Central Limit Theorem for Linear Eigenvalue Statistics for a Tensor Product Version of Sample Covariance Matrices

Journal of Theoretical Probability, Jan 2017

For \(k,m,n\in {\mathbb {N}}\), we consider \(n^k\times n^k\) random matrices of the form $$\begin{aligned} {\mathcal {M}}_{n,m,k}({\mathbf {y}})=\sum _{\alpha =1}^m\tau _\alpha {Y_\alpha }Y_\alpha ^T,\quad {Y}_\alpha ={\mathbf {y}}_\alpha ^{(1)}\otimes \cdots \otimes {\mathbf {y}}_\alpha ^{(k)}, \end{aligned}$$ where \(\tau _{\alpha }\), \(\alpha \in [m]\), are real numbers and \({\mathbf {y}}_\alpha ^{(j)}\), \(\alpha \in [m]\), \(j\in [k]\), are i.i.d. copies of a normalized isotropic random vector \({\mathbf {y}}\in {\mathbb {R}}^n\). For every fixed \(k\ge 1\), if the Normalized Counting Measures of \(\{\tau _{\alpha }\}_{\alpha }\) converge weakly as \(m,n\rightarrow \infty \), \(m/n^k\rightarrow c\in [0,\infty )\) and \({\mathbf {y}}\) is a good vector in the sense of Definition 1.1, then the Normalized Counting Measures of eigenvalues of \({\mathcal {M}}_{n,m,k}({\mathbf {y}})\) converge weakly in probability to a nonrandom limit found in Marchenko and Pastur (Math USSR Sb 1:457–483, 1967). For \(k=2\), we define a subclass of good vectors \({\mathbf {y}}\) for which the centered linear eigenvalue statistics \(n^{-1/2}{{\mathrm{Tr}}}\varphi ({\mathcal {M}}_{n,m,2}({\mathbf {y}}))^\circ \) converge in distribution to a Gaussian random variable, i.e., the Central Limit Theorem is valid.

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Central Limit Theorem for Linear Eigenvalue Statistics for a Tensor Product Version of Sample Covariance Matrices

Mathematics Subject Classification Central Limit Theorem for Linear Eigenvalue Statistics for a Tensor Product Version of Sample Covariance Matrices A. Lytova 0 1 Mn 0 1 m 0 1 k 0 1 0 Institute of Mathematics and Informatics, Opole University , 45 052 Opole , Poland 1 Department of Mathematical and Statistical Sciences, University of Alberta , Edmonton, AB T6G 2G1 , Canada For k, m, n ∈ N, we consider nk × nk random matrices of the form m Random matrices; Sample covariance matrices; Central Limit Theorem; Linear eigenvalue statistics 1 Introduction: Problem and Main Result Y = y(1) ⊗ · · · ⊗ y(k) ∈ (Rn)⊗k , where y(1),…, y(k) are i.i.d. copies of a normalized isotropic random vector y = (y1, . . . , yn) ∈ Rn, E{y j } = 0, E{yi y j } = δi j n−1, i, j ∈ [n], where we use the notation j for k-multiindex: j = { j1, . . . , jk }, j1, . . . , jk ∈ [n]. m m For every m ∈ N, let {Yα}α=1 be i.i.d. copies of Y , and let {τα}α=1 be a collection of real numbers. Consider an nk × nk real symmetric random matrix corresponding to a normalized isotropic random vector y, We suppose that Mn = Mn,m,k = Mn,m,k (y) = m → ∞ and m/nk → c ∈ (0, ∞) as n → ∞. Note that Mn,m,k can be also written in the form Mn,m,k = Bn,m,k Tm BnT,m,k , Bn,m,k = (Y1 Y2 . . . Ym ), Tm = {ταδαβ }αm,β=1. Such matrices with Tm ≥ 0 (not necessarily diagonal) are known as sample covariance matrices. The asymptotic behavior of their spectral statistics is well studied when all entries of Yα are independent. Much less is known in the case when columns Yα have dependence in their structure. The model constructed in (1.3) appeared in the quantum information theory and was introduced to random matrix theory by Hastings (see [3,14,15]). In [3], it was studied as a quantum analog of the classical probability problem on the allocation of p balls among q boxes (a quantum model of data hiding and correlation locking scheme). In particular, by combinatorial analysis of moments of n−k Tr Mnp, p ∈ N, it was proved that for the special cases of random vectors y uniformly distributed on the unit sphere in Cn or having Gaussian components, the expectations of the Normalized Counting Measures of eigenvalues of the corresponding matrices converge to the Marchenko– Pastur law [17]. The main goal of the present paper is to extend this result of [3] to a wider class of matrices Mn,m,k (y) and also to prove the Central Limit Theorem for linear eigenvalue statistics in the case k = 2. Let {λl(n)}ln=k 1 be the eigenvalues of Mn counting their multiplicity, and introduce their Normalized Counting Measure (NCM) Nn, setting for every ⊂ R Nn( ) = Card{l ∈ [nk ] : λl(n) ∈ m , Likewise, define the NCM σm of {τα}α=1 σm ( ) = Card{α ∈ [m] : τα ∈ In the case k = 1, there are a number of papers devoted to the convergence of the NCMs of the eigenvalues of Mn,m,1 and related matrices (see [1,6,12,17,20,27] and references therein). In particular, in [20] the convergence of NCMs of eigenvalues of Mn,m,1 was proved in the case when corresponding vectors {Yα}α are “good vectors” in the sense of the following definition. Definition 1.1 We say that a normalized isotropic vector y ∈ Rn is good, if for every n × n complex matrix Hn which does not depend on y, we have Var{(Hny, y)} ≤ ||Hn||2δn, δn = o(1), n → ∞, where ||Hn|| is the Euclidean operator norm of Hn. Following the scheme of the proof proposed in [20], we show that despite the fact that the number of independent parameters, kmn = O(nk+1) for k ≥ 2, is much less than the number of matrix entries, n2k , the limiting distribution of eigenvalues still obeys the Marchenko–Pastur law. We have: Theorem 1.2 Fix k ≥ 1. Let n and m be positive integers satisfying ( 1.4), let {τα}α be real numbers satisfying (1.7), and let y be a good vector in the sense of Definition 1.1. Then there exists a nonrandom measure N of total mass 1 such that the NCMs Nn of the eigenvalues of Mn (1.3) converge weakly in probability to N as n → ∞. The Stieltjes transform f of N , f (z) = z = 0, is the unique solution of the functional equation z f (z) = c − 1 − c Nn[ϕ] = j=1 is the linear eigenvalue statistic of Mn corresponding to a bounded continuous test function ϕ : R → C, then we have in probability This can be viewed as an analog of the Law of Large Numbers in probability theory for (1.11). Since the limit is nonrandom, the next natural step is to investigate the fluctuations of Nn[ϕ]. This corresponds to the question of validity of the Central Limit Theorem (CLT). The main goal of this paper is to prove the CLT for the linear eigenvalue statistics of the tensor version of the sample covariance matrix Mn,m,2 defined in (1.3). There are a considerable number of papers on the CLT for linear eigenvalue statistics of sample covariance matrices Mn,m,1 (1.5), where all entries of the matrix Bn,m,1 are independent (see [4,7–9,11,16,18,19,21,25] and references therein). Less is known in the case where the components of vector y are dependent. In [13], the CLT was proved for linear statistics of eigenvalues of Mn,m,1, corresponding to some special class of isotropic vectors defined below. Definition 1.3 The distribution of a random vector y ∈ Rn is called unconditional if n n its components {y j } j=1 have the same joint distribution as {±y j } j=1 for any choice of signs. Definition 1.4 We say that normalized isotropic vectors y ∈ Rn, n ∈ N, are very good if they have unconditional distributions, their mixed moments up to the fourth order do not depend on i, j, n, there exist n-independent a, b ∈ R such that as n → ∞, a2,2 := E{yi2 y 2j} = n−2 + an−3 + O(n−4), i = j, κ4 := E{y 4j} − 3a2,2 = bn−2 + O(n−3), and for every n × n complex matrix Hn which does not depend on y, E{|(Hny, y)◦|4} ≤ C ||Hn||4n−2. Here and in what follows we use the notation ξ ◦ = ξ − E{ξ }. An important step in proving the CLT for linear eigenvalue statistics is the asymptotic analysis of their variances Var{Nn[ϕ]} := E{|Nn◦[ϕ]|2}, in particular, the proof of the bound Var{Nn[ϕ]} ≤ Cn||ϕ||2H, where || . . . ||H is a functional norm and Cn depends only on n. This bound determines the normalization factor in front of Nn◦[ϕ] and the class H of the test functions for which the CLT, if any, is valid. It appears that for many random matrices normalized so that there exists a limit of their NCMs, in particular for sample covariance matrices Mn,m,1, the variance of the linear eigenvalue statistic corresponding to a smooth enough test function does not grow with n, and the CLT is valid for Nn◦[ϕ] itself without any n-dependent normalization factor in front. Consider the test functions ϕ : R → R from the Sobolev space Hs , possessing the norm ||ϕ||s2 = (1 + |t |)2s |ϕ(t )|2dt, ϕ(t ) = The following statement was proved in [13] (see Theorem 1.8 and Remark 1.11): and let y be a very good vector in the sense of Definition 1.4. Consider matrix Mn,m,1(y) (1.3) and the linear statistic of its eigenvalues Nn[ϕ] (1.11) corresponding to a test function ϕ ∈ Hs , s > 2. Then {Nn◦[ϕ]}n converges in distribution to a Gaussian