Convergence of empirical spectral distributions of large dimensional quaternion sample covariance matrices

Transcription

1 Ann Inst Stat Math : DOI 0007/s Convergence of empirical spectral distributions of large dimensional quaternion sample covariance matrices Huiqin Li Zhi Dong Bai Jiang Hu Received: 4 June 204 / Revised: 27 December 204 / Published online: 22 March 205 The Institute of Statistical Mathematics, Tokyo 205 Abstract In this paper, we establish the limit of empirical spectral distributions of quaternion sample covariance matrices Motivated by Bai and Silverstein Spectral analysis of large dimensional random matrices, Springer, New York, 200 and Marčenko and Pastur Matematicheskii Sb, 4: , 967, we can extend the results of the real or complex sample covariance matrix to the quaternion case Suppose X n = x n p n is a quaternion random matrix For each n, the entries {x n ij } are independent random quaternion variables with a common mean μ and variance σ 2 > 0 It is shown that the empirical spectral distribution of the quaternion sample covariance matrix S n = n X n Xn converges to the Marčenko Pastur law as p, n and p/n y 0, + Keywords Empirical spectral distribution LSD Quaternion matrices Sample covariance matrix Z D Bai was partially supported by CNSF 7057, the Fundamental Research Funds for the Central Universities, and PCSIRT; J Hu was partially supported by a grant CNSF H Li Z D Bai B J Hu KLASMOE and School of Mathematics and Statistics, Northeast Normal University, Changchun 30024, People s Republic of China baizd@nenueducn H Li lihq8@nenueducn J Hu huj56@nenueducn 23

2 766 H Li et al Introduction In 843, Hamilton described the hyper-complex number of rank 4, to which he gave the name of quaternion see Kuipers 999 Research on the quaternion matrices can be traced back to Wolf 936 After a long blank period, people gradually discovered that quaternions and quaternion matrices play important roles in quantum physics, robot technology and artificial satellite attitude control, among other applications, see Adler 995 and Finkelstein et al 962 Consequently, studies on quaternions have attracted considerable attention in recent years, see So et al 994, Zhang 995, Kanzieper 2002, Akemann 2005, and Akemann and Phillips 203, among others In the following, we introduce the quaternion notation A quaternion can be represented as a 2 2 complex matrix a + bi c + di x = a e + b i + c j + d k = c + di a bi, a, b, c, d R where i denotes the imaginary unit and the quaternion units can be represented as e = 0 i 0, i =, j = 0 0 i The conjugate of x is defined as 0 0 i, k = 0 i 0 a bi c di x = a e b i c j d k = c di a + bi and its norm as x = a 2 + b 2 + c 2 + d 2 More details can be found in Kuipers 999,Zhang 997, and Mehta 2004Using the matrix representation of quaternions, an n n quaternion matrix X can be rewritten as a 2n 2n complex matrix ψx, and so we can deal with quaternion matrices as complex matrices for convenience Denote S = n XX and ψs = n ψxψx It is known see Zhang 997 that the multiplicities of all the eigenvalues obviously they are all real of ψs are even Taking one from each of the n pairs of eigenvalues of ψs,then values are defined to be the eigenvalues of S In addition, wide application of computer science has increased a thousand fold in terms of computing speed and storage capability in the recent decades Due to the failure of the applications of many classical conclusions, we need a new theory to analyze very large data sets with high dimensions Luckily, the theory of random matrices RMT might be a possible route for dealing with these problems The sample covariance matrix is one of the most important random matrices in RMT, which can be traced back to Wishart 928 Marčenko and Pastur 967 proved that the empirical spectral distribution ESD of a large dimensional complex sample covariance matrix tends to the Marčenko Pastur M P law Since 23

