Zhidong Bai

METHODOLOGIES IN SPECTRAL ANALYSIS OF LARGE DIMENSIONAL RANDOM MATRICES, A REVIEW

In this paper, we give a brief review of the theory of spectral analysis of large dimensional random matrices. Most of the existing work in the literature has been stated for real matrices but the corresponding results for the complex case are also of interest, especially for researchers in Electrical and Electronic Engineering. Thus, we convert almost all results to the complex case, whenever possible. Only the latest results, including some new ones, are stated as theorems here. The main purpose of the paper is to show how important methodologies, or mathematical tools, have helped to develop the theory. Some unsolved problems are also stated.

On detection of the number of signals in presence of white noise

In this paper, the authors propose procedures for detection of the number of signals in presence of Gaussian white noise under an additive model. This problem is related to the problem of finding the multiplicity of the smallest eigenvalue of the covariance matrix of the observation vector. The methods used in this paper fall within the framework of the model selection procedures using information theoretic criteria. The strong consistency of the estimates of the number of signals, under different situations, is established. Extensions of the results are also discussed when the noise is not necessarily Gaussian. Also, certain information-theoretic criteria are investigated for determination of the multiplicities of various eigenvalues.

On the limit of the largest eigenvalue of the large dimensional sample covariance matrix

SummaryIn this paper the authors show that the largest eigenvalue of the sample covariance matrix tends to a limit under certain conditions when both the number of variables and the sample size tend to infinity. The above result is proved under the mild restriction that the fourth moment of the elements of the sample sums of squares and cross products (SP) matrix exist.

Corrections to LRT on large-dimensional covariance matrix by RMT

In this paper, we give an explanation to the failure of two likelihood ratio procedures for testing about covariance matrices from Gaussian populations when the dimension p is large compared to the sample size n. Next, using recent central limit theorems for linear spectral statistics of sample covariance matrices and of random F -matrices, we propose necessary corrections for these LR tests to cope with high-dimensional effects. The asymptotic distributions of these corrected tests under the null are given. Simulations demonstrate that the corrected LR tests yield a realized size close to nominal level for both moderate p (around 20) and high dimension, while the traditional LR tests with χ 2 approximation fails. Another contribution from the paper is that for testing the equality between two covariance matrices, the proposed correction applies equally for non-Gaussian populations yielding a valid pseudo-likelihood ratio test.

A note on the largest eigenvalue of a large dimensional sample covariance matrix

Let {vij; i, J = 1, 2, ...} be a family of i.i.d. random variables with E(v114) = [infinity]. For positive integers p, n with p = p(n) and p/n --> y > 0 as n --> [infinity], let Mn = (1/n) Vn VnT , where Vn = (vij)1

On detection of the number of signals when the noise covariance matrix is arbitrary

In this paper, the authors proposed model selection methods for determination of the number of signals in presence of noise with arbitrary covariance matrix. This problem is related to finding the multiplicity of the smallest eigenvalue of [Sigma]2[Sigma]1-1, where [Sigma]2 = [Gamma] + [lambda][Sigma]1, [Sigma]1 and [Sigma]2 are covariance matrices, [lambda] is a scalar, and [Gamma] is non-negative definite matrix and is not of full rank. Also, the authors proposed methods for determination of the multiplicities of various eigenvalues of [Sigma]2[Sigma]1-1. The methods used in these procedures are based upon certain information theoretic criteria. The strong consistency of these criteria is established in this paper.

ON ASYMPTOTICS OF EIGENVECTORS OF LARGE SAMPLE COVARIANCE MATRIX

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 <∞, and let An = (1 N T 1/2 n X n X* n (T 1/2 n , where T 1/2 n is the square root of a nonnegative definite matrix T n and X n is the n x N matrix of the upper-left comer of the double array. The matrix An 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 An, 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 An are proved to have Gaussian limits, which suggests that the eigenvector matrix of An is nearly Haar distributed when T n is a multiple of the identity matrix, an easy consequence for a Wishart matrix.

Marcinkiewicz strong laws for linear statistics

Strong laws are established for linear statistics that are weighted sums of a random sample. We show extensions of the Marcinkiewicz-Zygmund strong law under certain moment conditions on both the weights and the distribution. These complement the results of Cuzick (1995, J. Theoret. Probab. 8, 625-641) and Bai et al. (1997, Statist. Sinica, 923-928).

Asymptotic Performance of MMSE Receivers for Large Systems Using Random Matrix Theory

Random matrix theory is used to derive the limit and asymptotic distribution of signal-to-interference-plus-noise ratio (SIR) for a class of suboptimal minimum mean-square-error (MMSE) receivers applied to large random systems with unequal-power users. We prove that the limiting SIR converges to a deterministic value when K and N go to infinity with lim K/N = y being a positive constant, where K is the number of users and N is the number of degrees of freedom. We also prove that the SIR of each particular user is asymptotically Gaussian for large N and derive the closed-form expressions of the variance for the SIR variable under real-spreading and complex-spreading channel environments. It is revealed that for a given (K,N) pair, under certain mild conditions, the variance of the SIR for complex-spreading channels is half of that for the corresponding real-spreading channels. Since the suboptimal MMSE receiver becomes optimal for the case when the users are equally powered, our results show that the conjecture made by Tse and Zeitouni for the complex-spreading case is not affirmative. We also derive the asymptotic distribution for SIR in decibels which provides better description when N is small. Numerical results and computer simulations are provided to evaluate the accuracy of various limiting and asymptotic results obtained in this paper.

Asymptotic theorems for urn models with nonhomogeneous generating matrices

The generalized Friedmans urn (GFU) model has been extensively applied to biostatistics. However, in the literature, all the asymptotic results concerning the GFU are established under the assumption of a homogeneous generating matrix, whereas, in practical applications, the generating matrices are often nonhomogeneous. On the other hand, even for the homogeneous case, the generating matrix is assumed in the literature to have a diagonal Jordan form and satisfies [lambda]>2 Re([lambda]1), where [lambda] and [lambda]1 are the largest eigenvalue and the eigenvalue of the second largest real part of the generating matrix (see Smythe, 1996, Stochastic Process. Appl. 65, 115-137). In this paper, we study the asymptotic properties of the GFU model associated with nonhomogeneous generating matrices. The results are applicable to a variety of settings, such as the adaptive allocation rules with time trends in clinical trials and those with covariates. These results also apply to the case of a homogeneous generating matrix with a general Jordan form as well as the case where [lambda] = 2 Re([lambda]1).