random variable with zero mean and the variance V [ϕ] = limη↓0 Vη[ϕ], where L(z1, z2) − L(z1, z2) (ϕ(λ1) − ϕ(λ2))2dλ1dλ2 Lemma 1.6 Let {τα}α be a collection of real numbers satisfying (1.7) and (1.18), and let y be a normalized isotropic vector having an unconditional distribution, such that a2,2 = n−2 + O(n−3), κ4 = O(n−2). Consider the corresponding matrix Mn (1.3) and a linear statistic of its eigenvalues Nn[ϕ]. Then for every ϕ ∈ Hs , s > 5/2, and for all sufficiently large m and n, we have Var{Nn[ϕ]} ≤ C nk−1||ϕ||s2, where C does not depend on n and ϕ. It follows from Lemma 1.6 that in order to prove the CLT (if any) for linear eigenvalue statistics of Mn, one needs to normalize them by n−(k−1)/2. To formulate our main result we need more definitions. Definition 1.7 We say that the distribution of a random vector y ∈ Rn is permutationally invariant (or exchangeable) if it is invariant with respect to the permutations of entries of y. Definition 1.8 We say that normalized isotropic vectors y ∈ Rn, n ∈ N, are the CLTvectors if they have unconditional permutationally invariant distributions and satisfy the following conditions: (i) their fourth moments satisfy (1.13)–(1.14), (ii) their sixth moments satisfy conditions a2,2,2 := E{yi2 y 2j yk2} = n−3 + O(n−4), a2,4 := E{yi2 y 4j} = O(n−3), a6 := E{yi6} = O(n−3), (iii) for every n × n matrix Hn which does not depend on y, E{|(Hny, y)◦|6} ≤ C ||Hn||6n−3. It can be shown that a vector of the form y = x/n1/2, where x has i.i.d. components with even distribution and bounded twelfth moment is a CLT-vector as well as a vector uniformly distributed on the unit ball in Rn or a properly normalized vector uniformly distributed on the unit ball Bnp = x ∈ Rn : nj=1 |x j | p ≤ 1 in lnp (see [13], Section 2 for k = 1). The main result of the present paper is: Theorem 1.9 Let m and n be positive integers satisfying (1.4) with k = 2, and let m {Cτoαn}sαi=d1erbemaatsreicteosf Mreanl,mnu,2m(yb)er(s1.u3n)icfoorrmrelsypboonudnindgedtoinCαLTa-nvdecmtorasndy s∈atRisnfy.iInfgN(1n.[7ϕ)]. are the linear statistics of their eigenvalues (1.11) corresponding to a test function ϕ ∈ Hs , s > 5/2, then {n−1/2Nn◦[ϕ]}n converges in distribution to a Gaussian random variable with zero mean and the variance V [ϕ] = limη↓0 Vη[ϕ], where and f is given by (1.10). V [ϕ] = a− where a± = (1 ± √c)2 and am = 1 + c. (ii) We can replace the condition of the uniform boundedness of τα with the condition of uniform boundedness of eighth moments of the Normalized Counting Measures σn, or take {τα}α being real random variables independent of y with common probability law σ having finite eighth moment. In general, it is clear from (1.23) that it should be enough to have second moments of σn being uniformly bounded in n. (iii) If in (1.23) a + b + 2 = 0, then to prove the CLT one needs to renormalize linear eigenvalue statistics. In particular, it can be shown that if y in the definition of Mn,m,k (y) is uniformly distributed on the unit sphere in Rn, then a + b + 2 = 0 and under additional assumption m/n = c + O(n−1) the variance of the linear eigenvalue statistic corresponding to a smooth enough test function is of the order O(nk−2) (cf 1.20). The paper is organized as follows. Section 3 contains some known facts and auxiliary results. In Sect. 4, we prove Theorem 1.2 on the convergence of the NCMs of eigenvalues of Mn,m,k . Sections 5 and 7 present some asymptotic properties of bilinear forms (H Y, Y ), where Y is given by (1.1) and H does not depend on Y . In Sect. 6, we prove Lemma 1.6. In Sect. 8, the limit expression for the covariance of the resolvent traces is found. Section 9 contains the proof of the main result, Theorem 1.9. 2 Notations Here and in what follows gn(z) = n−k γn(z), fn(z) = E{gn(z)}. j=1 so that for the nonbold Latin and Greek indices the summations are from 1 to n and from 1 to m, respectively. For α ∈ [m], let Thus the upper index α indicates that the corresponding function does not depend on Yα. We use the notations Eα{. . .} and (. . .)◦α for the averaging and the centering with respect to Yα, so that (ξ )◦α = ξ − Eα{ξ }. In what follows we also need functions (see (4.5) below) Writing O(n− p) or o(n− p) we suppose that n → ∞ and that the coefficients in the corresponding relations are uniformly bounded in {τα}α, n ∈ N, and z ∈ K . We use the notation K for any compact set in C \ R. Given matrix H , ||H || and ||H ||H S are the Euclidean operator norm and the HilbertSchmidt norm, respectively. We use C for any absolute constant which can vary from place to place. 