Limiting laws for extreme eigenvalues of large-dimensional spiked Fisher matrices with a divergent number of spikes
LLimiting laws for extreme eigenvalues oflarge-dimensional spiked Fisher matrices with adivergent number of spikes
Junshan Xie ∗ , Yicheng Zeng † , Lixing Zhu ‡ Abstract
Consider the p × p matrix that is the product of a population covariance matrixand the inverse of another population covariance matrix. Suppose that their di ff erencehas a divergent rank with respect to p , when two samples of sizes n and T from thetwo populations are available, we construct its corresponding sample version. In theregime of high dimension where both n and T are proportional to p , we investigate thelimiting laws for extreme (spiked) eigenvalues of the sample (spiked) Fisher matrixwhen the number of spikes is divergent and these spikes are unbounded. Keywords:
Extreme eigenvalue, Fisher matrix, Phase transition phenomenon, Randommatrix theory, Spiked population model.
In the last few decades, as the remarkable development in storage devices and computingcapability, the demand for processing complex-structured data increases dramatically. Oneof the features as well as the challenges of these data sets is their high dimensions. Thedi ffi culty is that the classical limit theory for multivariate statistical analysis fails to en-sure reliable inference for high-dimensional data analysis. Classical limit theorems require“small p large n ” to keep their validity, which conflicts with the situation “large p large n ”in high-dimensional settings in the sense that p / n → c > ff erent. To attack the relevant issues, random matrix theory (RMT) serves as apowerful tool in addressing statistical problems in high dimensions. The first research ofrandom matrices in multivariate statistics was about the Wishart matrices in [18]. Abundantresearch has been established for various topics in this field during the past half century,especially in recent years. In the area of RMT in statistics, we refer to monographs [2] and[19] for systematical study and [12] for a comprehensive review. ∗ Co-first author. School of Mathematics and Statistics, Henan University, Kaifeng, China. † Co-first author. Department of Mathematics, Hong Kong Baptist University, Hong Kong. ‡ Corresponding author. Research Center for Statistics and Data Science, Beijing Normal University, Zhuhai,China and Department of Mathematics, Hong Kong Baptist University, Hong Kong. Email address: [email protected] . a r X i v : . [ m a t h . S T ] S e p relevant topic in multivariate statistics is about testing the equality of two covariancematrices: H : Σ = Σ vs. H : Σ = Σ + ∆ , (1.1)where Σ and Σ are two covariance matrices corresponding to two p -variate populations,and ∆ is a non-negative definite matrix with rank q . Let S and S be the sample covariancematrices from these two populations, respectively. When S is invertible, the random matrix F = S − S is called a Fisher matrix.The di ff erence between the null hypothesis and the alternative hypothesis relies on thoseextreme eigenvalues of F . Under the null hypothesis, Σ = Σ , [16] established the well-known Wacheter distribution as the limiting spectral distribution (LSD) of F . Some exten-sions were built later (see examples in [13], [14] and [15]). Furthermore, [1] pointed outthe fact that the largest eigenvalue of F converges to the upper bound of the support of theLSD of F . Under the alternative hypothesis, F is called a spiked Fisher matrix (see [17]),because Σ − Σ has a spiked structure similar to that of a spiked population model proposedby [10]. More specifically, the matrix Σ − Σ is assumed to have the spectrumspec( Σ − Σ ) = { λ , . . . , λ q , , . . . , } , (1.2)where λ ≥ . . . ≥ λ q >
1. When the rank q of ∆ is finite, [6] showed the phase transitionphenomenon of the extreme eigenvalues of F under Gaussian population assumption. Thatis, for 1 ≤ i ≤ q , the i -th largest eigenvalue of F will depart from the upper bound of thesupport of LSD of F if and only if λ exceeds certein phase transition point. [17] extendedit to the cases without Gaussian assumption and established central limit theorems for theoutlier eigenvalues of F .We in this paper consider, as a reasonable extension in theory and applications, the caseof divergent q with respect to the dimension p . We will investigate the convergence inprobability and central limit theorems for spiked eigenvalues of spiked Fisher matrices. Weformulate our problem as follows.Assume that Y = ( y , . . . , y T ) = ( y i j ) ≤ i ≤ p , ≤ j ≤ T ∈ R p × T and Z = ( z , . . . , z n ) = ( z i j ) ≤ i ≤ p , ≤ j ≤ n ∈ R p × n (1.3)are two independent arrays of independent real-valued random variables with zero meanand unit variance. We consider two samples { Σ / y i } ≤ i ≤ T and { Σ / z i } ≤ i ≤ n , then their cor-responding sample covariance matrices can respectively be written as S = T T (cid:88) i = Σ y i y (cid:62) i Σ = T Σ YY (cid:62) Σ and S = n n (cid:88) i = Σ z i z (cid:62) i Σ = n Σ ZZ (cid:62) Σ . Also, define the Fisher matrix F : = S − S , as the sample version of matrix Σ − Σ . We aimto investigate the limiting properties of the eigenvalues of F . As the eigenvalues of F remain2nvariant under the linear transformation( S , S ) → (cid:18) Σ − S Σ − , Σ − S Σ − (cid:19) , (1.4)thus we can assume Σ = I p throughout this paper without loss of generality. Under theassumption (1.2), eigenvalues of Σ are λ ≥ . . . ≥ λ q > λ q + = . . . = λ p =
1. Recalling(1.1) that Σ is a rank q pertubation of Σ = I p , we simply assume Σ = Σ I p − q . (1.5)For the sake of brevity and readability, we write the eigenvalues of F in descending order (cid:98) λ ≥ . . . ≥ (cid:98) λ p , simplifying the double subscripts as single ones. It should be noted that (cid:98) λ i isrelated to the sample size n .We then describe the related work and our contributions in this paper. When the num-ber of the spiked eigenvalues q is fixed, and all the spiked eigenvalue λ i , i = , . . . , q , arebounded, there are some results on the limiting properties of the eigenvalues of F in theliterature. Such as, the almost surely convergence (strong consistency) and central limittheorem (CLT) of spiked eigenvalues ([17]) and asymptotically Tracy-Widom distributionfor the largest non-spiked eigenvalue ([7] and [8]). In this paper, we consider the casethat the number of spiked eigenvalues q = q ( p ) → ∞ as p → ∞ , and spiked eigenvalues λ i , ≤ i ≤ q diverge as p → ∞ . To the best of our knowledge, there is no relevant re-sult in the literature. A relevant work is [5] who studied spiked population models, wherethe asymptotics for spiked eigenvalues, including convergence in probability (weak con-sistency) and CLT, as well as Tracy-Widom law for the largest nonspiked eigenvalue werebuilt under a quite general framework. Unlike the case of fixed q and bounded spikes λ i ,1 ≤ i ≤ q , normalizations for (cid:98) λ i , 1 ≤ i ≤ q are needed for the divergent q case. Consider thenormalized eigenvalues (cid:98) λ i /λ i in consistency and ( (cid:98) λ i − θ i ) /θ i in CLT, where θ i is a centeredparameter defined later.The basic approach behind the proofs of the asymptotics for spiked eigenvalues is theanalysis of an equation for the determinant of a q × q random matrix (indexed by n ). When q is bounded, [17] derived the almost sure entrywise convergence of the q × q matrix (andhence the convergence with respect to matrix norms) and then solving the equation to leadto the almost sure limits of spiked eigenvalues. This argument does not work in the diver-gent q case where the convergence of a q × q matrix with respect to some norm could not bedirectly implied by the entrywise convergence. Instead, we use the CLT for random sequi-linear forms in [3] to derive the convergence rate of each entry, and then use Chebyshev’sinequality to put all entries together to derive the convergence rate of the matrix in (cid:96) ∞ norm.In this way, we achieve the convergence in probability as well as the CLT of spiked eigen-values (after proper normalizations). This approach is similar to that used in [5], so sometechnical assumptions are also imposed similarly.The remaining parts of the paper are organized as follows. Section 2 establishes themain results, including the convergence in probability of (cid:98) λ i /λ i and central limit theorems3f ( (cid:98) λ i − θ i ) /θ i , for those spiked eigenvalues of spiked Fisher matrix F . Here, θ i , 1 ≤ i ≤ q ,is a sequence of centering parameters defined in this section. In the Section 3, we show theproofs of our main results in Section 2. Some important technical lemmas and their proofsare displayed in the Section 4. Considering the linear transformation (1.4), we assume that Σ = I p without loss of gen-erality, and then Σ has the structure as shown in (1.5). Further, we decompose the Σ in(1.5) as Σ = U (cid:62) Λ U . Here, U ≡ ( u , u , . . . , u q ) (cid:62) is a q × q orthogonal matrix and Λ = diag( λ , . . . , λ N (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) n , . . . , λ N (cid:96) − + , . . . , λ q (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) n (cid:96) ) , where λ = . . . = λ N > . . . > λ N (cid:96) − + = . . . = λ q and N i : = (cid:80) ij = n j for 1 ≤ i ≤ (cid:96) . In thiscase, Σ can be decomposed as Σ = U (cid:62) I p − q Λ I p − q U I p − q = : U (cid:62) I p − q Λ U I p − q . We give decompositions of the sample covariance matrices S and S as follows. Wefisrt decompose the matrices Y and Z defined in (1.3) as Y = ( Y (cid:62) , Y (cid:62) ) (cid:62) and Z = ( Z (cid:62) , Z (cid:62) ) (cid:62) ,where Y , Z ∈ R q × n and Y , Z ∈ R ( p − q ) × n . Let X : = Σ / Y . Then we can similarly write X = ( X (cid:62) , X (cid:62) ) (cid:62) , where X = Σ / Y = U (cid:62) Λ / UY ∈ R q × T and X = Y ∈ R ( p − q ) × T . Itfollows that S = T X X (cid:62) T X X (cid:62) T X X (cid:62) T X X (cid:62) and S = n Z Z (cid:62) n Z Z (cid:62) n Z Z (cid:62) n Z Z (cid:62) . (2.1)For λ ∈ R \ { } , we introduce F = (cid:32) n Z Z (cid:62) (cid:33) − (cid:32) T X X (cid:62) (cid:33) , M ( λ ) = I p − q − F λ , (cid:101) m θ ( z ) = p − q tr (cid:32) z I p − q − F θ (cid:33) − , θ ∈ R , z ∈ C + . (2.2)Let µ ≥ . . . ≥ µ p − q be the eigenvalues of the Fisher matrix F . Then the empirical spectraldistribution (ESD) of F can be defined as F n ( x ) = p − q p − q (cid:88) j = { µ j ≤ x } , x ∈ R .
4y the result in [17], under the assumption of p / n → y ∈ (0 ,
1) and p / T → c >
0, almostsurely, the empirical spectral distribution F n weakly converges to the limiting spectral dis-tribution F c , y , whose Stieltjes transform S ( z ) = (cid:82) ∞−∞ ( x − z ) − dF c , y ( x ) satisfies, for z (cid:60) [ a , b ] S ( z ) = − czc − c [ z (1 − y ) + − c ] + zy − c (cid:112) [ z (1 − y ) + − c ] − z zc ( c + zy ) , (2.3)where a = (1 − √ c + y − cy ) (1 − y ) − and b = (1 + √ c + y − cy ) (1 − y ) − .In the following, for any complex matrix A , we use s i ( A ) to denote the i -th largestsingular value, and (cid:107) A (cid:107) to denote the largest singular value throughout the paper. Write a n = O a . s . ( b n ) if it almost surely holds that a n = O ( b n ). Throughout this paper C is aconstant that may vary from place to place.The following assumptions are required. Assumption 2.1. y p : = p / n → y ∈ (0 , (cid:101) y p : = ( p − q ) / n ; c p : = p / T → c > (cid:101) c p : = ( p − q ) / T ; q = q ( n ) → ∞ as n → ∞ but q = o( n ). Assumption 2.2.
