i.e. Dirac measures) only. Furthermore we show that the set of indecomposable elements is dense in M.
A simple approach to global regime of the random matrix theory
 Mathematical Results in Statistical Mechanics. Singapore: World Scientific
, 1999
"... Abstract. We discuss a method of the asymptotic computation of moments of the normalized eigenvalue counting measure of random matrices of large order. The method is based on the resolvent identity and on some formulas relating expectations of certain matrix functions and the expectations including ..."
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Cited by 4 (1 self)
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Abstract. We discuss a method of the asymptotic computation of moments of the normalized eigenvalue counting measure of random matrices of large order. The method is based on the resolvent identity and on some formulas relating expectations of certain matrix functions and the expectations including their derivatives or, equivalently, on some simple formulas of the perturbation theory. In the framework of this unique approach we obtain functional equations for the Stieltjes transforms of the limiting normalized eigenvalue counting measure and the bounds for the rate of convergence for the majority known random matrix ensembles. 1.
Unified large system analysis of MMSE and adaptive least squares receivers for a class of random matrix channels
 IEEE TRANS. ON INFO. THEORY
, 2005
"... We present a unified large system analysis of linear receivers for a class of random matrix channels. The technique unifies the analysis of both the minimummeansquarederror (MMSE) receiver and the adaptive leastsquares (ALS) receiver, and also uses a common approach for both random i.i.d. and ra ..."
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Cited by 2 (2 self)
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We present a unified large system analysis of linear receivers for a class of random matrix channels. The technique unifies the analysis of both the minimummeansquarederror (MMSE) receiver and the adaptive leastsquares (ALS) receiver, and also uses a common approach for both random i.i.d. and random orthogonal precoding. We derive expressions for the asymptotic signaltointerferenceplusnoise (SINR) of the MMSE receiver, and both the transient and steadystate SINR of the ALS receiver, trained using either i.i.d. data sequences or orthogonal training sequences. The results are in terms of key system parameters, and allow for arbitrary distributions of the power of each of the data streams and the eigenvalues of the channel correlation matrix. In the case of the ALS receiver, we allow a diagonal loading constant and an arbitrary data windowing function. For i.i.d. training sequences and no diagonal loading, we give a fundamental relationship between the transient/steadystate SINR of the ALS and the MMSE receivers. We demonstrate that for a particular ratio of receive to transmit dimensions and window shape, all channels which have the same MMSE SINR have an identical transient ALS SINR response. We demonstrate several applications of the results, including an optimization of information throughput with respect to training sequence length in coded block transmission.
Unified LargeSystem Analysis of MMSE and Adaptive Least Squares Receivers for a class of Random Matrix Channels
"... Abstract — We present a unified largesystem analysis of linear receivers for a class of random matrix channels. The technique unifies the analysis of both the minimummeansquarederror (MMSE) receiver and the adaptive leastsquares (ALS) receiver, and also uses a common approach for both random i. ..."
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Abstract — We present a unified largesystem analysis of linear receivers for a class of random matrix channels. The technique unifies the analysis of both the minimummeansquarederror (MMSE) receiver and the adaptive leastsquares (ALS) receiver, and also uses a common approach for both random i.i.d. and random orthogonal precoding. We derive expressions for the asymptotic signaltointerferenceplusnoise (SINR) of the MMSE receiver, and both the transient and steadystate SINR of the ALS receiver, trained using either i.i.d. data sequences or orthogonal training sequences. The results are in terms of key system parameters, and allow for arbitrary distributions of the power of each of the data streams and the eigenvalues of the channel correlation matrix. In the case of the ALS receiver, we allow a diagonal loading constant and an arbitrary data windowing function. For i.i.d. training sequences and no diagonal loading, we give a fundamental relationship between the transient/steadystate SINR of the ALS and the MMSE receivers. We demonstrate that for a particular ratio of receive to transmit dimensions and window shape, all channels which have the same MMSE SINR have an identical transient ALS SINR response. We demonstrate several applications of the results, including an optimization of information throughput with respect to training sequence length in coded block transmission.
ASYMPTOTICALLY LIBERATING SEQUENCES OF RANDOM UNITARY MATRICES
"... Abstract. A fundamental result of free probability theory due to Voiculescu and subsequently refined by many authors states that conjugation by independent Haardistributed random unitary matrices delivers asymptotic freeness. In this paper we exhibit many other systems of random unitary matrices th ..."
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Abstract. A fundamental result of free probability theory due to Voiculescu and subsequently refined by many authors states that conjugation by independent Haardistributed random unitary matrices delivers asymptotic freeness. In this paper we exhibit many other systems of random unitary matrices that, when used for conjugation, lead to freeness. We do so by first proving a general result asserting “asymptotic liberation ” under quite mild conditions, and then we explain how to specialize these general results in a striking way by exploiting Hadamard matrices. In particular, we recover and generalize results of the secondnamed author concerning the limiting distribution of singular values of a randomly chosen submatrix of a discrete Fourier transform matrix. 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.