3 Some Facts and Auxiliary Results We need the following bound for the martingales moments, obtained in [10]: Proposition 3.1 Let {Sm }m≥1 be a martingale, i.e., ∀m, E{Sm+1 | S1, . . . , Sm } = Sm and E{|Sm |} < ∞. Let S0 = 0. Then for every ν ≥ 2, there exists an absolute constant Cν such that for all m = 1, 2 . . . Lemma 3.2 Let {ξα}α be independent random variables assuming values in Rnα and having probability laws Pα, α ∈ [m], and let : Rn1 × . . . × Rnm → C be a Borel measurable function. Then for every ν ≥ 2, there exists an absolute constant Cν such that for all m = 1, 2 . . . − E{ }|ν } ≤ Cν mν/2−1 Proof This simple statement is hidden in the proof of Proposition 1 in [25]. We give its proof for the sake of completeness. For α ∈ [m], denote E≥α = Eα . . . Em . Applying Proposition 3.1 with S0 = 0, Sα = E≥α+1{ } − E{ }, Sm = − E{ }, we get By the Ho¨ lder inequality which implies (3.2). − E{ }|ν } ≤ Cν mν/2−1 E{|E≥α+1{ } − E≥α { }|ν }. Lemma 3.3 Fix ≥ 2 and k ≥ 2. Let y ∈ Rn be a normalized isotropic random vector (1.2) such that for every n × n complex matrix H which does not depend on y, we have E{|( H y, y)◦| } ≤ || H || δn , δn = o(1), n → ∞. Then there exists an absolute constant C such that for every nk × nk complex matrix H which does not depend on y, we have E{|(HY, Y )◦| } ≤ C k /2||H|| δn , Proof It follows from (3.2) that E{|(HY, Y )◦| } ≤ C k /2−1 E{|(HY, Y )◦j | }, j =1 (HY, Y ) = Hp, qYpYq = ( H ( j )y( j ), y( j )), where H ( j ) is an n × n matrix with the entries (H ( j))st = This and (3.3) yield = E j {|( H ( j )y( j ), y( j ))◦| } ≤ || H ( j )|| δn . || H ( j )|| ≤ ||H|| i = j E j {|(HY, Y )◦j | } ≤ ||H|| E{||y(i)||2 }δn ≤ C ||H|| δn. i = j This and (3.5) lead to (3.4), which completes the proof of the lemma. The following statement was proved in [20]. Proposition 3.4 Let Nn be the NCM of the eigenvalues of Mn = α ταYαYα T , where m m {Yα}α=1 ∈ R p are i.i.d. random vectors and {τα}α=1 are real numbers. Then Var{Nn( )} ≤4m/ p2, ∀ ⊂ R, Var{gn(z)} ≤4m/( p| z|)2, ∀z ∈ C \ R. Also, we will need the following simple claim: Claim 3.5 If h1, h2 are bounded random variables, then Var{h1h2} ≤ C Var{h1} + Var{h2} . 4 Proof of Theorem 1.2 Theorem 1.2 essentially follows from Theorem 3.3 of [20] and Lemma 3.3; here we give a proof for the sake of completeness. In view of (3.6) with p = nk , it suffices to prove that the expectations N n = E{Nn} of the NCMs of the eigenvalues of Mn converge weakly to N . Due to the one-to-one correspondence between nonnegative measures and their Stieltjes transforms (see, e.g., [2]), it is enough to show that the Stieltjes transforms of N n, λ − z converge to the solution f of (1.10) uniformly on every compact set K ⊂ C \ R, and that fn(z) = In [20], it is proved that the solution of (1.10) satisfies (4.1), so it is enough to show that fn (z) ⇒ f (z), z ∈ K , (4.2) n→∞ where we use the double arrow notation for the uniform convergence. Assume first that all τα are bounded: Since Mn − Mn = ταYαYαT , the rank one perturbation formula α = − 1 + τα(GαYα, Yα) τα((Gα)2Yα, Yα) Bα γn − γnα = − 1 + τα(GαYα, Yα) = − Aα . It follows from the spectral theorem for the real symmetric matrices that there exists a nonnegative measure mα such that | Aα| ≥ | Aα| = |τα|| z| where we use ||G|| ≤ | z|−1. Let us show that It follows from (1.2) that | Aα−1| ≤ 1 + |τα| · ||Yα||2/| z|, |Eα{ Aα}|−1, |E{ Aα}|−1 ≤ 4(1 + |τα|/| z|). Eα{ Aα} = 1 + τα gnα(z), E{ Aα} = 1 + τα fnα(z). where Nnα is the counting measure of the eigenvalues of Mnα. For every η ∈ R \ {0}, consider < n−k n−k |Eα{ Aα}| ≥ | Eα{ Aα}| = |τα||η|n−k zgn(z) = −1 + n−k Tr Mn G = − 1 + mn−k − n−k This and the identity z fn(z) = − 1 + n−k − n−k rn(z) =n−k It follows from the Schwarz inequality that |E{ A◦α Aα−1}| ≤ E{| A◦α|2}1/2E{| Aα−2|}1/2. Note that since E{||Yα|| = 1}, we have by (1.8) E{||Yα||4} ≤ C . This and (4.8) imply that E{| Aα−2|} is uniformly bounded in |τα| ≤ L and z ∈ K . We also have A◦α = ( Aα)◦α + τα(gnα)◦ = τα (GαYα, Yα)◦α + (gnα)◦ , By (1.4) and (3.7) with p = nk , Var{gnα} ≤ C n−k | z|−2. It follows from (1.8) and Lemma 3.3 with