D. Jonsson. Some limit theorems for the eigenvalues of a sample covariance matrix. J. Multivariate Analysis, 12:1--38, 1982.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of the 2 Order linear MSMU receiver we will utilise so called large system analysis. We define a large system by taking the CDMA system parameters N and K to infinity but keeping their ratio (# = K N) held fixed. By doing this, a range of useful mathematical results can be applied (see [25, 26]) In related work, large system analysis has previously been used to determine the signal to interference plus noise ratio (SIR) of linear CDMA receivers including the conventional SUMF, the decorrelator and the direct LMMSE receiver [27, 28] As well, there also exist many other useful ....

....the following Lemmas. Lemma 1 If N with # = K N held f ixed, the random variable s s k converges in probability to the deterministic moment i (#) given as follows i (#) dG(#) where G(#) is the limiting empirical distribution function of the eigenvalues of S k S k (see [25, 28, 36]) The i th moment of the limiting empirical distribution function, # i (#) can be calculated recursively as # i (#) i 1 [ 2i 1) 1 #)# i 1 (#) i 2) 1 # i 2 (#) 15) where 0 (#) 1 and 1 (#) #. Proof See [3, 36] Lemma 2 The direct form of the matrix recursion of M k ....

D. Jonsson, "Some limit theorems for the eigenvalues of a sample covariance matrix," Journal of Multivariate Analysis, vol. 12, pp. 1--38, 1982.

.... 1 Let S be as in Theorem 1 and let the K K diagonal matrix A be such that AA = PI, then, for any l; k 2 N, R ) kk converges almost surely, as N;K 1 with N constant to the non random quantity a:s: m R : For AA = PI a closed form expression of the eigenvalue moments is in [8]. Theorems 1 and 2 and Corollary 1 can be extended to the case of Gaussian distribution for the elements of S and can be rewritten with hypothesis A.2 instead of hypothesis A.1 using the following lemma: Lemma 1 Let S be an N K matrix with statistically independent and identically Gaussian ....

D. Jonsson, \Some limit theorems for the eigenvalues of a sample covariance matrix," J. Multivariate Anal., vol. 12, pp. 1-38, 1982.

....the eigenvalues of the sample covariance matrix W. In the i.i.d. Rayleigh fading case W is also called a Wishart matrix. Wishart matrices have been studied since the 1920 s and a considerable amount is known about them. For general W matrices a wide range of limiting results are known [31] 22] [32], 33] 34] as M or N or both tend to infinity. In the particular case of Wishart matrices many exact results are also available [31] 35] There is not a great deal of information about intermediate results (neither limiting nor Wishart) but we are helped by the remarkable accuracy of some ....

D. Jonsson, "Some limit theorems for the eigenvalues of a sample covariance matrix," Journal of Multivariate Analysis, vol. 12, pp. 1--38, 1982. 37

....properties of the c.d.f. Fn(x) we will consider the dimensions m = re(N) and n = n(N) of matrix X to be functions of a variable N. We will consider asymptotics such that in the limit as N cx , we have re(N) m(N) and n(N) Q, where Q 1. Under these assumptions, it can be shown that [8] the empirical c.d.f. Fn(x) converges in probability to a continuous distribution function Fc2 (x) for every x, whose 1.5 0.5 0.5 1.5 1 0.5 0 0.5 I 1.5 Figure 1: Wigner s semi circle law: Distribution of the eigenvalues of v v where V is a random matrix takes the shape of a semi circle. ....

D. Jonsson. Some limit theorems for the eigenvalues of a sample covariance matrix. Journal of Multivariate Analysis, 12:1-38, 1982.

....x] 2#x dx, a(#) # b(#) 1, b(#) 10) where a(#) 1 # #) and b(#) 1 # #) Note that # i (#) is the i th moment of the limiting distribution function. We shall only give a brief discussion of the proof of Lemma 1, however, for more details the reader is referred to [21, 23, 24]. Now, let G (#) be the empirical distribution function of the eigenvalues of the N matrix S k S k . It is well known that with probability one, G converges in distribution to G as N with # = K N fixed (see [21] for example) The idea of the proof is that s s k is close to ....

....of Lemma 1, however, for more details the reader is referred to [21, 23, 24] Now, let G (#) be the empirical distribution function of the eigenvalues of the N matrix S k S k . It is well known that with probability one, G converges in distribution to G as N with # = K N fixed (see [21] for example) The idea of the proof is that s s k is close to trace(S k S dG (#) in a large system and that this latter quantity converges to dG(#) The result can be proved following identical lines to the proof of Lemma 4.3 in [23] see also Lemma 1 in [24] Corollary 1 ....

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D. Jonsson, "Some limit theorems for the eigenvalues of a sample covariance matrix," Journal of Multivariate Analysis, vol. 12, pp. 1--38, 1982.

....matrix theory is currently an attractive area because of its rich content of physics, statistics, and mathematics. Its motivations and applications can be found in several important areas: condensed matter physics, statistical mechanics and chaotic systems [14,26,27] multivariate statistics [8,10,13,15,19,28], the Riemann hypothesis [16,17] 2D potential theory and orthogonal polynomials [1,2,18] and numerical linear algebra [3 6,23] From the physics point of view, the dominance of spectral analysis for random matrices is mostly due to the significant physics meaning of eigenvalues, i.e. the ....

.... Ensemble W(N) and more generally, W(N,M) from Gaussian Ensemble G , M) see [5,15] namely the ensemble of N by N random positive matrices M AA , with A #Gp =Gp , N) The major advantage of such an approach is that one can immediately benefit from many works on Wishart Ensembles (see [8,10,13,12,15,19], for examples) in the literature of multivariate statistics. The pities are, if singular values (# k s) could indeed speak out for themselves, the merciless defiance of their independent civil rights in the kingdom of linear algebra and linear transforms. In the singular value decomposition ....

D. Jonsson, Some limit theorems for the eigenvalues of a sample covariance matrix, J. Multivariate Anal. 12 (1982) 1--38.

....ij = E # and # are the eigenvalues of R. Under the assumptions we made on S and A, the sequence of the empirical eigenvalue distribution of R converges almost surely, as = #, to a deterministic distribution [9] For AA a closed form expression of the eigenvalue moments is in [10]. The weighting (4) was proposed first in [7] as the one minimizing the ratio between the total useful power and the total noise and interference power for AA PI. The same expression can be obtained without any constraint on A minimizing the quantity: MSE k = lim trace (C # ) 5) ....

D. Jonsson, "Some limit theorems for the eigenvalues of a sample covariance matrix," J. Multivariate Anal., vol. 12, pp. 1--38, 1982.

....eigenvalue distribution of converges to the same limit as and . Proof: For the case of long sequences, it follows from the independence and circular symmetry of the signature sequence entries that the entries of are i.i.d. random variables with variance . By existing random matrix results [47] [49], it is known that the all the moments of , i.e. converge as . Moreover, it is known that the limiting moments are those of a distribution with bounded support. Therefore, the convergence of the moments implies the convergence of the expected empirical eigenvalue distribution to a limit . Let ....

....expected empirical eigenvalue distribution of . This convergence holds since is a bounded continuous function and converges to (in the weak topology) Since is also the limiting distribution for the case with long sequences, it can be given as the solution to the fixed point equation [47] [49] or, equivalently, 23) We observe that this is exactly the same as the in Theorem 2 (with ) for the long sequence case. From the facts that is independent of and that the entries of are uncorrelated, zero mean, variance , we get (24) as . Our goal now is to show from (21) that in fact ....

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D. Jonsson, "Some limit theorems for the eigenvalues of a sample covariance matrix," J. Multivariate Anal., vol. 12, pp. 1--38, 1982.

....extend to complex valued spreading sequences. The asymptotically optimum weights in (29) do not depend on the eigenvectors of the correlation matrix . Moreover, they are almost surely independent in total of the correlation matrix due to the following result from random matrix theory. Theorem 1 [14] : Let be an matrix whose entries are i.i.d. random variables with zero mean and variance , and let be the eigenvalues of . Moreover, let and ,but . Then, the moments of the eigenvalues (31) converge almost surely to the nonrandom limits (32) Note that, in context of CDMA, can be idenified ....