3 Convergence of ESD 767 then, several successive studies on large dimensional complex or real sample covariance matrices have been completed Here, the readers are referred to three books Anderson et al 200, Bai and Silverstein 200, and Mehta 2004 formore details Under the normality assumption, there are three classic random matrix models: Gaussian orthogonal ensemble GOE, for which all entries of the matrix are real normal random variables, Gaussian unitary ensemble GUE, for which all entries of the matrix are complex normal random variables, and Gaussian symplectic ensemble GSE, for which all entries of the matrix are normal quaternion random variables Benefiting from the density function of the ensemble and the joint density of the eigenvalues, the results have gotten their own style If we remove the normality assumption, the corresponding first two models have already had satisfactory results For quaternion matrices, there are only a few references see Yin and Bai 204; Yin et al 203, 204 In this paper, we prove that the ESD of the quaternion sample covariance matrix also converges to the M P law However, due to the multiplication of quaternions is not commutative, when the entries of X n are quaternion random variables, few works on the spectral properties are found in the literature unless the random variables are normality distributed, because in this case the joint density of the eigenvalues is available Thanks to the tool provided by Yin et al 203, it makes the quaternion case possible For the proof of this result, we first introduce two definitions about ESD and Stieltjes transform Let A be a p p Hermitian matrix and denote its eigenvalues by s j, j =, 2,,p The ESD of A is defined by F A x = p p I s j x, j= where I D is the indicator function of an event D and the Stieltjes transform of F A x is given by mz = + x z dfa x, where z = u + υi C + Letgx and m g x denote the density function and the Stieltjes transform of the M P law, which are gx = { 2π xyσ b xx a, 2 a x b; 0, otherwise, 2 and m g z = σ 2 y z + z σ 2 yσ 2 2 4yσ 2 2yzσ 2, 3 23

4 768 H Li et al respectively, where a = σ 2 y 2, b = σ 2 + y 2 Here, the constant y is the limit of dimension p to sample size n ratio and σ 2 is the scale parameter If y >, Gx, the distribution function of gx, has a point mass /y at the origin Now, our main theorem can be described as follows Theorem Let X n = x n, j =,,p, k =,,n Suppose for each n, { } x n are independent quaternion random variables with a common mean μ and variance σ 2 Assume that y n = p/n y 0, and for any constant η>0, E x n 2 I x n >η n 0 4 Then, with probability one, the ESD of the sample covariance matrix S n = n X nx n converges to the M P law in distribution which has density function 2 and a point mass /y at the origin when y > Here, superscript stands for the complex conjugate transpose Remark 2 Without loss of generality, in the proof of Theorem, we assume that σ 2 = Furthermore, one can see that removing the common mean of the entries of X n does not alter the LSD of sample covariance matrices In fact, let T n = n X n EX n X n EX n By Lemma 7, we have, for all large p, F S n F T n KS 2p rankex n p 0 where f KS = sup x f x Consequently, we assume that μ = 0 The paper is organized as follows In Sect 2, the structure of the inverse of some matrices about quaternions is established which is the key tool of proving Theorem Section 3 demonstrates the proof of the main theorem by two steps and in Sect 4,we outline some auxiliary lemmas that can be used in last section 2 Preliminaries We shall use Lemma 25 of Yin et al 203 to prove our main result in next section To keep this work self-contained, the lemma is now stated as follows 23

5 Convergence of ESD 769 Definition 3 A matrix is of Type-I, if it has the following structure: t 0 a 2 b 2 a n b n 0 t c 2 d 2 c n d n d 2 b 2 t 2 0 a 2n b 2n c 2 a 2 0 t 2 c 2n d 2n d n b n d 2n b 2n t n 0 c n a n c 2n a 2n 0 t n Here, all the entries are complex Definition 4 A matrix is of Type-II, if it has the following structure: t 0 a 2 + c 2 i b 2 + d 2 i a n + c n i b n + d n i 0 t b 2 d 2 i ā 2 + c 2 i b n d n i ā n + c n i ā 2 + c 2 i b 2 d 2 i t 2 0 a 2n + c 2n i b 2n + d 2n i b 2 + d 2 i a 2 + c 2 i 0 t 2 b 2n d 2n i ā 2n + c 2n i ā n + c n i b n d n i ā 2n + c 2n i b 2n d 2n i t n 0 b n + d n i a n + c n i b 2n + d 2n i a 2n + c 2n i 0 t n Here, i = denotes the usual imaginary unit and all the other entries are complex numbers Definition 5 A matrix is of Type-III, if it has the following structure: t 0 a 2 b 2 a n b n 0 t b 2 ā 2 b n ā n ā 2 b 2 t 2 0 a 2n b 2n b 2 a 2 0 t 2 b 2n ā 2n ā n b n ā 2n b 2n t n 0 b n a n b 2n a 2n 0 t n Here, all the entries are complex Lemma 6 For all n, if a complex matrix n is invertible and of Type-II, then is a Type-I matrix n The following corollary is immediate Corollary 7 For all n, if a complex matrix n is invertible and of Type-III, then is a Type-I matrix n 23

