Limiting behavior of eigenvectors of large dimensional random matrices

Based on the observation that the limiting spectral properties of large sional sample covariance matrix are asymptotically distribution free and the fact... dimen-Summary viithat the mat

LIMITING BEHAVIOR OF EIGENVECTORS

OF LARGE DIMENSIONAL RANDOM

MATRICES

XIA NINGNING

NATIONAL UNIVERSITY OF SINGAPORE

2013

LIMITING BEHAVIOR OF EIGENVECTORS

OF LARGE DIMENSIONAL RANDOM

MATRICES

XIA NINGNING

(B.Sc Bohai University of China)

A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF STATISTICS AND APPLIED

PROBABILITY NATIONAL UNIVERSITY OF SINGAPORE

2013

ACKNOWLEDGEMENTS

I would like to express my deep and sincere gratitude to my supervisor, ProfessorBai Zhidong His valuable guidance and continuous support are crucial to thecompletion of this thesis He is truly a great mentor not only in statistics but also

in daily life I have learned many things from him, especially regarding academicresearch and character building Next, I would like to thank Assistant ProfessorPan Guangming and Qin Yingli for discussion on various topics in research I alsothank all my friends who helped me to make life easier as a graduate student.Finally, I wish to express my gratitude to the university and the department forsupporting me through NUS Graduate Research Scholarship

CONTENTS

1.1 Large Dimensional Random Matrices 1

1.1.1 Spectral Analysis 3

1.1.2 Eigenvector 5

1.2 Methodologies 5

1.2.1 Moment Method 6

1.2.2 Stieltjes Transform 8

CONTENTS iv

1.2.3 Organization of the Thesis 13

Chapter 2 Literature Review for Sample Covariance Matrices 15 2.1 Spectral Analysis 15

2.1.1 Limiting Spectral Distribution 16

2.1.2 Limits of Extreme Eigenvalues 20

2.1.3 Convergence Rate 22

2.1.4 CLT of Linear Spectral Statistics 23

2.2 Eigenvector Properties 28

Chapter 3 Convergence Rate of VESD for Sample Covariance Ma-trices 32 3.1 Introduction and Main Result 32

3.1.1 Introduction 32

3.1.2 Main theorems 38

3.2 Methodology 41

3.2.1 Stieltjes transform 41

3.2.2 Inequalities for the distance between distributions via Stielt-jes transforms 43

3.3 Preliminary Formulae 43

3.4 Proofs 48

3.4.1 Truncation and Normalization 49

3.4.2 Proof of Theorem 3.1 53

3.4.3 Proof of Theorem 3.2 68

3.4.4 Proof of Theorem 3.3 69

3.4.5 Appendix 70

CONTENTS v

Chapter 4 Functional CLT of Eigenvectors for Sample Covariance

4.1 Introduction and Main Result 83

4.1.1 Introduction 83

4.1.2 Main Result 88

4.2 Bernstein Polynomial Strategy 90

4.3 Proof 93

4.3.1 Convergence of ∇1− E∇1 93

4.3.2 Mean Function 121

4.3.3 Appendix 122

Chapter 5 Conclusion and Future Research 130 5.1 Conclusion 130

5.2 Future Research 131

SUMMARY

All classical limiting theorems in multivariate statistical analysis assume thatthe number of variables is fixed and the sample size is much larger than the di-mension of the data In the light of the rapid development of computer science,

we are dealing with large dimensional data in most cases Moving from low mensional to large dimensional problems, random matrices theory (RMT) as anefficient approach has received much attention and developed significantly Theoriginal statistical issues in multivariate analysis have changed to the investigation

di-on limiting properties of eigenvalues and eigenvectors for large dimensidi-onal randommatrices in RMT

Based on the observation that the limiting spectral properties of large sional sample covariance matrix are asymptotically distribution free and the fact

dimen-Summary vii

that the matrix of eigenvectors (eigenmatrix) of the Wishart matrix is Haar tributed over the group of unitary matrices, it is conjectured that the behavior ofeigenmatrix of a large sample covariance matrix should asymptotically perform asHaar distributed under some moment conditions Thus, the thesis is concerned onfinding the limiting behavior of eigenvectors of large sample covariance matrices

dis-The main work in this thesis involves two parts In the first part (Chapter 3),

to investigate the limiting behavior of eigenvectors of a large sample covariancematrix, we define the eigenVector Empirical Spectral Distribution (VESD) withweights defined by eigenvectors and establish three types of convergence rates of

the VESD when data dimension n and sample size N proportionally tend to infinity.