....optimum weight as well as the asymptotic signal to total power ratio. For this purpose, it will be helpful to define (33) 35) found at the bottom of the page. In terms of and the asymptotic weights and the signal to total power ratio admit the expressions (36) 37) This theorem was shown in [14] for Gaussian random entries only. The present form of Theorem 1 which is not restricted to the distribution of the entries in SSS follows from more general results in [3] 15] 28) 33) 34) 35) TABLE I OPTIMUM WEIGHT VECTORS FOR L =1; 2; 3; 4 Any scalar ....

D. Jonsson, "Some limit theorems for the eigenvalues of a sample covariance matrix," J. Multivariate Anal., vol. 12, pp. 1--38, 1982.

....be the empirical cumulative distribution function (c.d.f. of the eigenvalues # ni 1#i#n , where U(x) is the unit step function. We will consider asymptotics such that in the limit as N we have m(N) ##, and m(N) Q, where Q 1. Under these assumptions, it can be shown that [11] the empirical c.d.f. F n (x) converges in probability to a continuous distribution function FQ (x) for every x, whose probability density function (p.d.f. is given by f Q (x) Q # (x #min ) #max x) 2#x # min x # max 0 otherwise, 1) where # min = 1 1 # Q) and # max = 1 ....

D. Jonsson. Some limit theorems for the eigenvalues of a sample covariance matrix. Journal of Multivariate Analysis, 12:1--38, 1982.

....channel estimation error variance. In contrast to the case of multiuser interference suppression without SIC, it is now necessary to regard the correlations of the channel estimation errors in e[#p] belonging to previously processed users in order to solve ## . Applying the findings given in [2, 3], it can be shown that it is the solution to n # 0 #dF (#) 1 ### (# #)# h G#(1 ### h ) ## . where F (#) denotes the cdf of the squared absolute values of the path weigh estimates plus # e , and G# (z) is solved from G#(z) z o 2a 2 4 o ....

D. Jonsson, "Some limit theorems for the eigenvalues of a sample covariance matrix," Journal of Multivariate Analysis, vol. 12, pp. 1--38, 1982.

.... C T ( 2 N 0 )C 1 T C: The asymptotically optimum weights in (19) do not depend on the eigenvectors of the correlation matrix R. Moreover, they are even almost surely totally independent of the correlation matrix due to the following result from random matrix theory: Theorem 1 [14] Let S be an N K matrix whose entries are independent identically distributed random variables with zero mean and variance 1=N , and let k be the eigenvalues of S H S. Moreover, let K 1 and N 1, but 0 4 = K=N 1. Then, the moments of the eigenvalues 1 K K X k=1 m k = 1 ....

D. Jonsson. Some limit theorems for the eigenvalues of a sample covariance matrix. J. Multivar. Analysis, 12:1-38, 1982.

....converges to the same limit as N; K 1 and K=N ff. Proof: For the case of long sequences, it follows from the independence and circular symmetry of the signature sequence entries that the entries of S are iid random variables with variance 1= N ) By existing random matrix results [47] 48] [49], it is known that the all the moments of F S S H , i.e. N ( S S H ] r ) converge as N 1. Moreover, it is known that the limiting moments are those of a distribution with bounded support. Therefore, the convergence of the moments implies the convergence of the expected empirical ....

....S H . This convergence holds since f( 1= v) is a bounded continuous function and F S S H converges to F (in the weak topology) Since F is also the limiting distribution for the case with long sequences, it can be given as the solution fi to the fixed point equation [47] 48] [49]: fi = v ff 1 1 fi Gamma1 March 6, 2000 DRAFT IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. XX, NO. Y, MONTH 2000 or equivalently ff (1 fi ) ff Gamma 1 vfi : 23) We observe that this fi is exactly the same as the fi c in Theorem 2 (with L = 1) for the long ....