6 770 H Li et al 3 Proof of Theorem In this section, we present the proof in two steps The first one is to truncate, centralize and rescale the random variables {x n ij }, then we may assume the additional conditions which are given in Remark 2 The other is the proof of the Theorem under the additional conditions Throughout the remainder of this paper, a local constant C may take different value at different place 3 Truncation, centralization and rescaling 3 Truncation Note that, condition 4 is equivalent to: for any η>0, lim n η 2 E x n 2 I x n >η n = 0 5 Applying Lemma 5, one can select a sequence η n 0 such that 5 remains true when η is replaced by η n Lemma 8 Suppose that the assumptions of Theorem refth: hold Truncate the variables x n n at η n n, and denote the resulting variables by x, ie, xn = x n I xn η n n Also denote Then, with probability, X n = Proof Using Lemma 7, one has x n and Ŝ n = n X n X n F S n FŜn KS 0 F S n FŜn KS 2p rank n X n X n n 2p Taking condition 5 into consideration, we get E I 2p 23 x n >η n n I 2η 2 n x n >η n n 6 E x n 2 I x n >η n n =o

7 Convergence of ESD 77 and Var I 2p 4η 2 n p2 n x n >η n n E x n 2 I x n >η n n = o p Then, by Bernstein s inequality see Lemma 8, for all small ε>0and large n, we obtain P I x n 2p >η n n ε 2e εp/2 which is summable Combining 6, the above inequality with the Borel Cantelli lemma, it follows that F S n FŜn KS as 0 This completes the proof of the lemma 32 Centralization Lemma 9 Suppose that the assumptions of Lemma 8 hold Denote x n = xn E xn, X n = x n and S n = n X n X n Then, we obtain where L, denotes the Lévy distance LFŜn, F S n = o, Proof Using Lemma 6 and condition 5, we have L 4 FŜn, F S n trŝn 2p 2 + S n tr n X n X n n X n X n n n = 2n 2 p 2 = x n 2 + x n E xn 2 x n 2 + x n E xn 2 E x n 2 2 E x n

8 772 H Li et al To complete the proof of this lemma, we need to show that the first parentheses of the right-hand side of 7 is almost surely bounded Applying Lemma 22, one has E x n 2 E x n 2 This indicates by the Borel Cantelli lemma Moreover, we can similarly obtain Now, turning to 7, for all large n, 4 C n 4 p 4 Cn 2 η 6 n n y 3 n x n 2 E x n 2 as 0 E x n 8 + E x n j,k + ηn 4 y 2 n 4 2 x n E xn 2 E x n E xn 2 as 0 8 L 4 FŜn, F S n E x n 2 + E x n E xn 2 + o as C C E x n 2 E x n 2 I x n >η n n 0 2 E x n 2 The proof of the lemma is complete 33 Rescaling Define σ 2 = E xn 2,ξ = { ζ, σ 2 < /2 x n, σ 2, = ξ, σ 2 /2 n = E ξ 2, where ζ is a bounded quaternion random variable with Eζ = 0, Varζ = and independent with all other variables 23

