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

For k,m,n∈N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k,m,n\in {\mathbb {N}}$$\end{document}, we consider nk×nk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n^k\times n^k$$\end{document} random matrices of the form Mn,m,k(y)=∑α=1mταYαYαT,Yα=yα(1)⊗⋯⊗yα(k),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\mathcal {M}}_{n,m,k}({\mathbf {y}})=\sum _{\alpha =1}^m\tau _\alpha {Y_\alpha }Y_\alpha ^T,\quad {Y}_\alpha ={\mathbf {y}}_\alpha ^{(1)}\otimes \cdots \otimes {\mathbf {y}}_\alpha ^{(k)}, \end{aligned}$$\end{document}where τα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{\alpha }$$\end{document}, α∈[m]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in [m]$$\end{document}, are real numbers and yα(j)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf {y}}_\alpha ^{(j)}$$\end{document}, α∈[m]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in [m]$$\end{document}, j∈[k]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j\in [k]$$\end{document}, are i.i.d. copies of a normalized isotropic random vector y∈Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf {y}}\in {\mathbb {R}}^n$$\end{document}. For every fixed k≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k\ge 1$$\end{document}, if the Normalized Counting Measures of {τα}α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{\tau _{\alpha }\}_{\alpha }$$\end{document} converge weakly as m,n→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m,n\rightarrow \infty $$\end{document}, m/nk→c∈[0,∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m/n^k\rightarrow c\in [0,\infty )$$\end{document} and y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf {y}}$$\end{document} is a good vector in the sense of Definition 1.1, then the Normalized Counting Measures of eigenvalues of Mn,m,k(y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {M}}_{n,m,k}({\mathbf {y}})$$\end{document} converge weakly in probability to a nonrandom limit found in Marchenko and Pastur (Math USSR Sb 1:457–483, 1967). For k=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=2$$\end{document}, we define a subclass of good vectors y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf {y}}$$\end{document} for which the centered linear eigenvalue statistics n-1/2Trφ(Mn,m,2(y))∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n^{-1/2}{{\mathrm{Tr}}}\varphi ({\mathcal {M}}_{n,m,2}({\mathbf {y}}))^\circ $$\end{document} converge in distribution to a Gaussian random variable, i.e., the Central Limit Theorem is valid.


Introduction: Problem and Main Result
For every k ∈ N, consider random vectors of the form Y = y (1) ⊗ · · · ⊗ y (k) ∈ (R n ) ⊗k , (1.1) where y (1) ,…, y (k) are i.i.d. copies of a normalized isotropic random vector y = (y 1 , . . . , y n ) ∈ R n , E{y j } = 0, E{y i y j } = δ i j n −1 , i, j ∈ [n], (1.2) [n] = {1, . . . , n}. The components of Y have the form where we use the notation j for k-multiindex: For every m ∈ N, let {Y α } m α=1 be i.i.d. copies of Y , and let {τ α } m α=1 be a collection of real numbers. Consider an n k × n k real symmetric random matrix corresponding to a normalized isotropic random vector y, where β=1 . Such matrices with T m ≥ 0 (not necessarily diagonal) are known as sample covariance matrices. The asymptotic behavior of their spectral statistics is well studied when all entries of Y α are independent. Much less is known in the case when columns Y α 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 M p n , p ∈ N, it was proved that for the special cases of random vectors y uniformly distributed on the unit sphere in C n 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 M n,m,k (y) and also to prove the Central Limit Theorem for linear eigenvalue statistics in the case k = 2.
Let {λ (n) l } n k l=1 be the eigenvalues of M n counting their multiplicity, and introduce their Normalized Counting Measure (NCM) N n , setting for every ⊂ R In the case k = 1, there are a number of papers devoted to the convergence of the NCMs of the eigenvalues of M n,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 M n,m,1 was proved in the case when corresponding vectors {Y α } α are "good vectors" in the sense of the following definition.
Definition 1. 1 We say that a normalized isotropic vector y ∈ R n is good, if for every n × n complex matrix H n which does not depend on y, we have Var{(H n y, y)} ≤ ||H n || 2 δ n , δ n = o(1), n → ∞, (1.8) where ||H n || is the Euclidean operator norm of H n .
Following the scheme of the proof proposed in [20], we show that despite the fact that the number of independent parameters, kmn = O(n k+1 ) for k ≥ 2, is much less than the number of matrix entries, n 2k , the limiting distribution of eigenvalues still obeys the Marchenko-Pastur law. We have: is the unique solution of the functional equation in the class of analytic in C \ R functions such that f (z) z ≥ 0, z = 0.
We use the notation for the integrals over R. Note that in [26] there was proved an analog of this statement for a deformed version of M n,m,2 .
It follows from Theorem 1.2 that if is the linear eigenvalue statistic of M n 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 N n [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 M n,m,2 defined in (1.3).
There are a considerable number of papers on the CLT for linear eigenvalue statistics of sample covariance matrices M n,m,1 (1.5), where all entries of the matrix B n,m,1 are independent (see [4,[7][8][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 M n,m,1 , corresponding to some special class of isotropic vectors defined below.