H = Gα and = 2 that Eα{|(GαYα, Yα)◦α|2} ≤ C2k||Gα||2δn ≤ C2k| z|−2δn. |rn| ≤ C (kδn + n−k )1/2. uniformly in |τα| ≤ L and z ∈ K . Hence z fn(z) = (−1 + mn−k ) − n−k It follows from (4.5) and (4.7) that | fn(z) − fnα(z)| ≤ n−k | z|−1. This and (4.9) imply that |1+τα fn(z)|−1 is uniformly bounded in |τα| ≤ L and z ∈ K . Hence, in (4.15) we can replace fnα with fn (the corresponding error term is of the order O(n−k )) and pass to the limit as n → ∞. Taking into account (1.7) we get that the limit of every convergent subsequence of { fn(z)}n satisfies (1.10). This finishes the proof of the theorem under assumption (4.3). Consider now the general case and take any sequence {σn} = {σm(n)} satisfying (1.7). For any L > 0, introduce the truncated random variables τα, |τα| < L , 0, otherwise. L rank(Mn − Mn ) ≤ Card{α ∈ [m] : |τα| ≥ L}. i → ∞. If NnLi is the NCM of the eigenvalues of MnLi and N nLi is its expectation, then the mini-max principle implies that for any interval ⊂ R: |N n( ) − N nLi ( )| ≤ uniformly on K . It follows from the first part of the proof that where cLi = cσ [−Li , Li ] → c as Li → ∞. Since N (R) = 1, there exists C > 0, such that Hence we have for all sufficiently big Li : Thus |τ /(1 + τ f Li (z))| ≤ | f Li (z)|−1 ≤ 2/C < ∞, z ∈ K . This allows us to pass to the limit Li → ∞ in (4.17) and to obtain (1.10) for f , which completes the proof of the theorem. Remark 4.1 It follows from the proof that in the model we can take k depending on n such that k → ∞ 5 Variance of Bilinear Forms Lemma 5.1 Let Y be defined in (1.1–1.2), where y has an unconditional distribution and satisfies (1.19). Then for every symmetric nk ×nk matrix H which does not depend on y and whose operator norm is uniformly bounded in n, there is an absolute constant C such that nVar{(H Y, Y )} ≤ C n−k ||H ||2H S ≤ C ||H ||2. If additionally y satisfies (1.13–1.14), then we have nVar{(H Y, Y )} = ka|n−k Tr H |2 + n−2k+1 + O(n−1), i=1 j,p where j( pi ) = { j1, . . . , ji−1, pi , ji+1, . . . , jk }. Proof Since y has an unconditional distribution, we have E{|(H Y, Y )|2} = i=1 For W ⊂ [k], W c = [k] \ W , denote For every fixed W, j, s, we have (W, j, s, p, q) = i∈W c ∈W (W, j, s, p, q) = O(n−k−|W |). Indeed, the number of pairs for which (W, j, s, p, q) = 0 does not exceed 2|W |nk−|W | (the number of choices of indices pi = qi for i ∈/ W equals to nk−|W |; all other indices p , q ( ∈ W ) must satisfy { p , q } = { j , s } and, therefore, can be chosen in at most two ways each). Since a2,2, wi = O(n−2), (5.4) follows. For every fixed W , |Hj, s||Hp, q| (W, j, s, p, q) ≤ (W, j, s, p, q)/2 = O(n−k−|W |)||H ||2H S. Var{(H Y, Y )} = r=0 |W |=r j,s,p,q This and (1.19) imply that i=1 T0 := (a2,2δ ji si δ pi qi ) = a2,2| Tr H |2. k n T0 − n−2k | Tr H |2 ≤ C n−k ||H ||2H S, and by (1.13), and by (1.13) T1 : = i=1 j,p i=1 j,p The term corresponding to Hj, s H p, q wi (j, s, p, q) =i a2k,2 Hj, j( pi ) H p, p( ji ) + a2k,−21κ4 Hj, j H p, pδ pi ji , Proof The proof follows the scheme proposed in [25] (see also Lemma 3.2 of [13]). For q = 1, 2, by (3.2) we have Also it follows from (5.5) that the terms corresponding to W : |W | ≥ 2 are less than C n−k−2||H ||2H S. Summarizing (5.6–5.8), we get (5.1) and (5.2) and complete the proof of the lemma. 6 Proof of Lemma 1.6 Lemma 6.1 Let {τα}α be a collection of real numbers satisfying (1.7), (1.18), and let y be a normalized isotropic vector having an unconditional distribution and satisfying (1.19). Consider the corresponding matrix Mn (1.3) and the trace of its resolvent γn(z) = Tr(Mn − z I )−1. We have Var{γn(z)} ≤ C nk−1| z|−6. E{|γn◦(z)|4} ≤ C n2k−2| z|−12. Applying (4.5), (4.7), and (4.9) we get E{|(γn)◦α|2q } = E{|γn − γnα − Eα{γn − γnα}|2q } ≤ C E ABαα − EEαα{{ABαα}} 2q = C E E(αB{αA)α◦α} − ABαα · E(αA{αA)α◦α} 2q ≤ C (1 + |τα|/| z|)2q E Eα{|(Bα)◦α|2q } + Eα{|( Aα)◦α|2q }/| z|2q . (6.4) Here by (5.1) nτα−2Eα{|(Bα)◦α|2} ≤ C n−k ||(Gα)2||2H S ≤ | z|−4. This and (6.3–6.4) lead to (6.1). Also it follows from (1.15) and Lemma 3.3 that Eα{|(Bα)◦α|4}, Eα{|( Aα)◦α|4}/| z|4 ≤ C τα4| z|−8n−2, which leads to (6.2). Proof of Lemma 1.6 The proof of (1.20) is based on the following inequality obtained in [25]: for ϕ ∈ Hs (see 1.17), Var{Nn[ϕ]} ≤ Cs ||ϕ||s2 ≤ C n−k−1 τα2(1 + η−2τα2)E{||(Gα)2||2H S + η−2||Gα||2H S}. By the spectral theorem for the real symmetric matrices, where Nnα is the expectation of the counting measure of the eigenvalues of Mnα. We have n−k Summarizing, we get Var{Nn[ϕ]} ≤ C nk−1||ϕ||s2 dηe−ηη2s−6 ≤ C nk−1||ϕ||s2 provided that s > 5/2. This finishes the proof of Lemma 1.6. 