9 Convergence of ESD 773 Lemma 0 Write x n = σ ξ, X n = x n, and S n = n X n X n Under the conditions assumed in Lemma 9, we have L F S n, F S n = o Proof a: Our first goal is to show that L F S n, F as 0 Let E n be the set of pairs j, k : σ 2 < 2 and N n = j,k E n I σ 2 < /2 be the number of such pairs Due to σ 2, we conclude that N n = o Owing to Lemma 6 and 8, we get L 4 F S n, F 2p 2 tr S n + tr n X n n X n = 2n 2 p 2 x n 2 + ξ 2 ξ x n 2 = 2n 2 p 2 E x n 2 + ξ 2 +o as ξ x n 2 C ξ x n 2 := C K u h 9 where K = N n and u h = ξ x n 2 Using the fact that for all l, l! l/3 l, we have m K E u h = n m p m h= n m p m C C m + +m K =m m l= m + +m l =m m t h= m! m! m K! Eum Eu m K K m! l!m! m l! l K t= m n m p m l m l! 2ηn 2 nm l 2 l K l l= m l= 6K l 2η 2 m l n l C p h= Eu m t h 6K + 2η2 n m m p 23

10 774 H Li et al By selecting m =[log p] that implies 2η2 n m p for any fixed t,ε >0, 0, and noticing 6K 0, one obtains m K E u h o p t ε h= From the inequality above with t = 2 and 9, it follows that L F S n, F as 0 b: Our next goal is to show that L F S n, F as 0 Applying Lemma 6, wehave L 4 F S n, F 2p 2 tr S n + tr n X n n X n Using the fact x n = 2n 2 p 2 = 2n 2 p 2 σ C σ 2 + ξ 2 ξ x n + σ 2 E ξ 2 + o as 2 ξ 2 2 ξ 2 E C 2 σ ξ 2 = C 2 C σ Cη2 n η 2 n 0 23 j,k/ E n 2 σ 2 [ E x n 2 I x n η n n + E x n I x n η ] 2 n n

11 Convergence of ESD 775 and Lemma 22, one gets E C C n 4 p 4 σ j,k Cn 2 [ n η 6 n y 3 n 2 ξ 2 E ξ 2 E x n 8 I x n η n n+ j,k ] + ηn 4 y 2 n 4 E x n 4 I x n η 2 n n which is summable Together with the Borel Cantelli lemma, it follows that L F S n, F as 0 c: Finally, from a and b, we can easily get the lemma Combining the results of Lemmas 8, 9, and 0, we have the following remarks Remark For brevity, we shall drop the superscript n from the variables Also the truncated and renormalized variables are still denoted by x Remark 2 Under the conditions assumed in Theorem, we can further assume that x η n n, 2 Ex = 0 and Varx = 32 Completion of the proof Denote where z = u + υi C + 32 Random part First, we should show that m n z = 2p tr S n zi 2p, 0 m n z Em n z as 0 Let π j denote the jth column of X n, S k n = S n n π kπ k and E k denote the conditional expectation given {π k+, π k+2,,π 2n } Then, m n z Em n z = 2p 2n k= γ k, 23

12 776 H Li et al where γ k = E k tr Ek S n zi 2p tr S n zi 2p = E k E k [tr ] S n zi 2p tr S k n zi 2p When k = 2t t =, 2,,n, due to k + th column is a function of the kth column, we obtain γ k = E k tr S n zi 2p Ek tr S n zi 2p = 0 2 When k = 2tt = 0,,,n, together with the formula A + αβ = A A αβ A + β A α, one finds [ ] γ k = E k E k trs n zi 2p tr S k n zi 2p n = E k E k π k S k 2π n zi 2p k + n π k S k π n zi 2p k Since n π k S k 2π n zi 2p k + n π k S k π n zi 2p k S n k π k 2 π n ui 2p + υ 2 I 2p k I + n π k Sk n zi 2p π k = υ, we can easily get γ k 2 υ Using Lemma 2, it follows that 2n 2 E m n z Em n z 4 K 4 2p 4 E γ k 2 4K 4n 2 p 4 v 4 k= = O n 2 Combining the Borel Cantelli lemma with the Chebyshev inequality, we conclude 23 m n z Em n z as 0

13 Convergence of ESD Mean convergence When σ 2 =, 3 turns into Next, we show that By Lemma 20 and 0, one has m n z = 2p k= mz = y z + z y 2 4yz 2 2yz Em n z mz p tr n φ k φ k zi 2 n 2 φ k X nk n X nkxnk zi 2p 2 X φ nk k where X nk is the matrix resulting from deleting the kth quaternion row of X n, and φ k is the vector obtained from the kth quaternion row of X n Here, superscript only stands for the transpose and φ k is a n quaternion matrix Write and ε k = n φ k φ k zi 2 n 2 φ k X nk n X nkxnk zi 2p 2 X φ nk k 3 z y n y n zem n zi 2 4 δ n = 2p z y n y n zem n z p { Etr ε k z y n y n zem n z I 2 + ε k } 5 k= where y n = p/n This implies that Em n z = z y n y n zem n z + δ n Solving Em n z from the equation above, we get Em n z = z y n + y n zδ n ± z y n y n zδ n 2y n z 2 4y n z 23

14 778 H Li et al As proved in the Eq 37 of Bai 993, we can assert that Em n z = z y n + y n zδ n + z y n y n zδ n 2 4y n z 6 2y n z Comparing 2 with 6, it suffices to show that δ n 0 For this purpose, we need the following two lemmas Lemma 3 Under the conditions of Remark 2, for any z = u + vi with v>0and for any k =,,p, we have Etrε k 0 7 Proof By calculation, we have Etrε k = n 2 EtrX nk n X nkxnk zi 2p 2 X nk + 2y n + 2y n zem n z = n Etr n X nkxnk zi 2p 2 n X nkxnk + 2y n + 2y n zem n z [ 2 n + z ] n E tr n X nxn zi 2p tr n X nkxnk zi 2p 2 2 n + 2 z nυ 0 where the last inequality has used Lemma 9 twice Then, the proof is complete Lemma 4 Under the conditions of Remark 2, for any z = u + vi with v>0and any k =,,p, we have E trε 2 k 0 Proof Write the form of S n zi 2p t 0 a 2 b 2 0 t b 2 ā 2 ā 2 b 2 t 2 0 b 2 a 2 0 t 2 By Corollary 7 and 3, we have n φ k φ k zi 2 n 2 φ k X nk R kx nk φ k is a scalar matrix Let R k = n X nk X nk zi 2p 2 Denote by αk the first column of φ k and by β k the second column of φ k, then combining 4 wehave 23

15 Convergence of ESD 779 where ε k = θ k I 2 8 θ k = n α kᾱk z n 2 α k X nk R kx nk ᾱ k z y n y n zem n z = n β k β k z n 2 β k X nk R kx nk β k z y n y n zem n z Let Ẽ denote the conditional expectation given {x jl, j =,,p, l =,,n; l = k}, then we get E trε 2 k = 2 E trε k 2 [ 2 E trε k Ẽtrε k 2 + E Ẽtrε k Etrε k 2 + Etrε k 2] 9 According to the inequality above, we proceed to complete the estimation of E trε 2 k by the following three steps a For the first term of the right-hand side of 9, denote T = t jl = I 2n n e X nk R kx nk where t jl = jl f jl Then, rewrite h jl g jl trε k Ẽtrε k = tr n φ k φ k n 2 φ k X nk R kx nk φ k tr I 2 n 2 X nk R kx nk = n tr φ k T φ k T = n tr x kj 2 t jj + trxkl n x kjt jl j= j =l By elementary calculation, we obtain Ẽ trε k Ẽtrε k 2 = n n 2 Ẽ tr x kj 2 t jj 2 + [ Ẽ tr xkl x kjt jl trx kj x kl t jl j= j =l +tr xkl x ] kjt jl tr xkl x kjtlj n n 2 Ẽ x kj 2 2 e jj + g jj Ẽ trxkl x kjt jl 2 j= j =l C n η 2 n 2 n n jj j= e 2 + g jj 2 + e jl 2 + f jl 2 + g jl 2 + h jl 2 j =l 23

16 780 H Li et al Cη2 n n n e jj 2 + g jj 2 + C n 2 e jl 2 + f jl 2 + g jl 2 + h jl 2 j= j,l Cη2 n n trtt + C n 2 trtt 20 For n X nk, there exists a 2p 2 q orthonormal matrix U anda2n q orthonormal matrix V such that n X nk = Udiags,,s q V where s,,s q are the singular values of n X nk and q = min{2p 2, 2n} Then, we get I 2n T = n X nk = Vdiag R k n X nk s 2 s 2 z,, sq 2 sq 2 z V which implies that T = Vdiag z s 2 z,, z sq 2 z V Consequently, it follows that trtt = q s 2 j= j z 2 z 2 2n z 2 υ 2 2 By 20 and 2, we obtain E trε k Ẽtrε k b Next, the second term of right-hand side of 9 is estimated Note that Ẽtrε k Etrε k = z n EtrR k trr k Using the martingale decomposition method, we have E Ẽtrε k Etrε k 2 = z 2 n 2 E EtrR k trr k 2 4 z nυ2 23

