Let \{$X_{ij}$\}, $i,j=...,$ be a double array of i.i.d. complex random variables with $EX_{11}=0,E|X_{11}|^2=1$ and $E|X_{11}|^4<\infty$, and let $A_n=\frac{1}{N}T_n^{{1}/{2}}X_nX_n^*T_n^{{1}/{2}}$, where $T_n^{{1}/{2}}$ is the square root of a nonnegative definite matrix $T_n$ and $X_n$ is the $n\times N$ matrix of the upper-left corner of the double array. The matrix $A_n$ can be considered as a sample covariance matrix of an i.i.d. sample from a population with mean zero and covariance matrix $T_n$, or as a multivariate $F$ matrix if $T_n$ is the inverse of another sample covariance matrix. To investigate the limiting behavior of the eigenvectors of $A_n$, a new form of empirical spectral distribution is defined with weights defined by eigenvectors and it is then shown that this has the same limiting spectral distribution as the empirical spectral distribution defined by equal weights. Moreover, if \{$X_{ij}$\} and $T_n$ are either real or complex and some additional moment assumptions are made then linear spectral statistics defined by the eigenvectors of $A_n$ are proved to have Gaussian limits, which suggests that the eigenvector matrix of $A_n$ is nearly Haar distributed when $T_n$ is a multiple of the identity matrix, an easy consequence for a Wishart matrix.Comment: Published at http://dx.doi.org/10.1214/009117906000001079 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

This paper is aimed at deriving the universality of the largest eigenvalue of a class of high-dimensional real or complex sample covariance matrices of the form WN = Σ 1/2 XX * Σ 1/2 . Here, X = (xij)M,N is an M ×N random matrix with independent entries xij,For a class of general deterministic positive-definite M × M matrices Σ, under some additional assumptions on the distribution of xij's, we show that the limiting behavior of the largest eigenvalue of WN is universal, via pursuing a Green function comparison strategy raised in [Probab. Theory Related Fields 154 (2012) 341-407, Adv. Math. 229 (2012) 1435-1515] by Erdős, Yau and Yin for Wigner matrices and extended by Pillai and Yin [Ann. Appl. Probab. 24 (2014) 935-1001] to sample covariance matrices in the null case (Σ = I). Consequently, in the standard complex case (Ex 2 ij = 0), combing this universality property and the results known for Gaussian matrices obtained by El Karoui in [Ann. Probab. 35 (2007) 663-714] (nonsingular case) and Onatski in [Ann. Appl. Probab. 18 (2008) 470-490] (singular case), we show that after an appropriate normalization the largest eigenvalue of WN converges weakly to the type 2 Tracy-Widom distribution TW2. Moreover, in the real case, we show that when Σ is spiked with a fixed number of subcritical spikes, the type 1 Tracy-Widom limit TW1 holds for the normalized largest eigenvalue of WN , which extends a result of Féral and Péché in [J. Math. Phys. 50 (2009) 073302] to the scenario of nondiagonal Σ and more generally distributed X.

Let s k = 1 √ N (v 1k , . . . , v Nk ) T , with {v ik , i, k = 1, . . .} independent and identically distributed complex random variables. Write S k = (s1, . . . , s k−1 , s k+1 , . . . , sK), P k = diag(p1, . . . , p k−1 , p k+1 , . . . , pK),to as the signalto-interference ratio (SIR) of user k under the multistage Wiener (MSW) receiver in a wireless communication system. It is proved that the output SIR under the MSW and the mutual information statistic under the matched filter (MF) are both asymptotic Gaussian when N/K → c > 0. Moreover, we provide a central limit theorem for linear spectral statistics of eigenvalues and eigenvectors of sample covariance matrices, which is a supplement of Theorem 2 in Bai, Miao and Pan [Ann. Probab. 35 (2007) 1532-1572. And we also improve Theorem 1.1 in Bai and Silverstein [Ann. Probab. 32 (2004) 553-605].

a b s t r a c tConsider the empirical spectral distribution of complex random n ×n matrix whose entries are independent and identically distributed random variables with mean zero and variance 1/n. In this paper, via applying potential theory in the complex plane and analyzing extreme singular values, we prove that this distribution converges, with probability one, to the uniform distribution over the unit disk in the complex plane, i.e. the well known circular law, under the finite fourth moment assumption on matrix elements.

Consider a Gaussian vector z = (x , y ) , consisting of two subvectors x and y with dimensions p and q respectively. With n independent observations of z, we study the correlation between x and y, from the perspective of the Canonical Correlation Analysis. We investigate the high-dimensional case: both p and q are proportional to the sample size n. Denote by Σuv the population crosscovariance matrix of random vectors u and v, and denote by Suv the sample counterpart. The canonical correlation coefficients between x and y are known as the square roots of the nonzero eigenvalues of the canonical correlation matrix Σ −1 xx ΣxyΣ −1 yy Σyx. In this paper, we focus on the case that Σxy is of finite rank k, i.e. there are k nonzero canonical correlation coefficients, whose squares are denoted by r1 ≥ · · · ≥ r k > 0. We study the sample counterparts of ri, i = 1, . . . , k, i.e. the largest k eigenvalues of the sample canonical correlation matrix S −1 xx SxyS −1 yy Syx, denoted by λ1 ≥ · · · ≥ λ k . We show that there exists a threshold rc ∈ (0, 1), such that for each i ∈ {1, . . . , k}, when ri ≤ rc, λi converges almost surely to the right edge of the limiting spectral distribution of the sample canonical correlation matrix, denoted by d+. When ri > rc, λi possesses an almost sure limit in (d+, 1], from which we can recover ri's in turn, thus provide an estimate of the latter in the high-dimensional scenario. We also obtain the limiting distribution of λi's under appropriate normalization. Specifically, λi possesses Gaussian type fluctuation if ri > rc, and follows Tracy-Widom distribution if ri < rc. Some applications of our results are also discussed.