For any 1 ≤ i ≤ q , λ i satisfies q /λ i → a ) . λ − i (cid:80) qj = λ j = o( q − n ) and λ − i (cid:80) qj = λ j = o( q − ); ( b ) . λ i (cid:80) qj = λ − j = o( q − n ). Assumption 2.3.
Random vectors in { y i : 1 ≤ i ≤ T } (cid:83) { z i : 1 ≤ i ≤ n } are independentidentically distributed, E z i j =
0, E | z i j | = ∀ ≤ i ≤ p , ≤ j ≤ n and sup ≤ i ≤ p E | z i j | < ∞ . Assumption 2.4.
There exists a constant C > λ N i /λ N i + ≥ C for any 1 ≤ i ≤ (cid:96) − Assumption 2.5.
Suppose that { λ i } ≤ i ≤ q are of bounded multiplicities, i.e., sup ≤ i ≤ (cid:96) n i < ∞ . The weak consistency of (cid:98) λ i is stated below. Due to the fact that λ i may go to infinity with n , consider the limit in probability for the ratio (cid:98) λ i /λ i , 1 ≤ i ≤ q . Theorem 2.6.
Assume that Assumptions 2.1, 2.2 and 2.3 hold. Then for all 1 ≤ i ≤ q , (cid:98) λ i λ i = − y + O (cid:16) y p − y (cid:17) + κ q · O p (cid:32) √ n + λ − i (cid:33) , where κ : = min { κ , κ } with κ : = q + λ − i (cid:80) qj = λ j and κ : = q + λ i (cid:80) qj = λ − j . Remark 2.7.
Note that the limit of the ratio (cid:98) λ i /λ i is 1 / (1 − y ) >
1, for all 1 ≤ i ≤ q . Thisis di ff erent from the relevant limit for spiked population model with divergent q , which is1 (see Theorem 2.1 in [5]). Roughly speaking, when we take y → / (1 − y ) → emark 2.8. In the case of fixed q and bounded spikes λ i , 1 ≤ i ≤ q , Theorem 3.1 in [17]shows that almost surely the spiked eigenvalue (cid:98) λ i converges to the limit λ i ( λ i + c − λ i − λ i y − − . Simply taking λ i → ∞ , the limit of λ i ( λ i + c − λ i − λ i y − − λ − i equals to1 / (1 − y ). Thus, Theorem 2.6 indicates that for the divergent q case, the result coincideswith the result for the fixed q case in [17]. Remark 2.9.
In Theorem 2.6 we only consider unbounded spikes, but actually it can bereadily extended to handle the case with both bounded and unbounded spikes. Consider themodel Σ = U (cid:62) I p − q Λ U I p − q , where Λ = diag( λ , . . . , λ q , λ q + , . . . , λ q + q , , . . . , q = o( n / ) and q is bounded. As-sume that spikes λ ≥ . . . ≥ λ q are unbounded as in Theorem 2.6 and λ q + ≥ . . . ≥ λ q + q are bounded. For q + ≤ i ≤ q + q , by Theorem A.10 in [2], we have (cid:98) λ i = s i (cid:16) S − S (cid:17) ≤ s i ( S ) s (cid:16) S − (cid:17) ≤ s i ( Σ ) s (cid:32) T YY (cid:62) (cid:33) s (cid:16) S − (cid:17) < ∞ almost surely. So it holds that det (cid:98) λ i n Z Z (cid:62) − T X X (cid:62) (cid:44) . Similar to the decomposition in (3.3), we have det (cid:98) λ i n Z Z (cid:62) − T X X (cid:62) − (cid:98) λ i n Z Z (cid:62) − T X X (cid:62) (cid:98) λ i n Z Z (cid:62) − T X X (cid:62) − (cid:98) λ i n Z Z (cid:62) − T X X (cid:62) = . (2.4) In the same manner as used in the proof of Theorem 2.6, it can be checked that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:98) λ i n Z Z (cid:62) − T X X (cid:62) (cid:98) λ i n Z Z (cid:62) − T X X (cid:62) − (cid:98) λ i n Z Z (cid:62) − T X X (cid:62) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ = o p (1) . Then the solution of equation (2.4) is close to that of the equationdet (cid:98) λ i n Z Z (cid:62) − T X X (cid:62) = . (2.5)Note that the solution of (2.5) is an eigenvlaue of the spiked Fisher matrix ( Z Z (cid:62) / n ) − ( X X (cid:62) / T )which has been well studied by [17]. Thus, the weak consistency for all outliers (cid:98) λ i , 1 ≤ i ≤ q + q , could be achieved by combining Theorem 3.1 in [17] and Theorem 2.6. Such a kindof extension could also be considered for the CLT in Theorem 2.10.6 .3 Central limit theorem As λ i , 1 ≤ i ≤ q , goes to infinity, the consistency of (cid:98) λ i /λ i in Theorem 2.6 does not meanthat (1 − y ) (cid:98) λ i is a good estimator of λ i . In this section, we establish the CLT for (cid:98) λ i to providefurther properties.We first introduce a centered parameter for (cid:98) λ i . Let θ i ∈ R , 1 ≤ i ≤ q , satisfy1 − n E (cid:104) tr (cid:110) M − ( θ i ) (cid:111)(cid:105) = λ i θ i (cid:32) + T E (cid:34) tr (cid:40) M − ( θ i ) F θ i (cid:41)(cid:35)(cid:33) , (2.6)and define δ i , for 1 ≤ i ≤ q , as δ i = (cid:98) λ i − θ i θ i . (2.7)By Lemma 4.1, when n → ∞ , we can easily see that1 p − q E (cid:104) tr (cid:110) M − ( θ i ) (cid:111)(cid:105) = E (cid:8)(cid:101) m θ i (1) (cid:9) → p − q E (cid:34) tr (cid:40) M − ( θ i ) F θ i (cid:41)(cid:35) → . It follows by (2.6) that λ i θ i = (cid:18) − p − qn (cid:19) + o(1) → − y . Since the equation in Definition 2.6 for θ i is hard to calculate, an alternative definition for θ i is proposed as follows. Recall the definition of (cid:101) m θ ( z ) in (2.2): (cid:101) m θ ( z ) = p − q tr (cid:32) z I p − q − F θ (cid:33) − , θ ∈ R , z ∈ C + . Denoting f θ ( x ) = θ/ ( θ − x ) for any fixed θ ∈ R , we have (cid:101) m θ (1) = p − q tr (cid:32) I p − q − F θ (cid:33) − = (cid:90) ∞−∞ θθ − x dF n ( x ) = : F n ( f θ ) , where F n denotes the ESD of the matrix F . By the CLT for linear spectral statistics (LSS)of Fisher matrices (see Theorem 3.10 in [19]), for any fixed θ , p { F n ( f θ ) − F (cid:101) c p , (cid:101) y p ( f θ ) } converges weakly to a Gaussian variable. It follows that (cid:101) m θ (1) = F (cid:101) c p , (cid:101) y p ( f θ ) + O p ( n − ) = − θ (cid:101) S ( θ ) + O p ( n − ) = (cid:101) c p − (cid:101) c p + (cid:101) c p { θ (1 − (cid:101) y p ) + − (cid:101) c p } + θ (cid:101) y p − (cid:101) c p (cid:113) { θ (1 − (cid:101) y p ) + − (cid:101) c p } − θ (cid:101) c p ( (cid:101) c p + θ (cid:101) y p ) + O p ( n − ) , where (cid:101) S ( · ) denotes the stieltjes transform of F (cid:101) c p , (cid:101) y p . This leads toE { (cid:101) m θ (1) } = − θ (cid:101) S ( θ ) + O( n − ) . (2.8)7he definition of θ i in (2.6) can be rewritten as1 − (cid:101) y p E (cid:8)(cid:101) m θ i (1) (cid:9) = λ i θ i (cid:104) − (cid:101) c p + (cid:101) c p E (cid:8)(cid:101) m θ i (1) (cid:9)(cid:105) . According to (2.8), it is equivalent to1 + (cid:101) y p θ i (cid:101) S ( θ i ) + O( n − ) = λ i θ i (cid:110) − (cid:101) c p − (cid:101) c p θ i (cid:101) S ( θ i ) + O( n − ) (cid:111) . (2.9)Thus, we give another definition of θ i by the following equation1 + (cid:101) y p θ i (cid:101) S ( θ i ) = λ i θ i (cid:110) − (cid:101) c p − (cid:101) c p θ i (cid:101) S ( θ i ) (cid:111) . (2.10)It is notable that the θ i defined by (2.10) is also applicable to the CLT of δ i in the later sec-tion. Comparing two equations (2.9) and (2.10), we can derive that the di ff erence betweentwo δ i ’s respectively derived from these two equations is at most O( n − ), which is smallerthan the scale n − / of δ i . Even Taylor’s expansion on the stieltjes transformantion (cid:101) S ( · ) canbe simply used to the equation (2.10) and then get the explicit forms of θ i , although someerrors would appear. In the remaining parts of this paper, we use θ i defined by (2.6) in allresults and their proofs.Consider the case where all the spiked eigenvalues are simple, that is, n i = ≤ i ≤ (cid:96) , which means that Λ = diag( λ , λ , . . . , λ q ). Theorem 2.10.
Under Assumptions 2.1, 2.2, 2.3, 2.4 and that n i = , ≤ i ≤ (cid:96) , i.e., (cid:96) = q ,it holds that, for all 1 ≤ i ≤ q , √ p δ i σ i d −→ N (0 , σ i : = ( y + c ) ν i − c − y (1 − y )(1 − y ) − , where ν i = E | u (cid:62) i Z e | , e = (1 , , . . . , (cid:62) ∈ R q and u i ∈ R q is the i -th column of the matrix U (cid:62) . Remark 2.11.