[Article contains additional citation context not shown here]

D. Jonsson, "Some limit theorems for the eigenvalues of a sample covariance matrix," Journal of Multivariate Analysis, vol. 12, pp. 1--38, 1982.

.... analysis of the eigenvalue distribution as the size of the multiuser system goes to infinity has been presented in [25] 27] Here, we demonstrate a method for deriving the exact expressions for the moments of the eigenvalues, which is also known as the moments of the correlation matrix [28]. The moments are found to be polynomials of the processing gain, the number of active users and the received signal energies. The computational complexity of calculating the weights increases only linearly with the number of users. Hence, it can be implemented on line given a moderate number of ....

....9 Here, A nk are independent random variables selected from a scaled q ary PSK 2 constellation with equal probability. Based on the statistical properties of the code set given in (22) and (23) only terms containing all complex conjugate pairs and or q powers of the variables A nk are relevant [28]. It is therefore possible to obtain M r through evaluation of the expectation over all combinations of indices. This involves a grouping of the indices into equivalence classes. Details of this grouping and evaluation can be found in [30] As the expectation is taken over all code sets, M r only ....

D. Jonsson, "Some limit theorems for the eigenvalues of a sample covariance matrix," Journal of Multivariate Analysis, Dec. 1982.

....matrix theory is currently an attractive area because of its rich content of physics, statistics, and mathematics. Its motivations and applications can be found in several important areas: condensed matter physics, statistical mechanics and chaotic systems [14, 26, 27] multivariate statistics [8, 10, 13, 15, 19, 28], the Riemann hypothesis [16, 17] 2 D potential theory and orthogonal polynomials [1, 2, 18] and numerical linear algebra [3, 4, 5, 6, 23] From the physics point of view, the dominance of spectral analysis for random matrices is mostly due to the signi cant physics meaning of eigenvalues, ....

.... generally, W (N; M) from Gaussian Ensemble G(N;M ) see Edelman [5] and Muirhead [15] namely the ensemble of N by N random positive matrices M = AA T , with A 2 G(N) G(N;N ) The major advantage of such a approach is that one can immediately bene t from many works on Wishart Ensembles (see [8, 10, 13, 12, 15, 19], for examples) in the literature of multivariate statistics. The pities are, if singular values ( k s) could indeed speak for themselves, the merciless de ance of their in2 dependent civil rights in the kingdom of linear algebra and linear transforms. In the singular value decomposition ....

D. Jonsson. Some limit theorems for the eigenvalues of a sample covariance matrix. J. Multivariate Anal., 12:1-38, 1982.

.... analysis of the eigenvalue distribution as the size of the multiuser system goes to infinity has been presented in [25] and [27] Here, we demonstrate a method for deriving the exact expressions for the moments of the eigenvalues, which is also known as the moments of the correlation matrix [28]. The moments are found to be polynomials of the processing gain, the number of active users, and the received signal energies. The computational complexity of calculating the weights increases only linearly with the number of users. Hence, it can be implemented online given a moderate number of ....

....E (27) E (28) Here, are independent random variables selected from a scaled ary PSK 2 constellation with equal probability. Based on the statistical properties of the code set given in (22) and (23) only terms containing all complex conjugate pairs and or powers of the variables are relevant [28]. It is therefore possible to obtain through evaluation of the expectation over all combinations of indexes. This involves a grouping of the indexes into equivalence classes. Details of this grouping and evaluation can be found in [30] As the expectation is taken over all code sets, only depends ....

D. Jonsson, "Some limit theorems for the eigenvalues of a sample covariance matrix," J. Multivariate Anal., vol. 12, pp. 1--38, Mar. 1982.

.... modification of the limit theorem in [11] Let Fn denote the empirical distribution function of the eigenvalues of Mn (that is, Fn (x) 1 n) number of eigenvalues of Mn # x) where we may as well assume x # 0) If Var(v 11 ) 1 (no other assumption on the moments) then it is known ( 4] [5], 12] 13] that, for every x # 0, 1.2) Fn (x) a.s. # F y ( x )asn##, where F y is a continuous, nonrandom probability distribution function depending only on y, having a density with support on [ 1 # y) 2 , 1 # y) 2 ] and for y 1, F y places mass 1 1 y at 0. Moreover, if ....

