Abstract
For \(k,m,n\in {\mathbb {N}}\), we consider \(n^k\times n^k\) random matrices of the form
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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1 Introduction: Problem and Main Result
For every \(k\in {\mathbb {N}}\), consider random vectors of the form
where \({\mathbf {y}}^{(1)}\),..., \({\mathbf {y}}^{(k)}\) are i.i.d. copies of a normalized isotropic random vector \({\mathbf {y}}=(y_1,\ldots ,y_n)\in {\mathbb {R}}^n\),
\([n]=\{1,\ldots ,n\}\). The components of \({Y}\) have the form
where we use the notation \({\mathbf {j}}\) for k-multiindex:
For every \(m\in {\mathbb {N}}\), let \(\{{Y}_\alpha \}_{\alpha =1}^{m}\) be i.i.d. copies of \({Y}\), and let \(\{\tau _{\alpha }\}_{\alpha =1}^{m}\) be a collection of real numbers. Consider an \(n^k\times n^k\) real symmetric random matrix corresponding to a normalized isotropic random vector \({\mathbf {y}}\),
We suppose that
Note that \({\mathcal {M}}_{n,m,k}\) can be also written in the form
where
Such matrices with \(T_m\ge 0\) (not necessarily diagonal) are known as sample covariance matrices. The asymptotic behavior of their spectral statistics is well studied when all entries of \(Y_\alpha \) are independent. Much less is known in the case when columns \(Y_\alpha \) 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 \(n^{-k}{{\mathrm{Tr}}}{\mathcal {M}}_n^p\), \(p\in {\mathbb {N}}\), it was proved that for the special cases of random vectors \({\mathbf {y}}\) uniformly distributed on the unit sphere in \({\mathbb {C}}^n\) 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 \(M_{n,m,k}({\mathbf {y}})\) and also to prove the Central Limit Theorem for linear eigenvalue statistics in the case \(k=2\).
Let \(\{\lambda ^{(n)}_{l}\}_{l=1}^{n^k}\) be the eigenvalues of \({\mathcal {M}}_n\) counting their multiplicity, and introduce their Normalized Counting Measure (NCM) \( N_{n}\), setting for every \(\Delta \subset {\mathbb {R}}\)
Likewise, define the NCM \(\sigma _{m}\) of \(\{\tau _{\alpha }\}_{\alpha =1}^{m}\),
We assume that the sequence \(\{\sigma _{m}\}_{m=1}^\infty \) converges weakly:
In the case \(k=1\), there are a number of papers devoted to the convergence of the NCMs of the eigenvalues of \({\mathcal {M}}_{n,m,1}\) and related matrices (see [1, 6, 12, 17, 20, 27] and references therein). In particular, in [20] the convergence of NCMs of eigenvalues of \({\mathcal {M}}_{n,m,1}\) was proved in the case when corresponding vectors \(\{Y_\alpha \}_\alpha \) are “good vectors” in the sense of the following definition.
Definition 1.1
We say that a normalized isotropic vector \({\mathbf {y}}\in {\mathbb {R}}^n\) is good, if for every \(n\times n\) complex matrix \(H_n\) which does not depend on \({\mathbf {y}}\), we have
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 \(k\ge 2\), is much less than the number of matrix entries, \(n^{2k}\), the limiting distribution of eigenvalues still obeys the Marchenko–Pastur law. We have:
Theorem 1.2
Fix \(k\ge 1\). Let n and m be positive integers satisfying ( 1.4), let \(\{\tau _{\alpha }\}_{\alpha }\) be real numbers satisfying (1.7), and let \({\mathbf {y}}\) 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 \(N_{n}\) of the eigenvalues of \({\mathcal {M}}_n\) (1.3) converge weakly in probability to N as \(n\rightarrow \infty \). The Stieltjes transform f of N,
is the unique solution of the functional equation
in the class of analytic in \(\mathbb {C\setminus \mathbb { \ R}}\) functions such that \(\mathfrak {I}f(z)\mathfrak {I}z\ge 0,\;\mathfrak {I}z\ne 0. \)
We use the notation \(\int \) for the integrals over \({\mathbb {R}}\). 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 \({\mathcal {M}}_n\) corresponding to a bounded continuous test function \(\varphi : {\mathbb {R}} \rightarrow {\mathbb {C}}\), 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 \({\mathcal {N}}_n[\varphi ]\). 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 \({\mathcal {M}}_{n,m,2}\) defined in (1.3).