In the second part (Chapter 4), the limiting behavior of eigenvectors of samplecovariance matrices is further discussed Using Bernstein polynomial approxima-tion and results obtained in Chapter 3, we prove the central limit theorem for thelinear spectral statistics associated with the VESD, indexed by a set of functionswith continuous second order derivatives This result provides us a strong evidence

to conjecture that the eigenmatrix of large sample covariance matrices is ically Haar distributed Thus, based on the result in Chapter 4, we have a betterview of the asymptotic property of eigenvectors for large general random matrices,such as Wigner matrices

LIST Of NOTATIONS

Xn (X ij)n ×N = (X1, · · · , X N)

Sn (1/N )X nX∗ n, the simplified sample covariance matrix

Sn (1/N )X ∗ nXn, the companion matrix of Sn

xn ∥x n ∥ = 1, unit vector in space C n

c n n/N → c, the dimension to sample size ratio index

F c (x), F c n (x) M-P law with ratio c and c n,

F S n (x), F S n (x) the empirical spectral distribution (ESD) of Sn , S n

F c n (x) (1− c n )I (0, ∞) (x) + c n F c n (x)

H S n (x) the eigenvector empirical spectral distribution

(VES-D) of Sn

List of Notations ix

m n (z), m H n (z) the Stieltjes transform of F S n (x) and H S n (x)

m0n (z), m(z) the Stieltjes transform of F c n (x) and F c (x)

m n (z) the Stieltjes transform of F S n (x)

m0

n (z), m(z) the Stieltjes transform of F c n (x) and F c (x)

U open interval including the support of M-P law

∥x n ∥ Euclidean norm for any vector xn ∈ C n

∥A∥ spectral norm of matrices, i.e ∥A∥ =√λAA∗

max

∥F (x)∥ norm of functions, i.e ∥F (x)∥ = sup x |F (x)|

Introduction

The development of Random Matrices Theory (RMT) comes from the fact thatthe classical multivariate analysis is no longer suitable for dealing with large di-mensional problems All classical multivariate analysis assumes that the dimension

of the data is small and fixed and the number of observations, or sample size, islarge and tends to infinity However, most of cases we are dealing with nowadaysare the data sets that the dimension increases together with the sample size, or inother words we can say that the dimension and sample size share the same order

1.1 Large Dimensional Random Matrices 2

The following two examples illustrate the serious effect of large dimensionalproblems solving by conventional statistical analysis Bai and Saranadasa (1996)showed that both Dempster’s non-exact test (Dempster, 1958) and their asymp-totically normally distributed test have higher power than classical Hotelling’s testwhen the data dimension is proportionally close to the sample degree of freedom

Another example was presented in Bai and Silverstein (2004) When dimension n increases proportionally to sample size N , an important statistics in multivariate analysis L n = ln(det Sn) performs in a complete different manner than it does ondata of very low dimension with large sample size Thus, a serious error is caused

when using classical limiting theory to show the asymptotic normality of L n underlarge dimensional case

Therefore, the theory of random matrices as a possible and effective method indealing with large dimensional data analysis has received much attention amongstatisticians in recent years For the same reason, the wide application of RMT can

be observed in many research areas, such as finance, engineering, signal processing,genetics, network security, image processing and wireless communication problems

From its inception, random matrix theory has been heavily influenced by itsapplications in physics, statistics and engineering The landmark contributions tothe theory of random matrices of Wishart (1928), Wigner (1958), and Marˇcenko

1.1 Large Dimensional Random Matrices 3

and Pastur (1967) were motivated to a large extent by practical experimental lems Nowadays, RMT finds applications in more fields as diverse as the Riemannhypothesis, stochastic differential equations, condensed matter physics, statisticalphysics, chaotic systems, numerical linear algebra, neural networks, multivariatestatistics, information theory, signal processing, and small-world networks

Definition 1.1 (Empirical Spectral Distribution)

Suppose A is an n ×n matrix with eigenvalues λ j , j = 1, 2, · · · , n If all these

eigen-values are real, e.g if A is Hermitian, we can define a one-dimensional distribution

called the empirical spectral distribution (ESD) of the matrix A, where I

denotes indicator function If the eigenvalues λ j’s are not all real, we can define a

two-dimensional empirical spectral distribution of the matrix A:

where ℜ and ℑ denote the real part and the imaginary part respectively.