Jonsson, D. (1982). Some limit theorems for the eigenvalues of a sample covariance matrix. J. Multivariate Anal. 12 1-38.

....independent columns, and T is n n Hermitian, independent of X. Several papers have dealt with the behavior of the eigenvalues of this matrix when N and n are both large but having the same order of magnitude (Marcenko and Pastur [4] Grenander and Silverstein [2] Wachter [6] Jonsson [3], Yin and Krishnaiah [8] Yin [7] The behavior is expressed in terms of limit theorems, as N ##, while n = n(N)withn N # c 0, on the empirical distribution function (e.d.f. F XTX # of the eigenvalues , that is, F XTX # (x)is the proportion of eigenvalues of XTX # # x) the conclusion ....

Jonsson, D. (1982). Some limit theorems for the eigenvalues of a sample covariance matrix. J. Multivariate Anal. 12 1-38.

....being the analyticity of F . Under condition 2) it is shown in [8] that F 0 has moments of all order satisfying Carleman s su#ciency condition, and are explicitly expressed. From the moments, F 0 has been derived in two cases: when Tn = I n (the n n identity matrix) that is, when H =1 [1,#) [1]) and when Tn = 1 m Y nY T n ) 1 ,whereY n is nm with n m,n m # y # (0, 1) and contains i.i.d. N(0, 1) entries ( 3] In both cases F 0 has a continuous density on (0, #) The moments of F 0 can also yield some qualitative behavior ( 5] namely: i) c and F 0 uniquely determine ....

....the resulting polynomial being at most 4) If no way of solving (1.6) is apparent, then a simple numerical scheme can be applied. It is remarked here that, even if F 0 = 0, it is still possible for f not to exist at 0. For example, for the case H =1 [1,#) and c =1,f(x) 1 (0,4) 1 2# q 4 x x ([1]) The proof of Theorem 1.1 relies on a result concerning the existence of a derivative of a p.d.f. whenever the imaginary part of its Stieltjes transform converges. It will be stated and proven in the next section, along with a result needed to establish the continuity of m 0 . The third section ....

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Jonsson, D. (1986). Some limit theorems for the eigenvalues of a sample covariance matrix. J. Multivariate Anal. 12 1-38.

....these moments uniquely determine F . 1 3 The following theorem pertains to the case when Tm is a multiple of the identity matrix. Theorem 2. When Tm = # 2 I m , F is known, having an algebraic density on the positive reals with support [# 2 (1 # y) 2 ,# 2 (1 # y) 2 ] 7] [8], 10] The largest eigenvalue of Mm converges almost surely [respectively in probability] to # 2 (1 # y) 2 as m# # if and only if EY 11 =0andE Y 11 4 # [respectively x 4 P Y 11 # x #0asx# #] 1] 6] 13] 16] Moreover, if Y 11 is standardized Gaussian, the smallest ....

D. Jonsson, "Some Limit Theorems for the Eigenvalues of a Sample Covariance Matrix," Journal of Multivariate Analysis, vol. 12, no. 1, pp. 1-38, March 1982.

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D. Jonsson. Some limit theorems for the eigenvalues of a sample covariance matrix. J. Multivariate Analysis, 12:1--38, 1982.

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D. Jonsson. Some limit theorems for the eigenvalues of a sample covariance matrix. Journal of Multivariate Analysis, 12:1--38, 1982.

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D. Jonsson, "Some limit theorems for the eigenvalues of a sample covariance matrix," J. Multivariate Anal., vol. 12, pp. 1--38, 1982.

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D. Jonsson, "Some limit theorems for the eigenvalues of a sample covariance matrix," Journal of Multivariate Analysis, vol. 12, pp. 1--38, 1982.

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D. Jonsson, "Some limit theorems for the eigenvalues of a sample covariance matrix," J. Multivariate Anal., vol. 12, pp. 1--38, 1982.

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