There are a considerable number of papers on the CLT for linear eigenvalue statistics of sample covariance matrices \({\mathcal {M}}_{n,m,1}\) (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 \({\mathbf {y}}\) are dependent. In [13], the CLT was proved for linear statistics of eigenvalues of \({\mathcal {M}}_{n,m,1}\), corresponding to some special class of isotropic vectors defined below.
Definition 1.3
The distribution of a random vector \({\mathbf {y}}\in {\mathbb {R}}^n\) is called unconditional if its components \(\{y_j\}_{j=1}^n\) have the same joint distribution as \(\{\pm y_j\}_{j=1}^n\) for any choice of signs.
Definition 1.4
We say that normalized isotropic vectors \({\mathbf {y}}\in {\mathbb {R}}^n\), \(n\in {\mathbb {N}}\), 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 \(a,b\in {\mathbb {R}}\) such that as \(n\rightarrow \infty \),
and for every \(n\times n\) complex matrix \(H_n\) which does not depend on \({\mathbf {y}}\),
Here and in what follows we use the notation \(\xi ^\circ =\xi -{\mathbf {E}}\{\xi \}\).
An important step in proving the CLT for linear eigenvalue statistics is the asymptotic analysis of their variances \(\mathbf {Var}\{{\mathcal {N}}_n[\varphi ]\} := {\mathbf {E}}\{|{\mathcal {N}}^\circ _n[\varphi ]|^2\}\), in particular, the proof of the bound
where \(||\ldots ||_{\mathcal {H}}\) is a functional norm and \(C_n\) depends only on n. This bound determines the normalization factor in front of \({\mathcal {N}}^\circ _n[\varphi ]\) and the class \({\mathcal {H}}\) 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 \({\mathcal {M}}_{n,m,1}\), 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 \({\mathcal {N}}^\circ _n[\varphi ]\) itself without any n-dependent normalization factor in front. Consider the test functions \(\varphi :{\mathbb {R}}\rightarrow {\mathbb {R}}\) from the Sobolev space \({\mathcal {H}}_s\), possessing the norm
The following statement was proved in [13] (see Theorem 1.8 and Remark 1.11):
Theorem 1.5
Let m and n be positive integers satisfying (1.4) with \(k=1\), let \(\{\tau _{\alpha }\}_{\alpha =1}^{m}\) be a collection of real numbers satisfying (1.7) and
and let \({\mathbf {y}}\) be a very good vector in the sense of Definition 1.4. Consider matrix \({\mathcal {M}}_{n,m,1}({\mathbf {y}})\) (1.3) and the linear statistic of its eigenvalues \({\mathcal {N}}_{n}[\varphi ]\) (1.11) corresponding to a test function \(\varphi \in {\mathcal {H}}_{s}\), \(s >2\). Then \(\{{\mathcal {N}}_{n}^{\circ }[\varphi ]\}_n\) converges in distribution to a Gaussian random variable with zero mean and the variance \(V[\varphi ]=\lim _{\eta \downarrow 0}V_\eta [\varphi ]\), where
\(z_{1,2}=\lambda _{1,2}+i\eta \), \(\Delta f=f(z_{1})-f(z_{2})\), \(\Delta z=z_{1}-z_{2}\), and f given by (1.10).
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 \(k\ge 1\):
Lemma 1.6
Let \(\{\tau _{\alpha }\}_{\alpha }\) be a collection of real numbers satisfying (1.7) and (1.18), and let \({\mathbf {y}}\) be a normalized isotropic vector having an unconditional distribution, such that
Consider the corresponding matrix \({\mathcal {M}}_n\) (1.3) and a linear statistic of its eigenvalues \({\mathcal {N}}_{n}[\varphi ]\). Then for every \(\varphi \in {\mathcal {H}}_{s}\), \(s>5/2\), and for all sufficiently large m and n, we have
where C does not depend on n and \(\varphi \).
It follows from Lemma 1.6 that in order to prove the CLT (if any) for linear eigenvalue statistics of \({\mathcal {M}}_n\), 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 \({\mathbf {y}}\in {\mathbb {R}}^n\) is permutationally invariant (or exchangeable) if it is invariant with respect to the permutations of entries of \({\mathbf {y}}\).