We are especially interested in sequences of random matrices with dimension

1.1 Large Dimensional Random Matrices 4

(number of rows) tending to infinity One of the main problems in RMT is to vestigate the convergence of the sequence of empirical spectral distributions{F A n }

in-for a given sequence of random matrices {A n } The limit distribution F , which is

usually nonrandom, is called the Limiting Spectral Distribution (LSD) of the

1.2 Methodologies 5

Besides the importance of spectral analysis, practical applications of RMT havealso raised the need for a better understanding to the limiting behavior of eigenvec-tors of large dimensional random matrices For example, in principal componentanalysis (PCA), the eigenvectors corresponding to a few of the largest eigenvalues

of random matrices (that is, the directions of the principal components) are of cial interest Therefore, the limiting behavior of eigenvectors of large dimensionalrandom matrices becomes an important issue in RMT However, the investigation

spe-on eigenvectors has been relatively weaker than that spe-on eigenvalues in the ture due to the difficulty of mathematical formulation since the dimension increaseswith the sample size

This section introduces two important methods in the spectral analysis of largedimensional random matrices, namely the moment method and Stieltjes transformmethod These two methods are widely used in random matrix theory In thissection, we give a detailed discussion on these two methods, especially on theinvestigation of the convergence rate using Stieltjes transform

1.2 Methodologies 6

Moment method is widely used in finding the existence of limiting spectral tributions and limiting theorems on extreme eigenvalues ever since it was firstlyused by Wigner in 1958 to prove the famous semicircle law Using moment method,Bai, Silverstein and Yin (1988) proved an important theorem that the existence

dis-of a finite fourth moment dis-of the entries dis-of both Wigner and sample covariancematrices is a sufficient and necessary condition to guarantee that the largest eigen-values converge almost surely to the largest number in the support of their limitingspectral distributions

Moment method is based on the moment convergence theorem Suppose{F n }

denotes a sequence of distribution functions with finite moments of all orders Let

the k-th moment of the distribution F n be denoted by

Lemma 1.1 (Moment Convergence Theorem).

A sequence of distribution functions {F n } converges weakly to a limit if the ing conditions are satisfied:

follow-1.2 Methodologies 7

(1) Each F n has finite moments of all orders.

(2) For each fixed integer k ≥ 0, β n,k converges to a finite limit β k as n → ∞.

(3) If two right continuous nondecreasing functions F , G have the same moment sequence {β k }, then F = G + constant.

The following two lemmas guarantee that a probability distribution function isuniquely determined by its moments

The moment convergence theorem shows that under what conditions the vergence of moments of all orders implies the weak convergence of the sequence of

con-1.2 Methodologies 8

the distributions{F n } In finding limiting spectral distributions, the foundamental

theory is the moment convergence theorem together with Carleman’s condition orRiesz’s condition

In this section, another important method in spectral analysis of random trix theory will be introduced - the Stieltjes transform method Stieltjes transformmethod is commonly used to investigate the limiting spectral properties of a class

ma-of random matrices Compared with the moment method, the Stieltjes transformmethod is more attractive to researchers This is mainly because the momen-

t method is always followed with sophisticated graph theory and combinatorics,which makes the proof much tedious and complex

First, we introduce basic concept and properties of the Stieltjes transform,referring to Bai and Silverstein (2010, Appendix B) Further, the use of Stieltjestransform will be demonstrated That is how to find limiting spectral distributionsand estimate convergence rate of empirical spectral distributions in terms of theirStieltjes transforms

Definition 1.2 (The Stieltjes Transform)

λ − z dG(λ), z ∈ C+,

where z ∈ C+≡ {z ∈ C : ℑz > 0} and ℑ denotes the imaginary part.

Remark 1.1 The Stieltjes transform is defined on C+

Remark 1.2 The imaginary part of z plays an important role in Stieltjes

transfor-m For all bounded variation functions, their Stieltjes transform always exist and

well defined since the absolute value of m G (z) is bounded by 1/v, where v = ℑz.

Theorem 1.1 (Inversion formula)

For any continuity points a < b of G, we have

ℑm G (x + iϵ)dx.