7 Case k = 2: Some Preliminary Results From now on we fix k = 2 and consider matrices Mn = Mn,m,2. For every j = { j1, j2} = j1 j2, j=1 Lemma 7.1 Under conditions of Theorem 1.9, Eα{|( Aα)◦α| p} ≤C (τα/| z|) pn− p/2, Eα{|(Bα)◦α| p} ≤C (τα/| z|2) pn− p/2, E{| A◦α| p}, E{|Bα◦| p} = O(n− p/2), 2 ≤ p ≤ 6. Eα{|( Aα)◦α|6} ≤ C (τα/| z|)6n−3, and by the Ho¨lder inequality we get the first estimate in (7.1). Analogously one can get the second estimate in (7.1). Also we have by (6.1) E{|(gnα)◦| p} ≤ | z|2− pE{|(gnα)◦|2} = O(n−3), p ≥ 2, which together with (4.13) and (7.1) leads to (7.2). It follows from (5.2) with k = 2 that nVar{(H Y, Y )} =2a|n−2 Tr H |2 + 2n−3 + bn−3 Hj, j1 p2 H p, p1 j2 + Hj, p1 j2 H p, j1 p2 Hj, j H p, p(δ p1 j1 + δ p2 j2 ) + O(n−1). Consider an n × n matrix of the form Gs, p = Since G = ||G|| ≤ We define functions Similarly, we introduce the matrix gn(1)(z1, z2) :=n−3 gn(2)(z1, z2) :=n−3 His, is (z1)H js, js (z2) = n−3 Gss (z1)Gss (z2). Gi, j = and define functions It follows from (7.3) that =2a(gnα(z))2 + 2(gn(1)(z, z) + gn(1)(z, z)) + b(gn(2)(z, z) + gn(2)(z, z)) + O(n−1). Var{gn(i)}, Var{gn(i)} =O(n−2), lim E{gn(i)(z1, z2)} = nl→im∞ E{gn(i)(z1, z2)} = f (z1) f (z2), n→∞ where f is the solution of (1.10). Proof We prove the lemma for gn(1); the cases of gn(2), gn(2), and gn(2) can be treated similarly. Without loss of generality we can assume that in the definitions of G and gn(1), H = G. It follows from (3.2) that Var{gn(1)} ≤ gn(1) − gn(1)α =n−3 Tr(G(z1) − Gα(z1))G(z2) + n−3 Tr Gα(z1)(G(z2) − Gα(z2)) =: Sn(1) + Sn(2). and to get (7.7), it is enough to show that Consider Sn(1). It follows from (4.4) that E{|Sn( j)|2} = O(n−4), j = 1, 2. Since for x , ξ ∈ Rn and an n × n matrix D Di j xi ξ j ≤ ||D|| · ||x || · ||ξ ||, This and following from (1.2) and (1.22) bound z1 fn(1)(z1, z2) = − fn(z2) + n−3 − n−3 = − fn(z2) + Tn(1) + Tn(2). j,p α τα2E Yαj(H Aα(αz(1z)1Y)α) j1 p2 · (H α(z2)Yα)Apα((Hz2α)(z2)Yα)p1 j2 imply (7.9) for j = 1. The case j = 2 can be treated similarly. So we get (7.7) for gn(1). (YαYαT H )j, q = Aα−1Yαj(H αYα)q. This and the resolvent identity yield Hj, q(z1) = −z1−1δj, q + z1−1 It follows from (1.2) that This and (4.12) yield Tn(1) = n−5 rn = n−3 Treating rn we note that By the Ho¨lder inequality, (4.8), and (7.13) ≤ C n−3 τα2E{||Yα||4| Aα(z1)|−1| Aα(z2)|−1} = O(n−1). Eα{Yαj(H αYα) j1 p2 } = n−2 Hjα, j1 p2 . E A◦α(z1) Yαj(H α(z1)Yα) j1 p2 Hpα, p1 j2 (z2) . Aα(z1) n−1 Hence, by the Schwarz inequality, (4.8), (4.9), (7.2), and (7.13) |rn| ≤ C n−2 ≤ C n−2 E{| A◦α|2}1/2E{| Aα|−2||Yα||4}1/2 = O(n−1/2). Also one can replace fnα and H α with fn and G (the error term is of the order O(n−1)). Hence, z1 fn(1)(z1, z2) = − fn(z2) + fn(1)(z1, z2)n−2 This, (1.4), (1.7), and (1.10) lead to f (1)(z1, z2) = f (z2) c −1 = f (z1) f (z2) and finishes the proof of the lemma. It follows from Lemmas 5.1 and 7.2 that under conditions of Theorem 1.9 lim nτα−2E{ A◦α(z1) Aα(z2)} = 2(a + b + 2) f (z1) f (z2), n→∞ where f is the solution of (1.10). Lemma 7.3 Under conditions of Theorem 1.9 Var{Eα{( A◦α) p}} = O(n−4), p = 2, 3. Proof Since τα, α ∈ [m], are uniformly bounded in α and n, then to get the desired bounds it is enough to consider the case τα = 1, α ∈ [m]. By (4.13), we have Eα{( A◦α)2} = Eα{(H Yα, Yα)◦α2} + (gnα)◦2, It also follows from (7.6) and Lemmas 6.1 and 7.2 that Var{Eα{(H Yα, Yα)◦α2}} = O(n−4), Var{Eα{(H Yα, Yα)◦α3}} = O(n−4). Var Eα{(H Yα, Yα)3} − gnα3 = O(n−4). Hi, j Hp, q Hs, t (i, j, p, q, s, t), k=1 (i, j, p, q, s, t) = − 3Eα{(H Yα, Yα)} · Eα{(H Yα, Yα)◦α2} = Eα{(H Yα, Yα)3} − gnα3 − 3gnα · Eα{(H Yα, Yα)◦α2}. + O(n−4), and by (1.21) (i, j, p, q, s, t) = O(n−6). Also, due to the unconditionality of the distribution, contains only even moments. Thus in the index pairs i, j, p, q, s, t ∈ [n]2, every index (both on the first positions and on the second positions) is repeated an even number of times. Hence, there are at most 6 independent indices: ≤ 3 on the first positions (call them i, j, k) and ≤ 3 on the second positions (call them u, v, w). For every fixed set of independent indices, consider maps from