Definition 1.8
We say that normalized isotropic vectors \({\mathbf {y}}\in {\mathbb {R}}^n\), \(n\in {\mathbb {N}}\), are the CLT-vectors if they have unconditional permutationally invariant distributions and satisfy the following conditions:
- (i)
-
(ii)
their sixth moments satisfy conditions
$$\begin{aligned} a_{2,2,2}:=&\,{\mathbf {E}}\{y_{ i}^2y_{ j}^2y_{ k}^2\}=n^{-3}+O(n^{-4}),\nonumber \\ a_{2,4}:=&\,{\mathbf {E}}\{y_{ i}^2y_{ j}^4\}=O(n^{-3}),\quad a_{6}:={\mathbf {E}}\{y_{ i}^6\}=O(n^{-3}), \end{aligned}$$(1.21) -
(iii)
for every \(n\times n\) matrix \(H_n\) which does not depend on \({\mathbf {y}}\),
$$\begin{aligned} {\mathbf {E}}\{|(H_n{\mathbf {y}},{\mathbf {y}})^{\circ }|^6\}\le C||H_{n}||^{6}n^{-3}. \end{aligned}$$(1.22)
It can be shown that a vector of the form \({\mathbf {y}}={\mathbf {x}}/n^{1/2}\), where \({\mathbf {x}}\) 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 \({\mathbb {R}}^n\) or a properly normalized vector uniformly distributed on the unit ball \(B_p^n=\big \{{\mathbf {x}}\in {\mathbb {R}}^n:\; \sum _{j=1}^n|x_j|^p\le 1\big \}\) in \(l_p^n\) (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 \(\{\tau _{\alpha }\}_{\alpha =1}^{m}\) be a set of real numbers uniformly bounded in \(\alpha \) and m and satisfying (1.7). Consider matrices \({\mathcal {M}}_{n,m,2}({\mathbf {y}})\) (1.3) corresponding to CLT-vectors \({\mathbf {y}}\in {\mathbb {R}}^n\). If \({\mathcal {N}}_{n}[\varphi ]\) are the linear statistics of their eigenvalues (1.11) corresponding to a test function \(\varphi \in {\mathcal {H}}_{s}\), \(s >5/2\), then \(\{n^{-1/2}{\mathcal {N}}_{n}^{\circ }[\varphi ]\}_n\) converges in distribution to a Gaussian random variable with zero mean and the variance \(V[\varphi ]=\lim _{\eta \downarrow 0}V_\eta [\varphi ]\), where
and f is given by (1.10).
Remark 1.10
-
(i)
In particular, if \(\tau _1=\cdots = \tau _m= 1\), then
$$\begin{aligned} V[\varphi ]=\frac{(a+b+2)}{2c\pi ^{2}}\left( \int _{a_{-}}^{a_{+}}\varphi (\mu )\frac{\mu -a_{m}}{\sqrt{(a_{+}-\mu )(\mu -a_{-})}}\hbox {d}\mu \right) ^{2}, \end{aligned}$$where \(a_{\pm }=(1\pm \sqrt{c})^{2}\) and \(a_{m}=1+c\).
-
(ii)
We can replace the condition of the uniform boundedness of \(\tau _\alpha \) with the condition of uniform boundedness of eighth moments of the Normalized Counting Measures \(\sigma _n\), or take \(\{\tau _\alpha \}_\alpha \) being real random variables independent of \({\mathbf {y}}\) with common probability law \(\sigma \) having finite eighth moment. In general, it is clear from (1.23) that it should be enough to have second moments of \(\sigma _n\) 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 \({\mathbf {y}}\) in the definition of \({\mathcal {M}}_{n,m,k}({\mathbf {y}})\) is uniformly distributed on the unit sphere in \({\mathbb {R}}^n\), 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 \(O(n^{k-2})\) (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 \({\mathcal {M}}_{n,m,k}\). 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.
2 Notations
Let I be the \(n^k\times n^k\) identity matrix. For \(z\in {\mathbb {C}}\), \(\mathfrak {I}z\ne 0\), let \(G(z)=({\mathcal {M}}_n-zI)^{-1}\) be the resolvent of \({\mathcal {M}}_n\), and
Here and in what follows
so that for the nonbold Latin and Greek indices the summations are from 1 to n and from 1 to m, respectively. For \(\alpha \in [m]\), let
Thus the upper index \(\alpha \) indicates that the corresponding function does not depend on \({Y}_\alpha \). We use the notations \({\mathbf {E}}_{\alpha }\{\ldots \}\) and \((\ldots )^\circ _\alpha \) for the averaging and the centering with respect to \({Y}_\alpha \), so that \((\xi )^\circ _\alpha =\xi -{\mathbf {E}}_{\alpha }\{\xi \}\).
In what follows we also need functions (see (4.5) below)
Writing \(O(n^{-p})\) or \(o(n^{-p})\) we suppose that \(n\rightarrow \infty \) and that the coefficients in the corresponding relations are uniformly bounded in \(\{\tau _\alpha \}_\alpha \), \(n\in {\mathbb {N}}\), and \(z\in K\). We use the notation K for any compact set in \({\mathbb {C}}\setminus {\mathbb {R}}\).