When the value of the variance σ i at the population level is unknown, forstatistical inference, estimating σ i is in need. A natural estimation way would be to es-timate the eigenvector u i first. For the spiked population model, [5] shows that when aleading eigenvalue of the sample covariance matrix is divergent, the corresponding sam-ple eigenvector is a good estimator for its population counterpart in terms of their in-ner product. However, the situation becomes much more di ffi cult when it comes to thespiked Fisher matrix. Recalling the assumed structure Σ − / Σ Σ − / = I p + ∆ , we sup-pose that v i : = ( u (cid:62) i , , . . . , (cid:62) ∈ R p is the eigenvector of Σ − / Σ Σ − / = I p + ∆ cor-responding to λ i and (cid:98) v i is that of S = Σ / YY (cid:62) Σ / . Then Σ / (cid:98) v i is the eigenvector of( I p + ∆ ) / YY (cid:62) ( I p + ∆ ) / corresponding to the i -th largest eigenvalue. If Σ is known orcan be consistently estimated, Σ / (cid:98) v i is a good estimator of v i , by Theorem 4.1 in [5]. Butactually Σ cannot be easily recovered based on S because of the delocalization of thoseeigenvectors for non-outliers (see [4]). Thus, how to construct a consistent estimation of8 becomes a challenging issue. As a special case, when entries of Y and Z are Gaus-sian, the parameter ν i equals to 3, which is independent of the value of u i . In practice, thebootstrap approximation would be an alternative way to achieve a reliable estimation of σ i . For estimation of the variance of the largest sample eigenvalue in a spiked populationmodel, spiked population model, [11] shows that the bootstrap approximation works whenthe largest eigenvalue is quite large. This deserves a further study.To check the practical applicability of Theorem 2.10, a simulation is conducted. Set p = T = n = q = (cid:100) p (cid:101) , λ i = (3 / q + − i (log p / for 1 ≤ i ≤ q , where (cid:100) x (cid:101) denotes the smallest integer greater than or equal to x . Let Σ = diag( λ , . . . , λ q , , . . . , Σ = I p . Draw a sample { x i } ≤ i ≤ T of size T from N (0 , Σ ) and a sample { z i } ≤ i ≤ n ofsize n from N (0 , Σ ). Compute the largest q eigenvalues (cid:98) λ i , 1 ≤ i ≤ q , of the Fisher matrix F = S − S and then δ i accordingly, where S = (cid:80) Ti = x i x (cid:62) i / T and S = (cid:80) ni = z i z (cid:62) i / n . Wedraw qq plots of √ p δ /σ and √ p δ q /σ q from 1000 independent replications in Figure 1. Itsuggests that both of √ p δ /σ and √ p δ q /σ q are well approximated by the standard normaldistribution. −3 −2 −1 0 1 2 3 − − − (a) −3 −2 −1 0 1 2 3 − − − (b) Figure 1: (a) The qq plot of the normalized largest spiked eigenvalue √ p δ /σ from 1000independent replications. (b) The qq plot of the normalized smallest spiked eigenvalue √ p δ q /σ q from 1000 independent replications.Next, consider the case where some spiked eigenvalues are possibly multiple: Λ = diag( λ , . . . , λ N (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) n , . . . , λ N (cid:96) − + , . . . , λ q (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) n (cid:96) ) , where λ = . . . = λ N > . . . > λ N (cid:96) − + = . . . = λ q , N i : = (cid:80) ij = n j for 1 ≤ i ≤ (cid:96) andthere exists a constant C < ∞ such that 1 ≤ n i ≤ C for all 1 ≤ i ≤ (cid:96) . According to9he multiplicities of spiked eigenvalues, we divide the index set { , . . . , q } into (cid:96) subsets, J i = { N i − + , . . . , N i } , ≤ i ≤ (cid:96) . Here we denote N =
0. For any 1 ≤ i ≤ (cid:96) , and1 ≤ h , k , h , k , h , k ≤ n i , define M N i , h , k : = E (cid:16) u (cid:62) N i − + h Z e u (cid:62) N i − + k Z e (cid:17) , M N i , h , k , h , k : = E (cid:16) u (cid:62) N i − + h Z e u (cid:62) N i − + k Z e u (cid:62) N i − + h Z e u (cid:62) N i − + k Z e (cid:17) . Theorem 2.12.
Suppose that Assumptions 2.1, 2.2, 2.3, 2.4 and 2.5 hold. Define φ i ( (cid:98) λ j ) = ( (cid:98) λ j − θ j ) /θ j , for 1 ≤ i ≤ (cid:96) and j ∈ J i . Then √ p { φ i ( (cid:98) λ j ) , j ∈ J i } converges weakly to the dis-tribution of the eigenvalues of the n i × n i random matrix (cid:60) ( i ) , where (cid:60) ( i ) = (cid:16) R ( i ) hk (cid:17) ≤ h , k ≤ n i is asymmetric matrix with independent Gaussian entries of mean zero and covariance structurecov (cid:16) R ( i ) h , k , R ( i ) h , k (cid:17) = (1 − y ) − ω (cid:0) M N i , h , k , h , k − M N i , h , k M N i , h , k (cid:1) + (1 − y ) − ( β − ω ) (cid:0) M N i , h , k M N i , h , k + M N i , h , h M N i , k , k (cid:1) , where ω = ( y + c ) (1 − y ) and β = y (1 − y ) + c (1 − y ) . We begin with a summary of the proofs. Roughly, the proof of Theorem 2.6 proceeds inthree steps. First, we prove that the spiked eigenvalue (cid:98) λ i , 1 ≤ i ≤ q , solves the equation(3.5) whose left-hand side is the determinant of a q × q matrix which can be decomposedinto four terms, namely U Ξ A U (cid:62) , U Ξ B U (cid:62) , U Ξ C U (cid:62) and U Ξ D U (cid:62) defined below. Second,we derive the limit of each entry of these four matrices and their convergence rates in (cid:96) ∞ norm, where the CLT for random sequilinear forms in [3] and Chebyshev’s inequality arerepeatedly used. Third, using eigenvalue perturbation theorems on (3.5), we estimate thefluctuation of the scaled eigenvalue (cid:98) λ i /λ i and reach the result. As for the proof of The-orem 2.10, we also work on the equation (3.5) in three main steps. First, we rewrite thematrix in (3.5) as the sum of U Θ n U (cid:62) , U δ i Θ n U (cid:62) and U Θ n U (cid:62) . See equation (3.30) below.Second, we prove the CLT for each diagonal entry of U Θ n U (cid:62) (Lemma 4.2) and estimatethe (cid:96) ∞ norm of U Θ n U (cid:62) (Lemma 4.3), U Θ n U (cid:62) (Lemma 4.4) and U Θ n U (cid:62) . Third, we ex-pand the determinant in (3.30) by Leibniz formula and then achieve the CLT for δ i . In thissection, we will cite the lemmas given in the next section without the proofs whose detailsare postponed to the next section. Proof of Theorem 2.6.
We first show that for 1 ≤ i ≤ q , (cid:98) λ i converges to infinity at the sameorder with λ i almost surely, i.e., there exists some constant C > C − < (cid:98) λ i /λ i < C almost surely.For any 1 ≤ i ≤ q , by Theorem A.10 in [2], we have that (cid:98) λ i = s i ( S − S ) ≤ s i ( S ) s ( S − ) = s i ( S ) s − p ( S ) and s i ( S ) ≤ s i ( S − S ) s ( S ) . s ( S ) → (1 + √ y ) and s p ( S ) → (1 − √ y ) > < C < (cid:98) λ i / s i ( S ) ≤ C < + ∞ almost surely for some constants C and C .Again, by Theorem A.10 in [2] and Weyl’s inequality, we have s i ( S ) ≤ s i ( Σ ) s (cid:32) T YY (cid:62) (cid:33) = λ i s (cid:32) T YY (cid:62) (cid:33) and s i ( S ) = s i (cid:32) T Y (cid:62) Σ Y (cid:33) = s i (cid:32) T Y (cid:62) Σ Y + T Y (cid:62) Y (cid:33) ≥ s i (cid:32) T Y (cid:62) Σ Y (cid:33) ≥ s i ( Σ ) s q (cid:32) T Y Y (cid:62) (cid:33) = λ i s q (cid:32) T Y Y (cid:62) (cid:33) . Due to the fact that s (cid:32) T YY (cid:62) (cid:33) → (1 + √ c ) and s q (cid:32) T Y Y (cid:62) (cid:33) → < C < s i ( S ) /λ i < C < + ∞ almost surely for some constants C and C .Thus, we conclude that C − < (cid:98) λ i /λ i < C almost surely for some constant C .For any 1 ≤ i ≤ q , by the definition of (cid:98) λ i , it solves the equation det (cid:16)(cid:98) λ i I − S − S (cid:17) = (cid:16)(cid:98) λ i S − S (cid:17) = . (3.1)By the decomposition of S and S in (2.1), the equation (3.1) can be rewritten asdet (cid:98) λ i n Z Z (cid:62) − T X X (cid:62) (cid:98) λ i n Z Z (cid:62) − T X X (cid:62) (cid:98) λ i n Z Z (cid:62) − T X X (cid:62) (cid:98) λ i n Z Z (cid:62) − T X X (cid:62) = . (3.2)By the formula of the determinant of partitioned matrices, we know that det A BC D = det( D ) det( A − BD − C ) when D is nonsingular. As for 1 ≤ i ≤ q , (cid:98) λ i is an outlier eigenvalueof S − S because (cid:98) λ i goes to infinity at the same order with λ i , which meansdet (cid:98) λ i n Z Z (cid:62) − T X X (cid:62) (cid:44) , then it follows by (3.2) that det (cid:98) λ i n Z Z (cid:62) − T X X (cid:62) − (cid:98) λ i n Z Z (cid:62) − T X X (cid:62) (cid:98) λ i n Z Z (cid:62) − T X X (cid:62) − (cid:98) λ i n Z Z (cid:62) − T X X (cid:62) = . (3.3) For λ ∈ R , defining A ( λ ) = Z (cid:62) M − ( λ ) (cid:32) n Z Z (cid:62) (cid:33) − n Z , ( λ ) = X (cid:62) M − ( λ ) (cid:32) n Z Z (cid:62) (cid:33) − λ T X , C ( λ ) = Z (cid:62) M − ( λ ) (cid:32) n Z Z (cid:62) (cid:33) − λ T X , D ( λ ) = X (cid:62) M − ( λ ) (cid:32) n Z Z (cid:62) (cid:33) − λ n Z , it holds that A ( λ ) = A ( λ ) (cid:62) , B ( λ ) = B ( λ ) (cid:62) and T C ( λ ) = n D ( λ ) (cid:62) . Then some elementarycalculations lead todet (cid:98) λ i Z (cid:110) I n − A ( (cid:98) λ i ) (cid:111) Z (cid:62) n − X (cid:110) I T + B ( (cid:98) λ i ) (cid:111) X (cid:62) T + (cid:98) λ i Z C ( (cid:98) λ i ) X (cid:62) n + (cid:98) λ i X D ( (cid:98) λ i ) Z (cid:62) T = . (3.4)To ease the notation, we define Ξ A : = (cid:98) λ i Z (cid:110) I n − A ( (cid:98) λ i ) (cid:111) Z (cid:62) n , Ξ B : = X (cid:110) I T + B ( (cid:98) λ i ) (cid:111) X (cid:62) T , Ξ C : = (cid:98) λ i Z C ( (cid:98) λ i ) X (cid:62) n , Ξ D : = (cid:98) λ i X D ( (cid:98) λ i ) Z (cid:62) T . Multiplying the matrix in (3.4) by U on the left side hand and by U (cid:62) on the right side, wehave det (cid:110) U ( Ξ A − Ξ B + Ξ C + Ξ D ) U (cid:62) (cid:111) = . (3.5)Next, we analyze these four terms in (3.5) in the following.For the term U Ξ A U (cid:62) , we first consider the decomposition1 n Z (cid:110) I n − A ( (cid:98) λ i ) (cid:111) Z (cid:62) = n Z { I n − A ( λ i ) } Z (cid:62) + n Z (cid:110) A ( (cid:98) λ i ) − A ( λ i ) (cid:111) Z (cid:62) . By Lemma 4.1 below, we have (cid:101) m λ i (1) − = O a . s . ( λ − i ), which implies1 n tr { I n − A ( λ i ) } = − p − qn (cid:101) m λ i (1) = − y p + qn + O a . s . ( λ − i ) . Note that E( Z Z (cid:62) / n ) = I q and that ( X , Z ) is independent of ( X , Z ). Under Assump-tion 2.3, by using Theorem 7.2 of [3], we have that, for all 1 ≤ j ≤ q , e (cid:62) j (cid:34) n Z { I n − A ( λ i ) } Z (cid:62) (cid:35) e j − (cid:26) − p − qn (cid:101) m λ i (1) (cid:27) = O p (cid:32) √ n (cid:33) (3.6)and E (cid:32) e (cid:62) j (cid:34) n Z { I n − A ( λ i ) } Z (cid:62) (cid:35) e j − (cid:26) − p − qn (cid:101) m λ i (1) (cid:27)(cid:33) = O (cid:32) n (cid:33) (3.7)12or all 1 ≤ j ≤ q . For those o ff -diagonal elements, we have that, for any 1 ≤ j (cid:44) j ≤ q , e (cid:62) j (cid:34) n Z { I n − A ( λ i ) } Z (cid:62) (cid:35) e j = O p (cid:32) √ n (cid:33) (3.8)and E (cid:32) e (cid:62) j (cid:34) n Z { I n − A ( λ i ) } Z (cid:62) (cid:35) e j (cid:33) = O (cid:32) n (cid:33) , (3.9)which is implied by Theorem 7.1 and Corollary 7.1 in [3]. Also we can write A ( λ i ) − A (cid:16)(cid:98) λ i (cid:17) = Z (cid:62) (cid:110) M − ( λ i ) − M − (cid:16)(cid:98) λ i (cid:17)(cid:111) (cid:32) n Z Z (cid:62) (cid:33) − n Z = Z (cid:62) M − ( λ i ) (cid:110) M (cid:16)(cid:98) λ i (cid:17) − M ( λ i ) (cid:111) M − (cid:16)(cid:98) λ i (cid:17) (cid:32) n Z Z (cid:62) (cid:33) − n Z = (cid:16) λ − i − (cid:98) λ − i (cid:17) Z (cid:62) M − ( λ i ) F M − (cid:16)(cid:98) λ i (cid:17) (cid:32) n Z Z (cid:62) (cid:33) − n Z . It can be bounded by (cid:13)(cid:13)(cid:13)(cid:13) A ( λ i ) − A (cid:16)(cid:98) λ i (cid:17)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:16) λ − i − (cid:98) λ − i (cid:17) Z (cid:62) M − ( λ i ) F M − (cid:16)(cid:98) λ i (cid:17) (cid:32) n Z Z (cid:62) (cid:33) − n Z (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:12)(cid:12)(cid:12)(cid:12) λ − i − (cid:98) λ − i (cid:12)(cid:12)(cid:12)(cid:12) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) √ n Z (cid:62) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) M − ( λ i ) (cid:13)(cid:13)(cid:13) (cid:107) F (cid:107) (cid:13)(cid:13)(cid:13)(cid:13) M − (cid:16)(cid:98) λ i (cid:17)(cid:13)(cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:32) n Z Z (cid:62) (cid:33) − (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) √ n Z (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = O( λ − i )almost surely. It follows that, for any 1 ≤ j , j ≤ q , e (cid:62) j (cid:34) n Z (cid:110) A ( (cid:98) λ i ) − A ( λ i ) (cid:111) Z (cid:62) (cid:35) e j = O a . s . ( λ − i ) . (3.10)Combining (3.6), (3.8)) and (3.10), we can get that, for any 1 ≤ j ≤ q , e (cid:62) j (cid:34) n Z (cid:110) I n − A ( (cid:98) λ i ) (cid:111) Z (cid:62) (cid:35) e j − (cid:18) − p − qn (cid:19) = O p (cid:32) √ n (cid:33) + O a . s . ( λ − i )and that, for any 1 ≤ j (cid:44) j ≤ q , e (cid:62) j (cid:34) n Z (cid:110) I n − A ( (cid:98) λ i ) (cid:111) Z (cid:62) (cid:35) e j = O p (cid:32) √ n (cid:33) + O a . s . ( λ − i ) . Replacing Z by UZ , it is easy to check that all the above conclusions still hold: e (cid:62) j (cid:34) n UZ (cid:110) I n − A ( (cid:98) λ i ) (cid:111) Z (cid:62) U (cid:62) (cid:35) e j = − p − qn + O p (cid:32) √ n (cid:33) + O a . s . ( λ − i ) (3.11)for all 1 ≤ j ≤ q , and e (cid:62) j (cid:34) n UZ (cid:110) I n − A ( (cid:98) λ i ) (cid:111) Z (cid:62) U (cid:62) (cid:35) e j = O p (cid:32) √ n (cid:33) + O a . s . ( λ − i ) (3.12)13or all 1 ≤ j (cid:44) j ≤ q . By the definition of Ξ A in (3.4), together with (3.11) and (3.12), wecan see that, for all 1 ≤ j ≤ q , e (cid:62) j U Ξ A U (cid:62) e j = (cid:98) λ i (cid:18) − p − qn (cid:19) + λ i · O p (cid:32) √ n (cid:33) + O a . s . (1) (3.13)and that, for all 1 ≤ j (cid:44) j ≤ q , e (cid:62) j U Ξ A U (cid:62) e j = λ i · O p (cid:32) √ n (cid:33) + O a . s . (1) . (3.14)For the term U Ξ B U (cid:62) , by the definition of X , we can derive that U Ξ B U (cid:62) = T Λ UY (cid:110) I T + B ( (cid:98) λ i ) (cid:111) Y (cid:62) U (cid:62) Λ = T Λ UY { I T + B ( λ i ) } Y (cid:62) U (cid:62) Λ + T Λ UY (cid:110) B ( (cid:98) λ i ) − B ( λ i ) (cid:111) Y (cid:62) U (cid:62) Λ , where 1 T tr { I T + B ( λ i ) } = T tr I T + X (cid:62) M − ( λ i ) (cid:32) n Z Z (cid:62) (cid:33) − λ i T X = + T tr M − ( λ i ) (cid:32) n Z Z (cid:62) (cid:33) − λ i T X X (cid:62) = + T tr (cid:32) I p − q − F λ i (cid:33) − F λ i = + p − qT (cid:8)(cid:101) m λ i (1) − (cid:9) and B ( (cid:98) λ i ) − B ( λ i ) = X (cid:62) (cid:110)(cid:98) λ − i M − (cid:16)(cid:98) λ i (cid:17) − λ − i M − ( λ i ) (cid:111) (cid:32) n Z Z (cid:62) (cid:33) − T X = (cid:98) λ − i λ − i X (cid:62) M − (cid:16)(cid:98) λ i (cid:17) (cid:110) λ i M ( λ i ) − (cid:98) λ i M (cid:16)(cid:98) λ i (cid:17)(cid:111) M − ( λ i ) (cid:32) n Z Z (cid:62) (cid:33) − T X = (cid:16)(cid:98) λ − i − λ − i (cid:17) X (cid:62) M − (cid:16)(cid:98) λ i (cid:17) M − ( λ i ) (cid:32) n Z Z (cid:62) (cid:33) − T X . The same arguments for deriving (3.13) and (3.14) lead to that, for all 1 ≤ j ≤ q , e (cid:62) j U Ξ B U (cid:62) e j = λ j + λ j · O p (cid:32) √ n (cid:33) + λ j · O a . s . ( λ − i ) (3.15)and that, for 1 ≤ j , j ≤ q , e (cid:62) j U Ξ B U (cid:62) e j = λ j λ j · O p (cid:32) √ n (cid:33) + λ j λ j · O a . s . ( λ − i ) (3.16)for all 1 ≤ j (cid:44) j ≤ q .For the term U ( Ξ C + Ξ D ) U (cid:62) , by using the fact that Y = U (cid:62) Λ − UX , we have that U ( Ξ C + Ξ D ) U (cid:62) U (cid:98) λ i Z C ( (cid:98) λ i ) X (cid:62) n + (cid:98) λ i X D ( (cid:98) λ i ) Z (cid:62) T U (cid:62) = U (cid:40) λ i Z C ( λ i ) X (cid:62) n + λ i X D ( λ i ) Z (cid:62) T (cid:41) U (cid:62) + UZ (cid:110)(cid:98) λ i C ( (cid:98) λ i ) − λ i C ( λ i ) (cid:111) n Y (cid:62) U (cid:62) Λ + Λ UY (cid:110)(cid:98) λ i D ( (cid:98) λ i ) − λ i D ( λ i ) (cid:111) T Z (cid:62) U (cid:62) , and that U (cid:40) λ i Z C ( λ i ) X (cid:62) n + λ i X D ( λ i ) Z (cid:62) T (cid:41) U (cid:62) = U (cid:16) Z X (cid:17) O λ i C ( λ i ) n λ i D ( λ i ) T O Z (cid:62) X (cid:62) U (cid:62) = (cid:18) UZ Λ UY (cid:19) O λ i C ( λ i ) n λ i D ( λ i ) T O Z (cid:62) U (cid:62) Y (cid:62) U (cid:62) Λ . Then we have that, for all 1 ≤ j , j ≤ q , e (cid:62) j U (cid:40) λ i Z C ( λ i ) X (cid:62) n + λ i X D ( λ i ) Z (cid:62) T (cid:41) U (cid:62) e j = e (cid:62) j (cid:18) UZ Λ UY (cid:19) O λ i C ( λ i ) n λ i D ( λ i ) T O Z (cid:62) U (cid:62) Y (cid:62) U (cid:62) Λ e j = e (cid:62) j (cid:18) UZ λ j UY (cid:19) O λ i C ( λ i ) n λ i D ( λ i ) T O Z (cid:62) U (cid:62) λ j Y (cid:62) U (cid:62) e j = λ j + λ j · e (cid:62) j (cid:16) UZ UY (cid:17) O λ i C ( λ i ) n λ i D ( λ i ) T O Z (cid:62) U (cid:62) Y (cid:62) U (cid:62) e j + λ j − λ j i · e (cid:62) j (cid:16) UZ UY (cid:17) O λ i C ( λ i ) n · i − λ i D ( λ i ) T · i O Z (cid:62) U (cid:62) Y (cid:62) U (cid:62) e j = λ j + λ j · O p (cid:32) √ n (cid:33) + λ j − λ j i · O p (cid:32) √ n (cid:33) = (cid:18) λ j + λ j (cid:19) · O p (cid:32) √ n (cid:33) , where i : = √− (cid:98) λ i C ( (cid:98) λ i ) − λ i C ( λ i ) = Z (cid:62) (cid:110) M − ( (cid:98) λ i ) − M − ( λ i ) (cid:111) (cid:32) n Z Z (cid:62) (cid:33) − T X = Z (cid:62) M − ( (cid:98) λ i ) (cid:110) M ( λ i ) − M ( (cid:98) λ i ) (cid:111) M − ( λ i ) (cid:32) n Z Z (cid:62) (cid:33) − T X = (cid:16)(cid:98) λ − i − λ − i (cid:17) Z (cid:62) M − ( (cid:98) λ i ) F M − ( λ i ) (cid:32) n Z Z (cid:62) (cid:33) − T X , (cid:98) λ i D ( (cid:98) λ i ) − λ i D ( λ i ) = X (cid:62) (cid:110) M − ( (cid:98) λ i ) − M − ( λ i ) (cid:111) (cid:32) n Z Z (cid:62) (cid:33) − n Z X (cid:62) M − ( (cid:98) λ i ) (cid:110) M ( λ i ) − M ( (cid:98) λ i ) (cid:111) M − ( λ i ) (cid:32) n Z Z (cid:62) (cid:33) − n Z = (cid:16)(cid:98) λ − i − λ − i (cid:17) X (cid:62) M − ( (cid:98) λ i ) F M − ( λ i ) (cid:32) n Z Z (cid:62) (cid:33) − n Z , we can get e (cid:62) j UZ (cid:110)(cid:98) λ i C ( (cid:98) λ i ) − λ i C ( λ i ) (cid:111) n Y (cid:62) U (cid:62) Λ e j = λ j · O a . s . (cid:16) λ − i (cid:17) and e (cid:62) j Λ UY (cid:110)(cid:98) λ i D ( (cid:98) λ i ) − λ i D ( λ i ) (cid:111) T Z (cid:62) U (cid:62) e j = λ j · O a . s . (cid:16) λ − i (cid:17) for any 1 ≤ j , j ≤ q . By using the similar arguments for proving (3.13) and (3.14), itholds that e (cid:62) j U ( Ξ C + Ξ D ) U (cid:62) e j = (cid:18) λ j + λ j (cid:19) · O p (cid:32) √ n (cid:33) + (cid:18) λ j + λ j (cid:19) · O a . s . (cid:16) λ − i (cid:17) (3.17)for any 1 ≤ j , j ≤ q .Combining (3.14)-(3.17) and the determinant (3.5), we can compute the limit of (cid:98) λ i /λ i for each 1 ≤ i ≤ q . We use a new notation to denote the matrix in the determinant (3.5).Define Ξ : = U ( Ξ A − Ξ B + Ξ C + Ξ D ) U (cid:62) , (cid:101) Ξ : = diag (cid:16) ξ , . . . , ξ qq (cid:17) , where ξ j j = (cid:98) λ i { − ( p − q ) / n } − λ j . Then by (3.14)-(3.17), we have that e (cid:62) j (cid:16) Ξ − (cid:101) Ξ (cid:17) e j = λ i · O p (cid:32) √ n (cid:33) + O p (1) + λ j λ j · O p (cid:32) √ n (cid:33) + λ j λ j · O a . s . ( λ − i ) + (cid:18) λ j + λ j (cid:19) · O p (cid:32) √ n (cid:33) + (cid:18) λ j + λ j (cid:19) · O a . s . (cid:16) λ − i (cid:17) = (cid:18) λ i + λ j λ j (cid:19) (cid:40) O p (cid:32) √ n (cid:33) + O a . s . (cid:16) λ − i (cid:17)(cid:41) for any 1 ≤ j , j ≤ q , which follows that e (cid:62) j λ − i (cid:16) Ξ − (cid:101) Ξ (cid:17) e j = (cid:18) + λ − i λ j λ j (cid:19) (cid:40) O p (cid:32) √ n (cid:33) + O a . s . (cid:16) λ − i (cid:17)(cid:41) . (3.18)According to (3.7) and (3.9) for Ξ A (similar results also hold for Ξ B , Ξ C and Ξ D ), it canbe easily checked that the variance of the term in (3.18) has the order (cid:18) + λ − i λ j λ j (cid:19) (cid:18) n − + λ − i (cid:19) . By Chebyshev’s inequality, we have that, for any (cid:15) > (cid:40) max ≤ j , j ≤ q (cid:12)(cid:12)(cid:12)(cid:12) e (cid:62) j λ − i (cid:16) Ξ − (cid:101) Ξ (cid:17) e j (cid:12)(cid:12)(cid:12)(cid:12) ≥ (cid:15) (cid:18) n − + λ − i (cid:19)(cid:41) (cid:88) ≤ j , j ≤ q Pr (cid:26)(cid:12)(cid:12)(cid:12)(cid:12) e (cid:62) j λ − i (cid:16) Ξ − (cid:101) Ξ (cid:17) e j (cid:12)(cid:12)(cid:12)(cid:12) ≥ (cid:15) (cid:18) n − + λ − i (cid:19)(cid:27) ≤ (cid:88) ≤ j , j ≤ q E (cid:110) e (cid:62) j λ − i (cid:16) Ξ − (cid:101) Ξ (cid:17) e j (cid:111) (cid:15) (cid:16) n − + λ − i (cid:17) = (cid:88) ≤ j , j ≤ q (cid:18) + λ − i λ j λ j (cid:19) · O (cid:16) (cid:15) − (cid:17) = q + λ − i q (cid:88) j = λ j · O (cid:16) (cid:15) − (cid:17) = κ · O (cid:16) (cid:15) − (cid:17) , which means (cid:13)(cid:13)(cid:13)(cid:13) λ − i (cid:16) Ξ − (cid:101) Ξ (cid:17)(cid:13)(cid:13)(cid:13)(cid:13) ∞ = max ≤ j , j ≤ q (cid:12)(cid:12)(cid:12)(cid:12) e (cid:62) j λ − i (cid:16) Ξ − (cid:101) Ξ (cid:17) e j (cid:12)(cid:12)(cid:12)(cid:12) = κ · O p (cid:32) √ n + λ − i (cid:33) and then ||| λ − i ( Ξ − (cid:101) Ξ ) ||| ∞ ≤ q (cid:107) λ − i ( Ξ − (cid:101) Ξ ) (cid:107) ∞ = κ q · O p (cid:32) √ n + λ − i (cid:33) . Note that the determinant equation det (cid:0)(cid:101) Ξ (cid:1) = (cid:16) λ − i (cid:101) Ξ (cid:17) =
0, that is,det (cid:98) λ i λ i (cid:18) − p − qn (cid:19) I q − λ − i Λ = . At the same time, the equation det (cid:0) Ξ (cid:1) = (cid:16) λ − i Ξ (cid:17) =
0, that is,det (cid:98) λ i λ i (cid:18) − p − qn (cid:19) I q − λ − i Λ + λ − i (cid:16) Ξ − (cid:101) Ξ (cid:17) = . By eigenvalue perturbation theorems (see Theorem 6.3.2 in Chapter 6, [9]), we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:98) λ i λ i (cid:18) − p − qn (cid:19) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ||| λ − i ( Ξ − (cid:101) Ξ ) ||| ∞ = κ q · O p (cid:32) √ n + λ − i (cid:33) , that is (cid:98) λ i λ i = − y + O (cid:16) y p − y (cid:17) + κ q · O p (cid:32) √ n + λ − i (cid:33) . (3.19)Instead, we can compare determinant equationsdet (cid:18) Λ − (cid:101) ΞΛ − (cid:19) = (cid:18) Λ − ΞΛ − (cid:19) = , and then repeat all the derivations above to achieve an upper bound of ||| Λ − / ( Ξ − (cid:101) Ξ ) Λ − / ||| ∞ .In this case, we can get (cid:98) λ i λ i = − y + O (cid:16) y p − y (cid:17) + κ q · O p (cid:32) √ n + λ − i (cid:33) . (3.20)Thus, (3.19) and (3.20) lead to (cid:98) λ i λ i = − y + O (cid:16) y p − y (cid:17) + κ q · O p (cid:32) √ n + λ − i (cid:33) , κ q ( n − / + λ − i ) = o(1) under Assumption 2.2. The proof is finished. (cid:3) Proof of Theorem 2.10.
We begin with the equation on (cid:98) λ i in (3.4). Recall that we haveexpressed (3.4) as det (cid:0) Ξ A − Ξ B + Ξ C + Ξ D (cid:1) = . (3.21)For the first term Ξ A , we can write Ξ A = (cid:98) λ i n Z (cid:104)(cid:110) I n − A ( (cid:98) λ i ) (cid:111) − { I n − A ( θ i ) } (cid:105) Z (cid:62) + (cid:98) λ i n (cid:16) Z { I n − A ( θ i ) } Z (cid:62) − E (cid:104) Z { I n − A ( θ i ) } Z (cid:62) (cid:105)(cid:17) + (cid:98) λ i n E (cid:104) Z { I n − A ( θ i ) } Z (cid:62) (cid:105) . Using the fact (cid:110) I n − A ( (cid:98) λ i ) (cid:111) − { I n − A ( θ i ) } = − δ i A ( θ i ) + δ i Z (cid:62) M − ( (cid:98) λ i ) M − ( θ i ) (cid:32) n Z Z (cid:62) (cid:33) − n Z and (cid:98) λ i = θ i (1 + δ i ) by (2.7), we can get Ξ A = θ i δ i (1 + δ i ) 1 n (cid:16) Z { I n − A ( θ i ) } Z (cid:62) − E (cid:104) Z { I n − A ( θ i ) } Z (cid:62) (cid:105)(cid:17) + θ i δ i (1 + δ i ) 1 n Z Z (cid:62) M − ( (cid:98) λ i ) M − ( θ i ) (cid:32) n Z Z (cid:62) (cid:33) − n Z Z (cid:62) − θ i δ i (1 + δ i ) 1 n Z Z (cid:62) + θ i δ i (1 + δ i ) 1 n E (cid:104) Z { I n − A ( θ i ) } Z (cid:62) (cid:105) + θ i (1 + δ i ) 1 n (cid:16) Z { I n − A ( θ i ) } Z (cid:62) − E (cid:104) Z { I n − A ( θ i ) } Z (cid:62) (cid:105)(cid:17) + θ i (1 + δ i ) 1 n E (cid:104) Z { I n − A ( θ i ) } Z (cid:62) (cid:105) = θ i (1 + δ i ) n (cid:16) Z { I n − A ( θ i ) } Z (cid:62) − E (cid:104) Z { I n − A ( θ i ) } Z (cid:62) (cid:105)(cid:17) + θ i δ i (1 + δ i ) 1 n Z Z (cid:62) M − ( (cid:98) λ i ) M − ( θ i ) (cid:32) n Z Z (cid:62) (cid:33) − n Z Z (cid:62) − θ i δ i (1 + δ i ) 1 n Z Z (cid:62) + θ i (1 + δ i ) n E (cid:104) Z { I n − A ( θ i ) } Z (cid:62) (cid:105) = : θ i (1 + δ i ) Ξ A + θ i δ i (1 + δ i ) Ξ A − θ i δ i (1 + δ i ) Ξ A + θ i (1 + δ i ) Ξ A . (3.22)For the second term Ξ B , we can similarly write Ξ B = T X (cid:110) B ( (cid:98) λ i ) − B ( θ i ) (cid:111) X (cid:62) + T (cid:16) X { I T + B ( θ i ) } X (cid:62) − E (cid:104) X { I T + B ( θ i ) } X (cid:62) (cid:105)(cid:17) + T E (cid:104) X { I T + B ( θ i ) } X (cid:62) (cid:105) = T (cid:16) X { I T + B ( θ i ) } X (cid:62) − E (cid:104) X { I T + B ( θ i ) } X (cid:62) (cid:105)(cid:17) − δ i (cid:98) λ i · T X X (cid:62) M − ( (cid:98) λ i ) M − ( θ i ) (cid:32) n Z Z (cid:62) (cid:33) − T X X (cid:62) + T E (cid:104) X { I T + B ( θ i ) } X (cid:62) (cid:105) : Ξ B − δ i (cid:98) λ i Ξ B + Ξ B , (3.23)where the second equality above uses the fact B ( (cid:98) λ i ) − B ( θ i ) = − δ i (cid:98) λ i X (cid:62) M − ( (cid:98) λ i ) M − ( θ i ) (cid:32) n Z Z (cid:62) (cid:33) − T X . For the term Ξ C , we have Ξ C = (cid:98) λ i n Z (cid:110) C ( (cid:98) λ i ) − C ( θ i ) (cid:111) X (cid:62) + (cid:98) λ i − θ i n Z C ( θ i ) X (cid:62) + θ i n (cid:104) Z C ( θ i ) X (cid:62) − E (cid:110) Z C ( θ i ) X (cid:62) (cid:111)(cid:105) . Using the fact C ( (cid:98) λ i ) − C ( θ i ) = − δ i (cid:98) λ i Z (cid:62) M − ( (cid:98) λ i ) M − ( θ i ) (cid:32) n Z Z (cid:62) (cid:33) − T X , we have the decomposition Ξ C = θ i n (cid:104) Z C ( θ i ) X (cid:62) − E (cid:110) Z C ( θ i ) X (cid:62) (cid:111)(cid:105) − δ i n Z Z (cid:62) M − ( (cid:98) λ i ) M − ( θ i ) (cid:32) n Z Z (cid:62) (cid:33) − T X X (cid:62) + θ i δ i n Z C ( θ i ) X (cid:62) = : θ i Ξ C − δ i Ξ C + θ i δ i Ξ C . (3.24)Similarly, we can write the last term Ξ D as Ξ D = θ i T (cid:104) X D ( θ i ) Z (cid:62) − E (cid:110) X D ( θ i ) Z (cid:62) (cid:111)(cid:105) − δ i T X X (cid:62) M − ( (cid:98) λ i ) M − ( θ i ) (cid:32) n Z Z (cid:62) (cid:33) − n Z Z (cid:62) + θ i δ i T X D ( θ i ) Z (cid:62) = : θ i Ξ D − δ i Ξ D + θ i δ i Ξ D . (3.25)Putting (3.22)-(3.25) into (3.21), we havedet( θ i Θ n + θ i δ i Θ n + θ i Θ n ) = , (3.26)where Θ n : = (1 + δ i ) Ξ A − θ − i Ξ B + Ξ C + Ξ D , (3.27) Θ n : = (1 + δ i ) Ξ A − (1 + δ i ) Ξ A + θ i (cid:98) λ i Ξ B − θ − i Ξ C + Ξ C − θ − i Ξ D + Ξ D , (3.28) Θ n : = (1 + δ i ) Ξ A − θ − i Ξ B . (3.29)Multiplying both sides of the matrix in (3.26) by θ − / i U from the left hand side and θ − / i U (cid:62) from the right hand side, we getdet (cid:110) U ( Θ n + δ i Θ n + Θ n ) U (cid:62) (cid:111) = . (3.30)19ecall that e i is the q -dimensional vector whose i -th element is 1 and others are 0. ByLemma 4.2 below, we have √ p (cid:98) S i : = √ p e (cid:62) i U Θ n U (cid:62) e i d −→ N (0 , (cid:101) σ i ) , (3.31)where (cid:101) σ i = ( y + c ) (1 − y ) ν i − y (1 − y ) (1 − y ) − c (1 − y ) . It follows by Lemma 4.3below that (cid:107) U Θ n U (cid:62) (cid:107) ∞ = O p (cid:32) q √ n + (cid:80) j λ j √ n λ i (cid:33) . (3.32)By Lemma 4.4 below, we also havemax ≤ j ≤ q (cid:12)(cid:12)(cid:12) e (cid:62) j U Θ n U (cid:62) e j − ( y − (cid:12)(cid:12)(cid:12) = O p √ q δ i λ i + √ q √ n + (cid:113)(cid:80) j λ j λ i + (cid:112)(cid:80) j λ j λ i , (3.33)max ≤ j (cid:44) j ≤ q (cid:12)(cid:12)(cid:12) e (cid:62) j U Θ n U (cid:62) e j (cid:12)(cid:12)(cid:12) = O p q δ i λ i + q √ n + (cid:80) j λ j λ i + (cid:112) q (cid:80) j λ j λ i . (3.34)For the term U Θ n U (cid:62) in (3.30), by considering its ( j , j ) entry for all 1 ≤ j , j ≤ q , wecan easily get that (1 + δ i ) U Ξ A U (cid:62) = (1 + δ i ) (cid:20) − p − qn E (cid:8)(cid:101) m θ i (1) (cid:9)(cid:21) I q , (3.35) U Ξ B U (cid:62) = (cid:18) + p − qT (cid:2) − + E (cid:8)(cid:101) m θ i (1) (cid:9)(cid:3)(cid:19) Λ . (3.36)By the definition of θ i in (2.6), we know1 − p − qn E (cid:8)(cid:101) m θ i (1) (cid:9) = λ i θ i (cid:18) + p − qT (cid:2) − + E (cid:8)(cid:101) m θ i (1) (cid:9)(cid:3)(cid:19) , which, together with the results in Lemma 4.1 below and Theorem 2.6, yields that(1 + δ i ) (cid:20) − p − qn E (cid:8)(cid:101) m θ i (1) (cid:9)(cid:21) − λ i θ i (cid:18) + p − qT (cid:2) − + E (cid:8)(cid:101) m θ i (1) (cid:9)(cid:3)(cid:19) = δ i (cid:20) − p − qn E (cid:8)(cid:101) m θ i (1) (cid:9)(cid:21) + δ i (cid:20) − p − qn E (cid:8)(cid:101) m θ i (1) (cid:9)(cid:21) = δ i (cid:26) − p − qn + o(1) (cid:27) . (3.37)Combining (3.35)-(3.37) and the definition of Θ n in (3.29), we can get that, for 1 ≤ j ≤ q , e (cid:62) j U Θ n U (cid:62) e (cid:62) j = (cid:40) (1 + δ i ) − λ j λ i (cid:41) (cid:20) − p − qn E (cid:8)(cid:101) m θ i (1) (cid:9)(cid:21) , which converges to zero if and only if λ j = λ i because (1 + δ i ) − λ j /λ i > C > C if λ j (cid:44) λ i under Assumption 2.4. When λ j = λ i , we have e (cid:62) j U Θ n U (cid:62) e (cid:62) j = δ i (cid:26) − p − qn + o(1) (cid:27) . (3.38)Note that all o ff -diagonal entries of the matrix U Θ n U (cid:62) is zero, i.e. e (cid:62) j U Θ n U (cid:62) e j = , ∀ ≤ j (cid:44) j ≤ q . (3.39)20nserting (3.31), (3.32), (3.33), (3.34), (3.38) and (3.39) into (3.30), we can solve thedeterminant equation (3.30) and get the limiting distribution of δ i (1 ≤ i ≤ q ) immediately.Since diagonal elements of U Θ n U (cid:62) are at least constant order, when e (cid:62) j U Θ n U (cid:62) e (cid:62) j goes toinfinity for some j ’s, we can divide these rows by e (cid:62) j U Θ n U (cid:62) e (cid:62) j . In this way, we can getdet O p (1) . . . O p ( ∗ ) . . . O p ( ∗ ) ... . . . ... . . . ... O p ( ∗ ) . . . (cid:98) S i + (1 − y + o p (1)) δ i . . . O p ( ∗ ) ... . . . ... . . . ... O p ( ∗ ) . . . O p ( ∗ ) . . . O p (1) = √ p (cid:98) S i d −→ N (0 , (cid:101) σ i ) and ∗ = q √ n + (cid:80) j λ j √ n λ i + q δ i λ i + δ i (cid:80) j λ j λ i + δ i (cid:112) q (cid:80) j λ j λ i . By Leibniz formula for determinants, we can get that (cid:98) S i + (cid:110) − y + o p (1) (cid:111) δ i + q O p (cid:16) ∗ (cid:17) = (cid:98) S i + (cid:110) − y + o p (1) (cid:111) δ i + O p q n + q ( (cid:80) j λ j ) n λ i + q δ i λ i + q δ i ( (cid:80) j λ j ) λ i + q δ i (cid:80) j λ j λ i = . Under Assumptions 2.1 and 2.2(a), we have q = o( n ) and λ − i (cid:80) j λ j = o( q − n ), then itfollows that q n = o( n − ) , q (cid:80) j λ j n λ i = o( n − ) , q δ i λ i = o p ( δ i n ) , q δ i ( (cid:80) j λ j ) λ i = o p ( δ i n ) , q δ i (cid:80) j λ j λ i = o p ( δ i n ) . It leads to (cid:98) S i + (cid:110) − y + o p (1) (cid:111) δ i + o p ( δ i n ) + o( n − ) = . By multiplying √ p on both sides, we further obtain that √ p (cid:98) S i + (cid:110) − y + o p (1) (cid:111) · √ p δ i + o p (1) · p δ i + o (1) = . Recalling that √ p (cid:98) S i d −→ N (0 , (cid:101) σ i ), we can reach to √ p δ i d −→ N (0 , σ i ), where σ i = (cid:101) σ i (1 − y ) = ( y + c ) ν i − c − y (1 − y )1 − y . Instead, we can consider the determinantdet (cid:26)(cid:101) Λ − U ( θ i Θ n + θ i δ i Θ n + θ i Θ n ) U (cid:62) (cid:101) Λ − (cid:27) = , (3.40)21here (cid:101) Λ = diag( θ , . . . , θ q ) ∈ R q × q . Repeating all the derivations above, we can get (cid:107) (cid:101) Λ − U θ i Θ n U (cid:62) (cid:101) Λ − (cid:107) ∞ = O p q √ n + λ i (cid:80) j λ − j √ n , (3.41)max ≤ j ≤ q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e (cid:62) j (cid:101) Λ − θ i U Θ n U (cid:62) (cid:101) Λ − e j − ( y − θ i θ j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O p δ i (cid:115)(cid:88) j λ − j + λ i (cid:113)(cid:80) j λ − j √ n + √ q λ i + (cid:115)(cid:88) j λ − j , (3.42)max ≤ j (cid:44) j ≤ q (cid:12)(cid:12)(cid:12)(cid:12) e (cid:62) j (cid:101) Λ − θ i U Θ n U (cid:62) (cid:101) Λ − e j (cid:12)(cid:12)(cid:12)(cid:12) = O p δ i (cid:88) j λ − j + λ i (cid:80) j λ − j √ n + q λ i + (cid:115) q (cid:88) j λ − j , (3.43) e (cid:62) j (cid:101) Λ − θ i U Θ n U (cid:62) (cid:101) Λ − e (cid:62) j = (cid:40) (1 + δ i ) − λ j λ i (cid:41) (cid:20) − p − qn E (cid:8)(cid:101) m θ i (1) (cid:9)(cid:21) , (3.44) e (cid:62) j (cid:101) Λ − θ i U Θ n U (cid:62) (cid:101) Λ − e j = , ∀ ≤ j (cid:44) j ≤ q . (3.45)Inserting (3.41)-(3.45) into (3.40), we can similarly prove √ p δ i d −→ N (0 , σ i ) under As-sumption 2.2(b). Thus the proof is completed. (cid:3) Proof of Theorem 2.12.
The proof of Theorem 2.12 is similar to that of Theorem 2.10, theonly di ff erence is that we take the J i × J i block as a typical object to analyse, some usefullemmas can also be obtained from Lemmas 4.2-4.4 below. Similar arguments for derivingthe proof of Theorem 4.1 in [17] can be used. Thus, we omit the details. (cid:3) Lemma 4.1.
Suppose that Assumptions 2.1 and 2.3 hold. For any θ → ∞ , we have (cid:101) m θ (1) − = O a . s . ( θ − ). Proof of lemma 4.1.
By the definition of (cid:101) m θ ( z ) in (2.2), (cid:101) m θ (1) = p − q tr (cid:32) I p − q − F θ (cid:33) − = + p − q tr F θ (cid:32) I p − q − F θ (cid:33) − , we have (cid:101) m θ (1) − = p − q tr F θ (cid:32) I p − q − F θ (cid:33) − = θ − p − q (cid:88) ≤ j ≤ p − q µ j − µ j /θ . Since all the eigenvalues of F , namely µ ≥ . . . ≥ µ p − q , are almost surely bounded, we canget that (cid:101) m θ (1) − = O a . s . ( θ − ). (cid:3) Recall that e i is the q -dimensional vector whose i -th element is 1 and others are 0, U (cid:62) = ( u , u , . . . , u q ), where u i ∈ R q is the i − th column of the matrix U (cid:62) . Then we get thefollowing lemma. 22 emma 4.2. For any fixed 1 ≤ i ≤ q , denote G ni = √ p U Θ n U (cid:62) . Under the assumptionsof Theorem 2.10, we have e (cid:62) i G ni e i d −→ N (0 , (cid:101) σ i ) , where (cid:101) σ i = ( y + c ) (1 − y ) ν i − y (1 − y ) (1 − y ) − c (1 − y ) and ν i = E | u (cid:62) i Z e | for 1 ≤ i ≤ q . Proof of Lemma 4.2.