Considering G as a finite signed measure, the above theorem shows a one-to-one

correspondence between the finite signed measures and their Stieltjes transforms

Another important advantage of Stieltjes transforms is that the density function

of a signed measure can be obtained easily via its Stieltjes transform We have thefollowing theorem

Theorem 1.2 (Differentiability)

Let G be function of bounded variation and x0 ∈ R Suppose that lim z ∈C+→x ℑm G (z)

de-of random matrices can be established by showing convergence de-of their Stieltjestransforms and the limiting spectral distribution can be found by the limit of asequence of Stieltjes transforms.

1.2 Methodologies 11

Let F A be the empirical spectral distribution function for any n × n Hermitian

matrix A Then, the Stieltjes transform of F A is given by

m F A (z) =

∫1

where a kk is the (k, k)th entry of A, ∗ denotes the conjugate transpose, A k is the

(n − 1) × (n − 1) submatrix of A with the k-th row and k-th column removed and

αk is the k-th column of A with the k-th entry removed.

If the denominator a kk −z−α ∗

k (A k−zI) −1 α

kcan be expressed as g(z, m F A (z))+

o(1) for some function g, then the limiting spectral distribution exists and its

Stieltjes transform m(z) is the solution to the equation

m(z) = 1

g(z, m(z)) .

For estimating convergence rates of empirical spectral distribution function

to its limiting spectral distribution, the following three lemmas (also called Baiinequality), which were first established by Bai in 1993, demonstrate the distancebetween two distribution functions in terms of their Stieltjes transforms

Lemma 1.4 (Theorem 2.1, Bai (1993a)).

γ = 1π

Sometimes, we can establish a bound for ∥F − G∥ on a finite interval in terms

of the integral of the absolute difference of their Stieltjes transform, when the

functions F and G have light tails or have bounded support We have the following

lemmas

Lemma 1.5 (Theorem 2.2, Bai (1993a))

Under the assumptions of Lemma 1.4, we have

κ = 4B π(A − B)(2γ − 1) < 1. (1.1)

1.2 Methodologies 13

Lemma 1.6 (Corollary 2.3, Bai (1993a))

In addition to the assumptions of Lemma 1.5, assume further that, for some constant B > 0, F ([ −B, B]) = 1 and |G|((−∞, −B)) = |G|((B, ∞)) = 0, where

|G|((a, b)) denotes the total variation of the signed measure G on the interval (a, b) Then, we have

Remark 1.3 From these lemmas, we can see that the convergence rate of∥F −G∥

has nothing to do with these constants h, γ, κ, A, B, but only depend on the rate

of v tending to 0 which will be illustrated in details in Chapter 3.

The thesis consists of five chapters and is organized as follows In Chapter 1,

we have provided a general introduction to the RMT theory including spectral andeigenvector analysis as well as two main methodologies in research

In the last chapter, we discuss future research.

ma-be expressed as a function of its eigenvalues Thus the spectral analysis of samplecovariance matrix has been well developed in the past decades We will introduce

2.1 Spectral Analysis 16

the detailed study of sample covariance matrix in the following subsections

The first study concerns the limiting spectral distribution of sample covariancematrices as the vector dimension and sample size proportionally tend to infinity.This aspect includes showing the convergence of empirical spectral distribution,identifying the limiting spectral distribution and further studying the analytic be-havior of the limiting spectral distribution

The conventional definition of a sample covariance matrix is as follows Let

Xn = (X ij)n ×N, 1 ≤ i ≤ n, 1 ≤ j ≤ N, is a double array of complex random

variables Write Xj = (X 1j , · · · , X nj)′ and Xn = (X1, · · · , X N) The samplecovariance matrix is defined as

2.1 Spectral Analysis 17

This is because, according to the following lemma, the removal of ¯X does not affect

the limiting spectral distribution since the rank of matrix ¯x¯ x∗ is 1

Lemma 2.1 (Theorem A.44, Bai and Silverstein (2010, Appendix A)).

Let A and B be two n × N complex matrices Then,

covari-spectral distributions of a class of random matrices having the form A + X∗TX

converge to some limiting distribution using Stieltjes transform method

original-ly And the limiting distribution is known as Marcˇenko-Pastur law (M-P law).Motivated by this paper, the generalization works were done by Grenander andSilverstein (1977), Johsson (1982) and Wachter (1978) In 1995, Silverstein andBai proposed a wilder conditions imposed on the underlying random variables andproved the convergence of sample covariance matrix with a general form in a easierand well understanding way

b = σ2(1 +√

c)2 The constant c is the dimension to sample size ratio index and

σ2 is the scale parameter If σ2 = 1, the M-P law is called as the standard M-Plaw

Lemma 2.2 (Theorem 3.10, Bai and Silverstein (2010)).