this set to the sets of index pairs {i, j, p, q, s, t}. We call such maps the index schemes. Let | | be the cardinality of the corresponding set of independent indices. For example, is an index scheme with 5 independent indices (i, j on the first positions and u, v, w on the second positions). The inclusion–exclusion principle allows to split the expression (7.19) into the sums over fixed sets of independent indices of cardinalities from 2 to 6 with the fixed coefficients depending on a2,2,2, a2,4, and a6 in front of every such sum. We have =2 S , S = : | |= where the last sum is taken over the set of independent indices of cardinality , is an index scheme constructing pairs {i, j, p, q, s, t} from this set, and ( ) is a certain expression, depending on , a2,2,2, a2,4, and a6. For example, S2 = F (a2,2,2, a2,4, a6) where F (a2,2,2, a2,4, a6) can be found by using the inclusion–exclusion formulas. As to ( ) in (7.21), the only thing we need to know is that ( ) = O(n−6), and that in the particular case of we have by (1.21) Tr : {i, j, k ; u, v, w} → {(i, u), (i, u); ( j, v), ( j, v); (k, w), (k, w)}, =2 = O(n−4). Hence to get (7.18) it suffices to consider terms with 5 and 6 independent indices and show that = O(n−4). Consider S5. In this case we have exactly 5 independent indices. By the symmetry we can suppose that there are two first independent indices, i, j , and three second independent indices, u, v, w, and that we have i on four places and j on two places. Thus, S5 is equal to the sum of terms of the form S5 =O(n−6) S5 =O(n−6) Here we suppose that there are some fixed indices on the dot places, which are different from explicitly mentioned ones. Note that S5 has a single “external” pairing with respect to j . While estimating the terms, our argument is essentially based on the simple relations and on the observation that the more the mixing of matrix entries we have the lower order of sums we get. Let V ⊂ Rn be the set of vectors of the form ξ = {ξ j }nj=1 = {H··, j·}nj=1 or ξ = {H··, ·u }un=1, and let W be the set of n × n matrices of the form It follows from (7.24) that ∀ξ ∈ V ||ξ || = O(1) and ∀D ∈ W ||D|| = O(1). |H··, j· H··, j·| =O(1), |H··, ·u H··, ·u | = O(1), Hi·, j· Hi·, ·· H··, j· =O(1), and Hi·, ·u Hi·, ·· H··, ·u = O(1). In particular, by (7.24) and (7.25), we have for S5 |S5| ≤ O(n−6) |Hi·, j· Hi·, j·| = O(n−2), so that Var{S5} = O(n−4). Consider S5 . Note that if in S5 we have a single “external” pairing with respect to at least one index on the second positions, then similar to S5, the variance of this term is of the order O(n−4). So we are left with the terms of the form S5 = O(n−6) Hiu, iu Hiv, iv H jw, jw. It follows from (7.5) that S5 = O(n−1) · gnα(z) · gn(2)(z, z). Now (3.8), (6.1), and (7.7) imply that Var{S5 } ≤ C n−2(Var{gn } + Var{gn(2)}) = O(n−4). α Summarizing we get Var{S5} = O(n−4). Consider S6 and show that Var{S6 − gα3 n } = O(n−4). In this case we have 6 independent indices, i, j, k for the first positions and u, v, w for the second positions. Suppose that we have two single external pairing with respect to two different first indices and consider terms of the form S6 =O(n−6) S6 =O(n−6) S6 = O(n−6) If the second indices in Hk·, k· are not equal, then we get the expression of the form S6 = O(n−6) It follows from (7.26) that S6 = O(n−2); hence Var{S6 } = O(n−4). If the second indices in Hk·, k· in (7.27) are equal, then we get the expressions of three types: O(n−6) O(n−6) O(n−6) Hiu, jv Hiu, jv Hkw, kw =gnαn−4 (Hiu, jv)2 = O(n−2), Hiu, jv Hiv, ju Hkw, kw =gnαn−4 Hiu, jv Hiv, ju = O(n−2), Hiu, ju Hiv, jv Hkw, kw =O(n−1)gn(1)(z, z)gnα(z), where we used (7.24) to estimate the first two expressions, so that their variances are of the order O(n−4). It also follows from (3.8), (6.1), and (7.7) that the variance of the third expression is of the order O(n−4). Hence, Var{S6 } = O(n−4). It remains to consider the term without external pairing, which corresponds to 2 3 Hiu, iu H jv, jv Hkw, kw = (a2,2,2) γn (see (7.22)). Summarizing we get = O(n−2)Var{gnα3} + O(n−4) = O(n−4), where we used (1.21) and (6.1). This leads to (7.23) and completes the proof of the lemma. 