Definition 1.3
The distribution of a random vector y ∈ R n is called unconditional if its components {y j } n j=1 have the same joint distribution as {±y j } n j=1 for any choice of signs. Definition 1. 4 We say that normalized isotropic vectors y ∈ R n , 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 → ∞, where || . . . || H is a functional norm and C n depends only on n. This bound determines the normalization factor in front of N • n [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 M n,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 N • n [ϕ] itself without any n-dependent normalization factor in front. Consider the test functions ϕ : R → R from the Sobolev space H s , possessing the norm The following statement was proved in [13]   Here we prove an analog of Theorem 1.5 in the case k = 2. We start with establishing a version of (1.16) in general case k ≥ 1:  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 M n , 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 ∈ R n 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 ∈ R n , n ∈ N, are the CLTvectors if they have unconditional permutationally invariant distributions and satisfy the following conditions: It can be shown that a vector of the form y = x/n 1/2 , 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 R n or a properly normalized vector uniformly distributed on the unit ball B n p = x ∈ R n : n j=1 |x j | p ≤ 1 in l n p (see [13], Section 2 for k = 1).
The main result of the present paper is: where a ± = (1 ± √ c) 2 and a m = 1 + c. (ii) We can replace the condition of the uniform boundedness of τ α with the condition of uniform boundedness of eighth moments of the Normalized Counting Measures σ n , or take {τ α } 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 σ n being uniformly bounded in n. 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 M n,m,k . Sections 5 and 7 present some asymptotic properties of bilinear forms (HY, 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.

Notations
Let I be the n k × n k identity matrix. For z ∈ C, z = 0, let G(z) = (M n − z I ) −1 be the resolvent of M n , and Here and in what follows , and α = m α=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 Y α . We use the notations E α {. . .} and (. . .) • α for the averaging and the centering with 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 {τ α } 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 Hilbert-Schmidt norm, respectively. We use C for any absolute constant which can vary from place to place.

Some Facts and Auxiliary Results
We need the following bound for the martingales moments, obtained in [10]: (3.1)

Lemma 3.2
Let {ξ α } α be independent random variables assuming values in R n α and having probability laws P α , α ∈ [m], and let : R n 1 × . . . × R n m → C be a Borel measurable function. Then for every ν ≥ 2, there exists an absolute constant C ν such that for all m = 1, 2 . . .
By the Hölder inequality