From the definition of Θ n in (3.27) and the fact that Y = Σ − X = U (cid:62) Λ − UX , we have the decomposition e (cid:62) i G ni e i = u (cid:62) i (cid:104) (1 + δ i ) √ pn Z { I n − A ( θ i ) } Z (cid:62) − λ i θ i √ pT Y { I T + B ( θ i ) } Y (cid:62) + √ p λ i n Z C ( θ i ) Y (cid:62) + √ p λ i T Y D ( θ i ) Z (cid:62) (cid:105) u i − E( · ) , (4.1)where E[ · ] is the expectation of all the preceding terms after the equal sign.By Theorem 2.6, δ i converges in probability to 0, thus we only need to consider thelimit of e (cid:62) i (cid:101) G ni e i : = u (cid:62) i (cid:104) √ pn Z { I n − A ( θ i ) } Z (cid:62) − λ i θ i √ pT Y { I T + B ( θ i ) } Y (cid:62) + √ p λ i n Z C ( θ i ) Y (cid:62) + √ p λ i T Y D ( θ i ) Z (cid:62) (cid:105) u i − E[ · ] . For the first two terms, Theorem 7.2 in [3] implies that, for any 1 ≤ i ≤ q ,1 √ n (cid:104) u (cid:62) i Z { I n − A ( θ i ) } Z (cid:62) u i − tr { I n − A ( θ i ) } (cid:105) d −→ N (0 , (cid:101) σ i A ) , √ T (cid:104) u (cid:62) i Y { I T + B ( θ i ) } Y (cid:62) u i − tr { I T + B ( θ i ) } (cid:105) d −→ N (0 , (cid:101) σ i B ) , with (cid:101) σ iA = ω I n − A ( θ i ) ( ν i − + β I n − A ( θ i ) and (cid:101) σ iB = ω I T + B ( θ i ) ( ν i − + β I T + B ( θ i ) , where ν i = E | u (cid:62) i Z e | = E | u (cid:62) i Y e | ,ω I n − A ( θ i ) = lim n →∞ n (cid:88) ≤ k ≤ n [ { I n − A ( θ i ) } ( k , k )] ,β I n − A ( θ i ) = lim n →∞ n tr { I n − A ( θ i ) } ,ω I T + B ( θ i ) and β I T + B ( θ i ) are similarly defined. Here the fact that E | u (cid:62) i Z e | = E | u (cid:62) i Y e | isimplied by Assumption 2.3. Based on the facts thatE (cid:104) u (cid:62) i Z { I n − A ( θ i ) } Z (cid:62) u i (cid:105) = E (cid:16) tr (cid:104) Z (cid:62) u i u (cid:62) i Z { I n − A ( θ i ) } (cid:105)(cid:17) = tr (cid:104) E (cid:16) Z (cid:62) u i u (cid:62) i Z (cid:17) E { I n − A ( θ i ) } (cid:105) = E [tr { I n − A ( θ i ) } ] = n − ( p − q )E (cid:8)(cid:101) m θ i (1) (cid:9) , and that (cid:101) m θ i (1) − E (cid:8)(cid:101) m θ i (1) (cid:9) = O p ( n − ), we can get that1 √ n (cid:16) E (cid:104) u (cid:62) i Z { I n − A ( θ i ) } Z (cid:62) u (cid:62) i (cid:105) − tr { I n − A ( θ i ) } (cid:17) = o p (1) . √ n (cid:16) u (cid:62) i Z { I n − A ( θ i ) } Z (cid:62) u i − E (cid:104) u (cid:62) i Z { I n − A ( θ i ) } Z (cid:62) u (cid:62) i (cid:105)(cid:17) d −→ N (0 , (cid:101) σ i A ) , and similarly,1 √ T (cid:16) u (cid:62) i Y { I T + B ( θ i ) } Y (cid:62) u i − E (cid:104) u (cid:62) i Y { I T + B ( θ i ) } Y (cid:62) u i (cid:105)(cid:17) d −→ N (0 , (cid:101) σ i B ) . For other two terms, by the same approach in the proof of Theorem 2.6, we have that u (cid:62) i (cid:40) √ p λ i n Z C ( θ i ) Y (cid:62) + √ p λ i T Y D ( θ i ) Z (cid:62) (cid:41) u i = O p (cid:32) √ λ i (cid:33) . By all these arguments above, we can derive that e (cid:62) i G ni e i d −→ N (0 , (cid:101) σ i ) with (cid:101) σ i = y σ i A + c (1 − y ) σ i B .We compute ω I n − A ( θ i ) , β I n − A ( θ i ) , ω I T + B ( θ i ) and β I T + B ( θ i ) in the following. By the deriva-tions in the proof of Lemma 6 in [17], { I n − A ( θ i ) } ( k , k ) = − Z (cid:62) M ( θ i ) − (cid:32) n Z Z (cid:62) (cid:33) − n Z ( k , k ) = − θ i n Z (cid:62) (cid:32) θ i · n Z Z (cid:62) − T X X (cid:62) (cid:33) − Z ( k , k ) = + θ i n (cid:26) η (cid:62) k (cid:16) θ i n Z k Z (cid:62) k − T X X (cid:62) (cid:17) − η k (cid:27) , where η k is the k -th column of Z and Z k is defined by removing the k -th column of Z .Note that (cid:32) n Z k Z (cid:62) k − θ i T X X (cid:62) (cid:33) − − (cid:32) n Z k Z (cid:62) k (cid:33) − = (cid:32) n Z i Z (cid:62) i − θ i T X X (cid:62) (cid:33) − (cid:40) n Z k Z (cid:62) k − (cid:32) n Z k Z (cid:62) k − θ i T X X (cid:62) (cid:33)(cid:41) (cid:32) n Z k Z (cid:62) k (cid:33) − = θ − i (cid:32) n Z i Z (cid:62) i − θ i T X X (cid:62) (cid:33) − (cid:32) T X X (cid:62) (cid:33) (cid:32) n Z k Z (cid:62) k (cid:33) − (4.2)and 1 p − q − (cid:32) n Z k Z (cid:62) k (cid:33) − = S MP (0) + O p (cid:16) p − (cid:17) = − y + O p (cid:16) p − (cid:17) , (4.3)where S MP denotes the Stieltjes transform of the Marcenko-Pastur law. Then we have that1 p − q θ i E tr (cid:32) θ i n Z k Z (cid:62) k − T X X (cid:62) (cid:33) − = E p − q tr (cid:32) n Z k Z (cid:62) k (cid:33) − + O a . s . (cid:16) θ − i (cid:17) = E (cid:40) − y + O a . s . (cid:16) θ − i (cid:17) + O p (cid:16) p − (cid:17)(cid:41) → − y .
24y Lemma A.2. in [17], it holds that { I n − A ( θ i ) } ( k , k ) → + y (1 − y ) − = − y , which implies ω I n − A ( θ i ) = lim n →∞ n (cid:88) ≤ k ≤ n [ { I n − A ( θ i ) } ( k , k )] = (1 − y ) . By the similar argument, we can obtain that ω I T + B ( θ i ) = lim T →∞ T (cid:88) ≤ k ≤ T [ { I T + B ( θ i ) } ( k , k )] = . Now we come to the calculation of β I n − A ( θ i ) and β I T + B ( θ i ) . Since θ i → + ∞ as n goes toinfinity, we havelim n →∞ (cid:90) ∞−∞ θ i θ i − x dF n ( x ) = , lim n →∞ (cid:90) ∞−∞ θ i ( θ i − x ) dF n ( x ) = , lim T →∞ (cid:90) ∞−∞ x θ i − x dF n ( x ) = , lim T →∞ (cid:90) ∞−∞ x ( θ i − x ) dF n ( x ) = . Then these calculations lead to β I n − A ( θ i ) = lim n →∞ n tr { I n − A ( θ i ) } = lim n →∞ n tr (cid:110) I n − A ( θ i ) + A ( θ i ) (cid:111) = − n →∞ (cid:32) p − qn (cid:90) ∞−∞ θ i θ i − x dF n ( x ) (cid:33) + lim n →∞ p − qn (cid:90) ∞−∞ θ i ( θ i − x ) dF n ( x ) = − y + y = − y ,β I T + B ( θ i ) = lim T →∞ T tr { I T + B ( θ i ) } = lim T →∞ T tr (cid:110) I T + B ( θ i ) + B ( θ i ) (cid:111) = + T →∞ (cid:40) p − qT (cid:90) ∞−∞ x θ i − x dF n ( x ) (cid:41) + lim T →∞ (cid:40) p − qT (cid:90) ∞−∞ x ( θ i − x ) dF n ( x ) (cid:41) = + + = . Thus, we can write (cid:101) σ i = y (cid:101) σ i A + c (1 − y ) (cid:101) σ i B = y { ω I n − A ( θ i ) ( ν i − + β I n − A ( θ i ) } + c (1 − y ) { ω I T + B ( θ i ) ( ν i − + β I T + B ( θ i ) } = ( y + c ) (1 − y ) ν i − y (1 − y ) (1 − y ) − c (1 − y ) . Thus the proof is completed. (cid:3)
Lemma 4.3.