Suppose that, for each N , the entries of X n are independent complex variables with a common mean µ and variance σ2 Assume that n/N → c ∈ (0, ∞) and that, for any η > 0,

Lemma 2.3 (Theorem 1, Silverstein and Bai (1995)).

Suppose that the entries of X n (n × N) are complex random variables that are

2.1 Spectral Analysis 19

independent for each n and identically distributed for all n and satisfy E( |X11−

EX11|2) = 1 Also, assume that T n = diag(τ1, · · · , τ N ), τ i is real, and the ical distribution function of {τ1, · · · , τ N } converges almost surely to a probability distribution function H as n → ∞ The entries of both X n and T n may depend

empir-on n, which is suppressed for brevity Set B n = An+ 1

nXnTnX

∗

n , where A n is Hermitian, n × n satisfying F A n → F A almost surely, where F A is a distribution

function (possibly defective) on the real line Assume also that X n , T n and A n

are independent When N = N (n) with N/n → c > 0 as n → ∞, then, almost surely, F B n , the empirical spectral distribution of the eigenvalues of B n , converges vaguely, as n → ∞, to a (nonrandom) distribution function F , where for any

z ∈ C+ ≡ {z ∈ C : ℑz > 0}, its Stieltjes transform m = m(z) is the unique solution in C+ to the equation

Based on the above equation, the analytic properties of limiting spectral bution of sample covariance matrix was developed by Silverstein and Choi in 1995

distri-The result includes the continuous dependence of F on c and H, the continuous density of F on R+ and a method of determining its support

2.1 Spectral Analysis 20

The second study on the spectral analysis of sample covariance matrices isthe investigation on the limits of extreme eigenvalues The first work in this di-rection was done by Geman (1980) He proved that the largest eigenvalue of a

large dimensional sample covariance matrix goes to b, which is defined in (2.1),

under strong conditions on the underlying distribution Later, Bai, Silverstein andYin (1988) weaken the conditions into the assumption of the existence of fourthmoment And in the same year, they further illustrated that the fourth momentcondition is a necessary and sufficient condition for the existence of the limit ofthe largest eigenvalue of a large dimensional sample covariance matrix Identifyingthe limit of smallest eigenvalue of a large dimensional sample covariance matrixproves difficult Until 1993, Bai and Yin (1993) proved that the smallest eigenvalue

tends to a defined in (2.1), also under the existence of fourth moment assumption.

Another contributing result was made by Bai and Silverstein (1998) They showedthat there are no eigenvalues in any closed interval outside the support of the lim-iting spectral distribution of large dimensional sample covariance matrices, withprobability one We will show these results in the following lemmas

Lemma 2.4 (Theorem 5.8, Bai and Silverstein (2010)).

Suppose that {X jk , j, k = 1, 2, · · · } is a double array of i.i.d random variables

2.1 Spectral Analysis 21

with mean zero and variance σ2 and finite fourth moment Let X n = (X jk , j ≤

n, k ≤ N) and S n = N −1XnX∗ n Then the largest eigenvalue of S n tends to σ2(1 +

√

c)2almost surely If the fourth moment of the underlying distribution is not finite,

then with probability one, the limsup of the largest eigenvalue of S n is infinity.

Lemma 2.5 (Theorem 2, Bai and Yin (1993)).

Assume that the entries of {X ij } is a double array of i.i.d complex random variables with mean zero, variance σ2 and finite 4-th moment Let X n = (X ij ) be

the n × N matrix of the upper-left corner of the double array If n/N → c ∈ (0, 1), then, we have almost surely,

lim

n →∞ λ min(Sn ) = σ

2(1− √ c)2and

2.1 Spectral Analysis 22

(d) ∥T n ∥, the spectral norm of T n , is bounded in n.

(e) B n = (1/N )T 1/2 n XnX∗ nT1/2 n , T 1/2 n any Hermitian square root of T n , B n =

(1/N )X ∗ nTnXn , where X n = (X ij ), i = 1, 2, · · · , n, j = 1, 2, · · · , N.