Lemma 3.3
Fix ≥ 2 and k ≥ 2. Let y ∈ R n be a normalized isotropic random vector (1.2) such that for every n × n complex matrix H which does not depend on y, Then there exists an absolute constant C such that for every n k × n k complex matrix H which does not depend on y, we have where H ( j) is an n × n matrix with the entries This and (3.3) yield We have This and (3.5) lead to (3.4), which completes the proof of the lemma.
The following statement was proved in [20].
Also, we will need the following simple claim: 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 = n k , it suffices to prove that the expectations N n = E{N n } of the NCMs of the eigenvalues of M n 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 , converge to the solution f of (1.10) uniformly on every compact set K ⊂ C \ R, and that lim In [20], it is proved that the solution of (1.10) satisfies (4.1), so it is enough to show that where we use the double arrow notation for the uniform convergence. Assume first that all τ α are bounded: It follows from the spectral theorem for the real symmetric matrices that there exists a nonnegative measure m α such that This yields It also follows from (4.4) that where we use ||G|| ≤ | z| −1 . Let us show that It follows from (1.2) that Consider E α {A α }. By the spectral theorem for the real symmetric matrices, where N α n is the counting measure of the eigenvalues of M α n . For every η ∈ R \ {0}, consider This leads to (4.9) for E α {A α }. Replacing in our argument N α n with N α n , we get (4.9) for E{A α }.
Consider now the general case and take any sequence {σ n } = {σ m(n) } satisfying (1.7). For any L > 0, introduce the truncated random variables Take any sequence {L i } i which does not contain atoms of σ and tends to infinity as i → ∞. If N L i n is the NCM of the eigenvalues of M L i n and N L i n is its expectation, then the mini-max principle implies that for any interval ⊂ R: We have where by (1.7) the first term on the r.h.s. tends to zero as n → ∞. Hence, Thus if f and f L i are the Stieltjes transforms of N and lim n→∞ N L i n , then uniformly on K . It follows from the first part of the proof that where Hence we have for all sufficiently big L i : This allows us to pass to the limit L i → ∞ 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 → ∞ and kδ n → 0 as n → ∞, and the theorem remains valid (see 4.14).

Variance of Bilinear Forms
Proof Since y has an unconditional distribution, we have E{y j y s y p y q } = a 2,2 (δ js δ pq + δ j p δ sq + δ jq δ sp ) + κ 4 δ js δ j p δ jq .
Proof of Lemma 1.6 The proof of (1.20) is based on the following inequality obtained in [25]: for ϕ ∈ H s (see 1.17), Let z = μ + iη, η > 0. It follows from (6.3) -(6.6) that By the spectral theorem for the real symmetric matrices, where N α n is the expectation of the counting measure of the eigenvalues of M α n . We have