8 Covariance of the Resolvent Traces Lemma 8.1 Suppose that the conditions of Theorem 1.9 are fulfilled. Let C (z1, z2) = 2(a + b + 2)c f (z1) f (z2) (1 + τ f (z1))2 (1 + τ f (z2))2 Proof For a convergent subsequence {Cni }, denote C (z1, z2) := nil→im∞ Cni (z1, z2). We will show that for every converging subsequence, its limit satisfies (8.1). Applying the resolvent identity, we get (see (4.11)) E{ Aα−1(z1)(γn − γnα)◦(z2)} =: Tn(1) + Tn(2). Consider Tn(1). Iterating (4.12) four times, we get =: Sn(1) + Sn(2) + Sn(3) + Sn(4). It follows from (4.9), (6.1), and (7.15) that Sn(i) = O(n−1/2), i = 2, 3. Also, by (4.8) we have where by the Schwarz inequality, (6.2), (7.1), and (7.13) ≤ E{|γnα◦(z2)|}/|z1| + E{|γn◦(z1)|}/|z2| = O(n1/2). 1 τα = Cn(z1, z2) n2z1 α (1 + τα fnα(z1))2 + O(n−1/2). and we have Summarizing, we get Consider now Tn(2) of (8.2). By (4.5), 1 ∂ E{ A◦α(z1) A◦α(z2)} E{ Aα−1(z1)(Bα/ Aα)◦(z2)} = − E{ Aα(z1)}2 ∂ z2 E{ Aα(z2)} + O(n−3/2). For shortness let for the moment Ai = Aα(zi ), i = 1, 2, B2 = Bα(z2). Iterating (4.12) with respect to A1 and A2 two times we get E{(1/ A1)◦(B2/ A2)◦} E{ A1}2E{ A2}2 −E{ A1◦ B2}E{ A2} + E{B2}E{ A1◦ A2} E{ A1}2E{ A2}2 E{ A1}2E{ A2}2 Applying (1.22), (7.13), and using bounds (4.7), (4.8), (4.9) for |B2/ A2|, | Ai |−1, |E{ Ai }|−1, i = 1, 2, one can show that the terms containing at least three centered factors A◦1, A◦2, B2◦ are of the order O(n−3/2). This implies that E{(1/ A1)◦(B2/ A2)◦} = E{ A1}2E{ A2}2 Returning to the original notations and taking into account that D = 2(a + b + 2). It follows from (7.14) and (8.4–8.5) that Denote for the moment This and (8.2–8.3) yield Note that by (1.10), (1 + τ f (z1))2 ∂ z2 1 + τ f (z2) Dc C (z1, z2) = c τ (1+τ f (z1))−2dσ (τ )−z1 (1+τ f (z1))2 ∂ z2 1 + τ f (z2) which completes the proof of the lemma. 9 Proof of Theorem 1.9 τ dσ (τ ) f (z) (1 + τ f (z))2 − z = f (z) C (z1, z2) = Dc f (z1) f (z2) (1 + τ f (z1))2 (1 + τ f (z2))2 and “∗” denotes the convolution. We have Denote for the moment the characteristic function (9.1) by Zn[ϕ], to make explicit its dependence on the test function. Take any converging subsequence {Zn j [ϕ]} ∞j=1 Without loss of generality assume that the whole sequence {Zn j [ϕη]} converges as n j → ∞. By (1.20), we have |Zn j [ϕ] − Zn j [ϕη]| ≤ |x |n−1/2 Var{Nn j [ϕ] − Nn j [ϕη]} lim Zn j [ϕ] = lηi↓m0 n j →∞ lim Zn j [ϕη]. n j →∞ Thus it suffices to find the limit of as n → ∞. It follows from (9.2) – (9.3) that This allows to write ϕ(μ) γn(z)dμ, z = μ + i η. Yn(z, x ) = n−1/2E{γn(z)eη◦n (x )}. Since |Yn(z, x )| ≤ 2n−1/2Var{γn(z)}1/2, it follows from the proof of Lemma 1.6 that for every η > 0 the integrals of |Yn(z, x )| over μ are uniformly bounded in n. This and the fact that ϕ ∈ L2 together with Lemma 9.1 below show that to find the limit of integrals in (9.7) it is enough to find the pointwise limit of Yn(μ + i η, x ). We have ϕ(λ1) (γn − γnα)◦(z1)dλ1 ϕ(λ1) (γn − γnα)◦(z1)dλ1 i x eηαn E{ Aα−n1(z)(eηn − eηαn )◦(x )} = √ ϕ(λ1) (γn − γnα)◦(z1)dλ1 where z j = λ j + i η, j = 1, 2, and Using the argument of the proof of the Lemma 8.1, it can be shown that Rn = O(n−5/2). Hence, 1 Yn(z, x ) = − zn1/2 Treating the r.h.s. similarly to Tn(1) and Tn(2) of (8.2), we get Yn(z, x ) = where C (z, z1) is defined in (8.1). It follows from (9.7) and (9.8) that E{eηαn (x )( Aα−1(z))◦ (γn − γnα)◦(z1)}dλ1 + O(n−1). (see (1.23)) and finally Taking into account (9.5), we pass to the limit η ↓ 0 and complete the proof of the theorem. It remains to prove the following lemma. Lemma 9.1 Let g ∈ L 2(R) and let {hn } ⊂ L 2(R) be a sequence of complex-valued functions such that and hn → h a.e. as n → ∞, where |h(x )| ≤ ∞ g(x )hn (x )dx → g(x )h(x )dx as n → ∞. Proof According to the convergence theorem of Vitali (see, e.g., [24]), if ( X, F , μ) is a positive measure space and then F ∈ L 1(μ) and limn→∞ that g(x ) = 0, x ∈ R, and take {Fn }n is uniformly integrable, ∞, |F (x )| ≤ ∞ Fn = ghn /|g|2, F = gh/|g|2. |g(x )|2dx < ∞, ∞, |F (x )| ≤ ∞ Hence, the conditions of Vitali’s theorem are fulfilled and we get |hn − h||g|dx = 0, Acknowledgements The author would like to thank Leonid Pastur for an introduction to the problem and for fruitful discussions. Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. 1. 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A. Lytova. Central Limit Theorem for Linear Eigenvalue Statistics for a Tensor Product Version of Sample Covariance Matrices, Journal of Theoretical Probability, 2017, 1-34, DOI: 10.1007/s10959-017-0741-9