17 Convergence of ESD 78 c Finally, combining 7, 9, 22, and 23, we conclude that E trε 2 k 0 These indicate that we complete the proof of the lemma Now, we are in a position to show that δ n 0 By 5 and 8, we can write δ n = 2p z y n y n zem n z 2 Note that + 2p z y n y n zem n z 2 p Etrε k k= p trε 2 k E z y n y n zem n z + θ k k= I z y n y n zem n z < υ, which implies that z y n y n zem n z >υ 24 By Lemma 3 and 24, we have 2p z y n y n zem n z 2 Together with Lemma 4, 24, and p k= Etrε k 2pυ 2 p Etrε k 0 25 k= Iθ k + z y n y n zem n z =I n α kᾱk z n 2 α k X nk n X nkxnk zi 2p 2 X nkᾱ k [ ] 2 = υ + α k X nk n X nkxnk ui 2p 2 + υ 2 I 2p 2 X nkᾱ k < υ 23

18 782 H Li et al one finds that 2p z y n y n zem n z 2 2pυ 3 p k= Combining 25 with 26, we get p trε 2 k E z y n y n zem n z + θ k k= E trε 2 k 0 26 p δ n 2p z y n y n zem n z 2 Etrε k k= p trε 2 k + 2p z y n y n zem n z 2 E z y k= n y n zem n z + θ k 0 So far, we have completed the proof of the mean convergence Em n z m z 323 Completion of the proof of Theorem By Sects 32 and 322, for any fixed z C +,wehave m n z as m z To complete the proof Theorem, we need the last part of Chapter 2 of Bai and Silverstein 200 For the readers convenience, we repeat here That is, for each z C +, there exists a null set N z ie, P N z = 0 such that m n z,w m z, for all w N c z Now, let C + 0 be a dense subset of C+ eg, all z of rational real and imaginary parts and let N = z C + N z Then, 0 m n z,w m z, for all w N c and z C + 0 Let C + m ={z C+ : Iz > /m, z m} When z C + m,wehave m n z m Applying Lemma 23, wehave 23 m n z,w m z, for all w N c and z C + m

19 Convergence of ESD 783 Since the convergence above holds for every m, we conclude that m n z,w m z, for all w N c and z C + Applying Lemma 24, we conclude that F S n w F, as 4 Appendix In this section, some results are listed which are used in the proof of the main theorem Lemma 5 Suppose for any η>0 n= f η, n <, then we can select a slowly decreasing sequence of constants η n 0 such that f η n, n < n= where f is a nonnegative function Similarly, if f η, n 0 for any fixed η > 0, then there exists a decreasing sequence η n 0 such that f η n, n 0 Proof Letting η = m, one has n= f m, n < Moreover, there exists a increasing sequence N m such that n=nm f m, n 2 m Define a sequence η n = m when N m n < N m+ We get f η n, n = n= m= N n= N m+ n=n m f f, n + m, n m= m= n=n m 2 m < f m, n This completes the proof of this lemma Lemma 6 Corollary A42 of Bai and Silverstein 200 Let A and B be two p n matrices and denote the ESD of S = AA and S = BB by F S and F S, respectively Then, L 4 F S, F S 2 p 2 traa + BB tr[a BA B ], where L, denotes the Lévy distance, that is, LF S, F S = inf{ε : F S x ε, y ε ε F S F S x + ε, y + ε + ε} 23