Case k = 2: Some Preliminary Results
From now on we fix k = 2 and consider matrices M n = M n,m,2 . For every j = { j 1 , j 2 } = j 1 j 2 , In this section we establish some asymptotic properties of A α , (G α Y α , Y α ), and their central moments. We start with Lemma 7.1 Under conditions of Theorem 1.9, and by the Hölder 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) which together with (4.13) and (7.1) leads to (7.2). Let It follows from (5.2) with k = 2 that Consider an n × n matrix of the form We define functions Similarly, we introduce the matrix and define functions It follows from (7.3) that n (z, z) + g (1) n (z, z)) + b(g (2) n (z, z) + g (2) n (z, z)) + O(n −1 ). (7.6) We have: where f is the solution of (1.10).
Proof We prove the lemma for g (1) n ; the cases of g (2) n , g (2) n , and g (2) n can be treated similarly. Without loss of generality we can assume that in the definitions of G and g (1) n , H = G. It follows from (3.2) that We have Hence and to get (7.7), it is enough to show that n . It follows from (4.4) that Since for x, ξ ∈ R n and an n × n matrix D i, j D i j x i ξ j ≤ ||D|| · ||x|| · ||ξ ||, (7.11) taking into account ||H || ≤ 1/| z|, (4.8), and (7.4) we get This and following from (1.2) and (1.22) bound E{||Y α || p } ≤ C, p ≤ 12 (7.13) imply (7.9) for j = 1. The case j = 2 can be treated similarly. So we get (7.7) for g (1) n . Let us prove (7.8) for g (1) n . Let f This and the resolvent identity yield Hence, By the Hölder inequality, (4.8), and (7.13) It follows from (1.2) that This and (4.12) yield Treating r n we note that Hence, by the Schwarz inequality, (4.8), (4.9), (7.2), and (7.13) Also one can replace f α n and H α with f n and G (the error term is of the order O(n −1 )). Hence, This, (1.4), (1.7), and (1.10) lead to and finishes the proof of the lemma.
It follows from Lemmas 5.1 and 7.2 that under conditions of Theorem 1.9 where f is the solution of (1.10). Proof Since τ α , α ∈ [m], are uniformly bounded in α and n, then to get the desired bounds it is enough to consider the case τ α = 1, α ∈ [m]. By (4.13), we have where by (6.2) E{|(g α n ) • | 2 p } = O(n −6 ), p = 2, 3, and by (7.1) and (6.1) Hence, It also follows from (7.6) and Lemmas 6.1 and 7.2 that (7.16) which leads to (7.15) for p = 2. To get (7.15) for p = 3, it is enough to show that We have It follows from (6.1), (7.16), and (3. Hence, and to get (7.17) for p = 3 it is enough to show that We have i,j,p,q,s,t H i, j H p, q H s, t (i, j, p, q, s, t), (7.19) where  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 a 2,2,2 , a 2,4 , and a 6 in front of every such sum. We have (7.21) 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 , a 2,2,2 , a 2,4 , and a 6 . For example, where F(a 2,2,2 , a 2,4 , a 6 ) 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 ), (7.22) and that in the particular case of we have by (1.21) ( ) = a 2 2,2,2 = n −6 + O(n −7 ), and the corresponding term in S 6 has the form a 2 2,2,2 (Tr H ) 3 . Note that by (7.20), S 2 is of the order O(n −4 ). By the same reason so that Hence to get (7.18) it suffices to consider terms with 5 and 6 independent indices and show that Consider S 5 . 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, S 5 is equal to the sum of terms of the form Here we suppose that there are some fixed indices on the dot places, which are different from explicitly mentioned ones. Note that S 5 has a single "external" pairing with respect to j. While estimating the terms, our argument is essentially based on the simple relations It follows from (7.5) that (2) n (z, z). Now (3.8), (6.1), and (7.7) imply that Summarizing we get Var{S 5 } = O(n −4 ). Consider S 6 and show that Var{S 6 − g α3 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 Summarizing, we get Consider now T (2) n of (8.2). By (4.5), For shortness let for the moment . Iterating (4.12) with respect to A 1 and A 2 two times we get Applying (1.22), (7.13), and using bounds (4.7), (4.8), (4.9) for |B 2 /A 2 |, |A i | −1 , |E{A i }| −1 , i = 1, 2, one can show that the terms containing at least three centered factors A • 1 , A • 2 , B • 2 are of the order O(n −3/2 ). This implies that Returning to the original notations and taking into account that Thus it suffices to find the limit of where Y n (z, x) = n −1/2 E{γ n (z)e • ηn (x)}.
Since |Y n (z, x)| ≤ 2n −1/2 Var{γ n (z)} 1/2 , it follows from the proof of Lemma 1.6 that for every η > 0 the integrals of |Y n (z, x)| over μ are uniformly bounded in n. This and the fact that ϕ ∈ L 2 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 Y n (μ + iη, x). We have Treating the r.h.s. similarly to T (1) n and T (2) n of (8.2), we get Y n (z, x) = x Z ηn (x) 2π ϕ(λ 1 ) [C(z, z 1 ) − C(z, z 1 )]dλ 1 + o(1), (9.8) where C(z, z 1 ) is defined in (8.1). It follows from (9.7) and (9.8) that 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 {h n } ⊂ L 2 (R) be a sequence of complex-valued functions such that |h n | 2 dx < C and h n → h a.e. as n → ∞, where |h(x)| ≤ ∞ a.e.
Then g(x)h n (x)dx → g(x)h(x)dx as n → ∞.