(f ) The interval [a, b] with a > 0 lies outside the support of F c,H and F c n ,H n for all large n Here F c n ,H n is the limiting nonrandom distribution function associated with the limiting ratio c n and distribution function H n

Then P (no eigenvalue of B n appears in [a, b] for all large n) = 1.

The third study on the spectral analysis of large dimensional random ces goes to the investigation on its convergence rate Using moment method, wecan only establish the existence of the limiting spectral distribution of a class oflarge dimensional random matrices This method gives no contribution to theconvergence rate As an open problem, the convergence rate of empirical spectraldistribution puzzled statisticians for decades until 1993 For the first time, Bai es-tablished a Berry-Essen type inequality of the difference of two empirical spectraldistributions in terms of their Stieltjes transforms Referring to lemmas 1.4, 1.5and 1.6 Later, using this method, lots of papers were published to make the con-vergence rate of empirical spectral distribution more accurate In 2003, Bai, Miaoand Yao (2003) improved the rate for the expected spectral distribution to the

matri-2.1 Spectral Analysis 23

order O(n −1/2) G¨otze and Tikhomirov (2007) showed that the empirical spectral

distribution converges to M-P law with rate O p (n −1/2) The recent result refers

to Pillai and Yin (2012) they proved that the difference between eigenvalues of

sample covariance matrices and M-P law is of order O p ((log n) O(log log n) /n) under

sub-exponential decay assumption As for Wigner matrix, a great breakthroughwas made by Tao and Vu (2012) They offered a new method, Lindeberg replace-ment, to solve the convergence rate problem They proved that the convergence

rate of empirical spectral distribution of Wigner matrix is O p(logO(1) n/n) under

exponential decay condition However, the optimal convergence rate for samplecovariance matrix and Wigner matrix are still open

The third study on the spectral analysis of large dimensional random matricesconcerns the central limit theory for linear spectral statistics The main reason

is because many important statistics in multivariate statistical analysis can beexpressed as functionals of the empirical spectral distribution of some randommatrices Thus, a deeper investigation on the convergence of the empirical spectraldistribution is needed for more efficient statistical inferences, such as the test ofhypotheses, confidence regions, etc

Generalization the above example, we have the definition of linear spectral statistic.

Definition 2.2 (Linear Spectral Statistic (LSS))

let F nbe the empirical spectral distribution of a random matrix which has a limiting

spectral distribution F We call

a linear spectral statistic (LSS).

Associated with the given random matrix, a linear spectral statistic can beconsidered as an estimator of

G n (f ) tends to a nondegenerate distribution.

A natural idea is to pursue the properties of linear functionals by proving results

on the process X n (x) = α n (F n (x) − F (x)) when viewed as a random element in

2.1 Spectral Analysis 25

D(0, ∞), the metric space of functions with discontinuities of the first kind, along

with the Skorohod metric The limiting distribution of all LSS G n (f ) can be derived if X n (x) tends to a limiting process in space C or D equipped with the

Skorokhod metric However, work done by Diaconis and Evans (2001) proved that

all finite dimensional distributions of X n (x) converge in distribution to independent Gaussian variables when α n = n/ √

log n This result shows that the process X n (x) cannot be tight in D(0, ∞) with α n = n/ √

log n Besides, Bai and silverstein (2004) showed that X n (x) cannot converge weakly to any nontrivial process for any choice

of α n

Therefore, instead of looking for the limiting process of X n (x), we shall consider the convergence of G n (f ) with suitable α n The earliest work was done by Jonsson

(1982) in which he showed the central limit theorem for the centralized sum of the

r-th power of eigenvalues of a normalized Wishart matrix Later, Sinai and Soshnikov(1998) did a similar work for Wigner matrix and Johansson (2000) proved thecentral limit theorem of linear spectral statistics of Wigner matrix under densityassumptions

An example showed by Diaconis and Evans (2001) tells us that the convergence

of G n (f ) cannot be true for all functions f , at least for indicator functions In

2004, Bai and Silverstein (2004) weakened the conditions on f , assuming that f

are analytic functions on a open set including the support of the corresponding

2.1 Spectral Analysis 26

limit distributions And they proved that with such conditions on f and constant

α n = n, G n (f ) converges to Gaussian under certain assumptions The result is

nTnXn, and denote its limiting spectral

distri-bution and limiting Stieltjes transform as F c,H and m(z) Write

G n (f ) =

∫

f (x)dX n (x), where X n (x) = n(F B n (x) − F c n H n (x)).