Under the assumptions of Theorem 2.10, (cid:107) U Θ n U (cid:62) (cid:107) ∞ = O p (cid:32) q √ n + (cid:80) j λ j √ n λ i (cid:33) . roof of Lemma 4.3. By the definition of Θ n in (3.27) again, we know Θ n = (1 + δ i ) n Z { I n − A ( θ i ) } Z (cid:62) − θ − i T X { I T + B ( θ i ) } X (cid:62) + n Z C ( θ i ) X (cid:62) + T X D ( θ i ) Z (cid:62) − E( · ) , (4.4)where E( · ) is the expectation of all the preceding terms.Denote η n = n Z { I n − A ( θ i ) } Z (cid:62) − E (cid:34) n Z { I n − A ( θ i ) } Z (cid:62) (cid:35) ,η n = T Y { I T + B ( θ i ) } Y (cid:62) − E (cid:34) T Y { I T + B ( θ i ) } Y (cid:62) (cid:35) ,η n = (cid:112) θ i n Z C ( θ i ) Y (cid:62) − E (cid:40) (cid:112) θ i n Z C ( θ i ) Y (cid:62) (cid:41) ,η n = (cid:112) θ i T Y D ( θ i ) Z (cid:62) − E (cid:40) (cid:112) θ i T Y D ( θ i ) Z (cid:62) (cid:41) . By the fact X = U (cid:62) Λ UY , we can write U Θ n U (cid:62) : = (cid:88) i = V ni , (4.5)where V n = (1 + δ i ) U (cid:32) n Z { I n − A ( θ i ) } Z (cid:62) − E (cid:34) n Z { I n − A ( θ i ) } Z (cid:62) (cid:35)(cid:33) U (cid:62) = (1 + δ i ) U η n U (cid:62) , (4.6) V n = − θ − i U (cid:32) T X { I T + B ( θ i ) } X (cid:62) − E (cid:34) T X { I T + B ( θ i ) } X (cid:62) (cid:35)(cid:33) U (cid:62) = − θ − i Λ U (cid:32) T Y { I T + B ( θ i ) } Y (cid:62) − E (cid:34) T Y { I T + B ( θ i ) } Y (cid:62) (cid:35)(cid:33) U (cid:62) Λ = − θ − i Λ U η n U (cid:62) Λ , (4.7) V n = U (cid:34) n Z C ( θ i ) X (cid:62) − E (cid:40) n Z C ( θ i ) X (cid:62) (cid:41)(cid:35) U (cid:62) = U (cid:34) n Z C ( θ i ) Y (cid:62) − E (cid:40) n Z C ( θ i ) Y (cid:62) (cid:41)(cid:35) U (cid:62) Λ = U (cid:34) n Z C ( θ i ) Y (cid:62) − E (cid:40) n Z C ( θ i ) Y (cid:62) (cid:41)(cid:35) U (cid:62) Λ = θ − i U η n U (cid:62) Λ (4.8) V n = U (cid:34) T X D ( θ i ) Z (cid:62) − E (cid:40) T X D ( θ i ) Z (cid:62) (cid:41)(cid:35) U (cid:62) = θ − i Λ U η n U (cid:62) . (4.9)Similarly as the arguments in the proof of Lemma 4.2, it holds that, for 1 ≤ j , j ≤ q , e (cid:62) j η n e j = O p (cid:32) √ n (cid:33) , e (cid:62) j η n e j = O p (cid:32) √ n (cid:33) , (cid:62) j η n e j = O p (cid:32) √ n λ i (cid:33) , e (cid:62) j η n e j = O p (cid:32) √ n λ i (cid:33) . Noting that U is an orthogonal matrix, we have that e (cid:62) j V n e j = e (cid:62) j (1 + δ i ) U η n U (cid:62) e j = O p (cid:32) √ n (cid:33) , e (cid:62) j V n e j = − e (cid:62) j θ − i Λ U η n U (cid:62) Λ e j = λ − i λ j λ j · O p (cid:32) √ n (cid:33) , e (cid:62) j V n e j = e (cid:62) j θ − i U η n U (cid:62) Λ e j = λ − i λ j · O p (cid:32) √ n (cid:33) , e (cid:62) j V n e j = e (cid:62) j θ − i Λ U η n U (cid:62) e j = λ − i λ j · O p (cid:32) √ n (cid:33) . Then by Chebyshev’s inequality, we can deduce that (cid:107) V n (cid:107) ∞ = O p (cid:32) q √ n (cid:33) , (cid:107) V n (cid:107) ∞ = O p (cid:32) (cid:80) j λ j √ n λ i (cid:33) , (cid:107) V n + V n (cid:107) ∞ = O p (cid:112) q (cid:80) j λ j √ n λ i , where (cid:112) q (cid:80) j λ j = o( (cid:80) j λ j ). Thus we complete the proof by (4.5). (cid:3) Lemma 4.4.
Under the assumptions of Theorem 2.10,max ≤ j ≤ q (cid:12)(cid:12)(cid:12) e (cid:62) j U Θ n U (cid:62) e j − ( y − (cid:12)(cid:12)(cid:12) = O p √ q δ i λ i + √ q √ n + (cid:113)(cid:80) j λ j λ i + (cid:112)(cid:80) j λ j λ i , (4.10)max ≤ j (cid:44) j ≤ q | e (cid:62) j U Θ n U (cid:62) e j | = O p q δ i λ i + q √ n + (cid:80) j λ j λ i + (cid:112) q (cid:80) j λ j λ i . (4.11) Proof of Lemma 4.4.
Recall the definition of Θ n in (3.28): Θ n = (1 + δ i ) 1 n Z Z (cid:62) M − ( (cid:98) λ i ) M − ( θ i ) (cid:32) n Z Z (cid:62) (cid:33) − n Z Z (cid:62) − (1 + δ i ) 1 n Z Z (cid:62) + (cid:98) λ i θ i T X X (cid:62) M − ( (cid:98) λ i ) M − ( θ i ) (cid:32) n Z Z (cid:62) (cid:33) − T X X (cid:62) − (1 + δ i ) (cid:98) λ i n Z Z (cid:62) M − ( (cid:98) λ i ) M − ( θ i ) (cid:32) n Z Z (cid:62) (cid:33) − T X X (cid:62) + n Z C ( θ i ) X (cid:62) − (1 + δ i ) (cid:98) λ i T X X (cid:62) M − ( (cid:98) λ i ) M − ( θ i ) (cid:32) n Z Z (cid:62) (cid:33) − n Z Z (cid:62) + T X D ( θ i ) Z (cid:62) . Noting that M − ( (cid:98) λ i ) − M − ( θ i ) = M − ( (cid:98) λ i ) (cid:110) M ( θ i ) − M ( (cid:98) λ i ) (cid:111) M − ( θ i ) = − δ i (cid:98) λ i M − ( (cid:98) λ i ) F M − ( θ i ) , we decompose the first term in Θ n as1 n Z Z (cid:62) M − ( (cid:98) λ i ) M − ( θ i ) (cid:32) n Z Z (cid:62) (cid:33) − n Z Z (cid:62) n Z Z (cid:62) (cid:110) M − ( (cid:98) λ i ) − M − ( θ i ) (cid:111) M − ( θ i ) (cid:32) n Z Z (cid:62) (cid:33) − n Z Z (cid:62) + n Z Z (cid:62) M − ( θ i ) (cid:32) n Z Z (cid:62) (cid:33) − n Z Z (cid:62) = − δ i (cid:98) λ i n Z Z (cid:62) M − ( (cid:98) λ i ) F M − ( θ i ) (cid:32) n Z Z (cid:62) (cid:33) − n Z Z (cid:62) + n Z Z (cid:62) M − ( θ i ) (cid:32) n Z Z (cid:62) (cid:33) − n Z Z (cid:62) . On one hand, similar to the arguments in the proof of Theorem 2.6, we can derive thatmax ≤ j ≤ q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e (cid:62) j δ i (cid:98) λ i n Z Z (cid:62) M − ( (cid:98) λ i ) F M − ( θ i ) (cid:32) n Z Z (cid:62) (cid:33) − n Z Z (cid:62) e j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O p (cid:32) √ q δ i λ i (cid:33) , max ≤ j (cid:44) j ≤ q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e (cid:62) j δ i (cid:98) λ i n Z Z (cid:62) M − ( (cid:98) λ i ) F M − ( θ i ) (cid:32) n Z Z (cid:62) (cid:33) − n Z Z (cid:62) e j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O p (cid:32) q δ i λ i (cid:33) . On the other hand, similar to the proof of Lemma 4.2, we can get that1 n e (cid:62) j Z Z (cid:62) M − ( θ i ) (cid:32) n Z Z (cid:62) (cid:33) − n Z Z (cid:62) e j − E (cid:104) tr (cid:110) M − ( θ i ) (cid:111)(cid:105) = O p (cid:32) √ n (cid:33) , n e (cid:62) j Z Z (cid:62) M − ( θ i ) (cid:32) n Z Z (cid:62) (cid:33) − n Z Z (cid:62) e j = O p (cid:32) √ n (cid:33) , where n E (cid:110) tr M − ( θ i ) (cid:111) → y . It follows thatmax ≤ j ≤ q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n e (cid:62) j (1 + δ i ) Z Z (cid:62) M − ( θ i ) (cid:32) n Z Z (cid:62) (cid:33) − n Z Z (cid:62) e j − n E (cid:104) tr (cid:110) M − ( θ i ) (cid:111)(cid:105)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O p (cid:32) √ q √ n (cid:33) max ≤ j (cid:44) j ≤ q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n e (cid:62) j (1 + δ i ) Z Z (cid:62) M − ( θ i ) (cid:32) n Z Z (cid:62) (cid:33) − n Z Z (cid:62) e j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O p (cid:32) q √ n (cid:33) . Similarly, we can get the following for other terms:max ≤ j ≤ q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e (cid:62) j (1 + δ i ) 1 n Z Z (cid:62) e j − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O p (cid:32) √ q √ n (cid:33) , max ≤ j (cid:44) j ≤ q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e (cid:62) j (1 + δ i ) 1 n Z Z (cid:62) e j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O p (cid:32) q √ n (cid:33) , max ≤ j ≤ q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e (cid:62) j (cid:98) λ i θ i T X X (cid:62) M − ( (cid:98) λ i ) M − ( θ i ) (cid:32) n Z Z (cid:62) (cid:33) − T X X (cid:62) e j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O p (cid:113)(cid:80) j λ j λ i , max ≤ j (cid:44) j ≤ q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e (cid:62) j (cid:98) λ i θ i T X X (cid:62) M − ( (cid:98) λ i ) M − ( θ i ) (cid:32) n Z Z (cid:62) (cid:33) − T X X (cid:62) e j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O p (cid:80) j λ j λ i , max ≤ j ≤ q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e (cid:62) j (1 + δ i ) (cid:98) λ i n Z Z (cid:62) M − ( (cid:98) λ i ) M − ( θ i ) (cid:32) n Z Z (cid:62) (cid:33) − T X X (cid:62) e j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O p (cid:112)(cid:80) j λ j λ i , max ≤ j (cid:44) j ≤ q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e (cid:62) j (1 + δ i ) (cid:98) λ i n Z Z (cid:62) M − ( (cid:98) λ i ) M − ( θ i ) (cid:32) n Z Z (cid:62) (cid:33) − T X X (cid:62) e j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O p (cid:112) q (cid:80) j λ j λ i , max ≤ j ≤ q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e (cid:62) j n Z C ( θ i ) X (cid:62) e j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O p (cid:112)(cid:80) j λ j √ n λ i , max ≤ j (cid:44) j ≤ q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e (cid:62) j n Z C ( θ i ) X (cid:62) e j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O p (cid:112) q (cid:80) j λ j √ n λ i , max ≤ j ≤ q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e (cid:62) j (1 + δ i ) (cid:98) λ i T X X (cid:62) M − ( (cid:98) λ i ) M − ( θ i ) (cid:32) n Z Z (cid:62) (cid:33) − n Z Z (cid:62) e j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O p (cid:112)(cid:80) j λ j λ i , ≤ j (cid:44) j ≤ q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e (cid:62) j (1 + δ i ) (cid:98) λ i T X X (cid:62) M − ( (cid:98) λ i ) M − ( θ i ) (cid:32) n Z Z (cid:62) (cid:33) − n Z Z (cid:62) e j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O p (cid:112) q (cid:80) j λ j λ i , max ≤ j ≤ q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e (cid:62) j T X D ( θ i ) Z (cid:62) e j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O p (cid:112)(cid:80) j λ j √ n λ i , max ≤ j (cid:44) j ≤ q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e (cid:62) j T X D ( θ i ) Z (cid:62) e j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O p (cid:112) q (cid:80) j λ j √ n λ i . Thus, all these inequalities lead tomax ≤ j ≤ q (cid:12)(cid:12)(cid:12) e (cid:62) j Θ n e j − ( y − (cid:12)(cid:12)(cid:12) = O p √ q δ i λ i + √ q √ n + (cid:113)(cid:80) j λ j λ i + (cid:112)(cid:80) j λ j λ i , max ≤ j (cid:44) j ≤ q (cid:12)(cid:12)(cid:12) e (cid:62) j Θ n e j (cid:12)(cid:12)(cid:12) = O p q δ i λ i + q √ n + (cid:80) j λ j λ i + (cid:112) q (cid:80) j λ j λ i . The proof is completed. (cid:3)
Acknowledgments
The authors gratefully acknowledge a grant from the University Grants Council of HongKong and a NSFC grant (NSFC11671042). Drs Xie and Zeng are co-first authors.
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