20 784 H Li et al Lemma 7 Theorem A44 of Bai and Silverstein 200 Let A and B be p n complex matrices Then, F AA F BB KS rank A B p Lemma 8 Bernstein s inequality If τ,,τ n are independent random variables with means zero and uniformly bounded by b, then, for any ε>0, n P j= τ j where B 2 n = Eτ + +τ n 2 ] ε 2exp ε 2 / [2B n 2 + bε Lemma 9 see A2 of Bai and Silverstein 200 Let z = u + iv, v > 0, and A be an n n Hermitian matrix Denote by A k the kth major sub-matrix of A of order n, to be the matrix resulting from deleting the kth row and column from A Then, tra zi n tra k zi n υ Lemma 20 Suppose that the matrix has the partition as given by 2 If 2 22 and are nonsingular, then the inverse of has the form = where 22 = Lemma 2 Burkholder s inequality Let {X k } be a complex martingale difference sequence with respect to the increasing σ -field Then, for p >, p p/2 E X k K p E X k 2 k Lemma 22 Rosenthal s inequality Let X i be independent with zero means, then we have, for some constant C k, 2k E X i C k i i k E X i 2k + i k E X i 2 Lemma 23 Lemma 24 of Bai and Silverstein 200 Let f, f 2,be analytic in D, a connected open set of C, satisfying f n z M for every n and z in D, and 23

21 Convergence of ESD 785 f n z converges as n for each z in a subset of D having a limit point in D Then, there exists a function f analytic in D for which f n z f z and f n f z for all z D Moreover, on any set bounded by a contour interior to D, the convergence is uniform and { f n z} is uniformly bounded Lemma 24 Theorem B9 of Bai and Silverstein 200 Assume that {G n } is a sequence of functions of bounded variation and G n = 0 for all n Then, lim n m G n z = m z z D where D {z C :Iz > 0} if and only if there is a function of bounded variation G with G = 0 and Stieltjes transform m z and such that G n G vaguely References Adler, S L 995 Quaternionic quantum mechanics and quantum fields Vol Oxford: Oxford University Press Akemann, G 2005 The complex Laguerre symplectic ensemble of non-hermitian matrices Nuclear Physics B, 7303, Akemann, G, Phillips, M 203 The interpolating airy kernels for the beta = and beta = 4 elliptic Ginibre ensembles arxiv: Anderson, G W, Guionnet, A, Zeitouni, O 200 An introduction to random matrices Vol 8 Cambridge: Cambridge University Press Bai, Z D 993 Convergence rate of expected spectral distributions of large random matrices Part II Sample covariance matrices The Annals of Probability, 22, Bai, Z D, Silverstein, J W 200 Spectral analysis of large dimensional random matrices 2nd ed New York: Springer Finkelstein, D, Jauch, J M, Schiminovich, S, Speiser, D 962 Foundations of quaternion quantum mechanics Journal of Mathematical Physics, 32, 207 Kanzieper, E 2002 Eigenvalue correlations in non-hermitean symplectic random matrices Journal of Physics A: Mathematical and General, 353, 663 Kuipers, J B 999 Quaternions and rotation sequences Princeton: Princeton University Press Marčenko, V A, Pastur, L A 967 Distribution of eigenvalues for some sets of random matrices Matematicheskii Sbornik, 44, Mehta, M L 2004 Random matrices Vol 42 Access online via Elsevier So, W, Thompson, R C, Zhang, F 994 The numerical range of normal matrices with quaternion entries Linear and Multilinear Algebra, 37 3, Wishart, J 928 The generalised product moment distribution in samples from a normal multivariate population Biometrika, 20/2, Wolf, L A 936 Similarity of matrices in which the elements are real quaternions Bulletin of the American Mathematical Society, 420, Yin, Y, Bai, Z D 204 Convergence rates of the spectral distributions of large random quaternion self-dual Hermitian matrices Journal of Statistical Physics, 576, Yin, Y, Bai, Z D, Hu, J 203 On the semicircular law of large dimensional random quaternion matrices Journal of Theoretical Probability to appear arxiv: Yin, Y, Bai, Z D, Hu, J 204 On the limit of extreme eigenvalues of large dimensional random quaternion matrices Physics Letters A, 37867, Zhang, F 995 On numerical range of normal matrices of quaternions Journal of Mathmatical and Physical Science, 296, Zhang, F 997 Quaternions and matrices of quaternions Linear Algebra and Its Applications, 25,