Lemma 2.7 (Theorem 1.1, Bai and Silverstein (2004)).

2.1 Spectral Analysis 27

(1) the random vector

(G n (f1), · · · , G n (f k)) (2.2)

forms a tight sequence in n.

(2) If X11 and T n are real and EX4

11= 3, then (2.2) converges weakly to a Gaussian

vector (G f1, · · · , G f k ) with means

(3) If X11 is complex with EX2

11 = 0 and E( |X11|4) = 2, then (2) also holds, except

the means are zero and the covariance function is 1/2 times the function give in (2.4).

2.2 Eigenvector Properties 28

So far, all results mentioned above are concerned with the limiting behavior ofeigenvalues of large dimensional random matrices As mentioned in the introduc-tion, properties of eigenvectors of large dimensional random matrices also play animportant role in practical applications In principle component analysis, the direc-tions of principle components are of special interest In wireless communications,

an expression of signal-to-interference ratio (SIR) for the decorrelator receiver isderived in terms of eigenvalues and eigenvectors of random matrices, referring toEldar and Chan (2003) And in recent years, delocalization of eigenvectors hasbeen discussed by Erd¨os, Schlein and Yau (2009), and Erd¨os, Yau and Yin (2011)

However, the investigation on eigenvectors of a large dimensional random

ma-trix An are much more difficult than on eigenvalues due to the nondeterminacy

property of eigenvectors even if we assume that the spectrum of An is simple,

λ1(An ) < · · · < λ n(An) To eliminate this ambiguity, some viewpoints are

adopt-ed, such as forcing the first coefficient of eigenvectors to be positive or considering

the equivalence class [u i(An)] := {e √ −1θ u

i(An ) : θ ∈ R}, where u i(An) is an

eigenvector of An

Still, some excellent works in this direction have been done recently Such as,

2.2 Eigenvector Properties 29

Knowles, Yin (2011), Tao and Vu (2011), they investigated the probability bution of the components of eigenvectors for Wigner matrices Due to the factthat the eigenvector matrix of the Wishart matrix is Haar distributed over thegroup of unitary matrices, Jiang (2006) showed that how many entries of Haardistributed matrices can be replaced by standard normal random variables Sil-verstein (1981,1984,1989,1990) investigated the behavior of the eigenvector matrix

distri-in a whole picture He gave a conjecture that the Borel probability measure distri-duced by the eigenvector matrix of a large sample covariance matrix should beclose to Haar measure He also proposed a terminology ”asymptotical Haar” tomeasure this closeness Next, we will introduce the definition of Haar measure andits properties in the real case

in-For an n × n sample covariance matrix S n, we can write S n in its spectral

de-composition OnΛ n O′n with Λ n diagonal, its diagonal entries being the eigenvalues

of Sn arranged in ascending order, and On orthogonal, columns containing the

eigenvectors of Sn, which we will call the eigenmatrix of Sn We shall consider

the eigenmatrix On as a random element inO n , the space of n × n orthogonal

ma-trices There is a natural way to define a measure on O n, despite of the multiple

choices of eigenmatrix On

Note that O n forms a compact topological group under matrix multiplication

The mappings f1: O n × O n → O n and f2: O n → O n defined by f1(O1, O2) =

2.2 Eigenvector Properties 30

O1O2 and f2(O) = O−1 are continuous Typically, the space O n is called the

n × n orthogonal group According to these properties on O n, there is a unique

probability measure h nonO n, called the uniform or Haar measure, which is defined

as follows

Definition 2.3 (Haar Measure)

The probability measure h n defined on the Borel σ-field B on of Borel subsets of

On is called Haar measure if ,for any Borel set A ∈ B on and orthogonal matrix

O∈ O n , h n(OA) = h n(O), where OA denotes the set of all OA, A ∈ A.

If a n-dimensional random orthogonal matrix H n is Haar distributed, then it

is called a n-dimensional Haar matrix.

Remark 2.1 Haar measures defined on general topological groups can be found

in Halmos’s book, Halmos (1950)

Remark 2.2 For the complex case, we will have unitary matrices and unitary

groups instead of orthogonal ones

The following properties of Haar matrices are quoted in Bai and Silverstein(2010, Chapter 10)

Property 2.1 If H n is Haar distributed, then for any unit n-vector x n , y n =

Hnxn is uniformly distributed on the unit n-sphere.