Given matrix H, ||H|| and \(||H||_{HS}\) 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.
3 Some Facts and Auxiliary Results
We need the following bound for the martingales moments, obtained in [10]:
Proposition 3.1
Let \(\{S_m\}_{m\ge 1}\) be a martingale, i.e., \(\forall m\), \({\mathbf {E}}\{S_{m+1}\,\vert \,S_1,\ldots ,S_m\}=S_m\) and \({\mathbf {E}}\{|S_m|\}<\infty \). Let \(S_0=0\). Then for every \(\nu \ge 2\), there exists an absolute constant \(C_\nu \) such that for all \(m=1,2\ldots \)
Lemma 3.2
Let \(\{\xi _\alpha \}_\alpha \) be independent random variables assuming values in \({\mathbb {R}}^{n_\alpha }\) and having probability laws \(P_\alpha \), \(\alpha \in [m]\), and let \(\Phi : {\mathbb {R}}^{n_1}\times \ldots \times {\mathbb {R}}^{n_m}\rightarrow {\mathbb {C}}\) be a Borel measurable function. Then for every \(\nu \ge 2\), there exists an absolute constant \(C_\nu \) such that for all \(m=1,2\ldots \)
where \((\Phi )^\circ _\alpha =\Phi -{\mathbf {E}}_\alpha \{\Phi \}\), and \({\mathbf {E}}_\alpha \) is the averaging with respect to \(\xi _\alpha \).
Proof
This simple statement is hidden in the proof of Proposition 1 in [25]. We give its proof for the sake of completeness. For \(\alpha \in [m]\), denote \({\mathbf {E}}_{\ge \alpha }={\mathbf {E}}_{\alpha }\ldots {\mathbf {E}}_m\). Applying Proposition 3.1 with \(S_0=0\), \(S_\alpha ={\mathbf {E}}_{\ge \alpha +1}\{\Phi \}-{\mathbf {E}}\{\Phi \}\), \(S_m=\Phi -{\mathbf {E}}\{\Phi \}\), we get
By the H\(\ddot{\text {o}}\)lder inequality
which implies (3.2). \(\square \)
Lemma 3.3
Fix \(\ell \ge 2\) and \(k\ge 2\). Let \({\mathbf {y}}\in {\mathbb {R}}^n\) be a normalized isotropic random vector (1.2) such that for every \(n\times n\) complex matrix H which does not depend on \({\mathbf {y}}\), we have
Then there exists an absolute constant \(C_\ell \) such that for every \(n^k\times n^k\) complex matrix \({\mathcal {H}}\) which does not depend on \({\mathbf {y}}\), we have
where \({Y}={\mathbf {y}}^{(1)}\otimes \ldots \otimes {\mathbf {y}}^{(k)}\), and \({\mathbf {y}}^{(j)}\), \(j\in [k]\), are i.i.d. copies of \({\mathbf {y}}\).
Proof
It follows from (3.2) that
where \(\xi ^\circ _{j}=\xi -{\mathbf {E}}_{j}\{\xi \}\) and \({\mathbf {E}}_j\) is the averaging w.r.t. \(y^{(j)}\). We have
where \(H^{(j)}\) is an \(n\times n\) matrix with the entries
This and (3.3) yield
We have
For \(i\in [k]\), since by (1.2) \({\mathbf {E}}\{||y^{(i)}||\}=1\), we have by (3.3) \({\mathbf {E}}\{||y^{(i)}||^{2\ell }\}\le C\).
Hence
This and (3.5) lead to (3.4), which completes the proof of the lemma. \(\square \)
The following statement was proved in [20].
Proposition 3.4
Let \(N_{n}\) be the NCM of the eigenvalues of \({M}_n=\sum _{\alpha }\tau _\alpha {{{Y}}_\alpha }{{{Y}}_\alpha }^T\), where \(\{{Y}_{\alpha }\}_{\alpha =1}^{m}\in {\mathbb {R}}^{p}\) are i.i.d. random vectors and \(\{\tau _{\alpha }\}_{\alpha =1}^{m}\) are real numbers. Then
Also, we will need the following simple claim:
Claim 3.5
If \(h_1\), \(h_2\) are bounded random variables, then
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 \(p=n^k\), it suffices to prove that the expectations \( {\overline{N}}_{n}={\mathbf {E}}\{N_{n}\} \) of the NCMs of the eigenvalues of \({\mathcal {M}}_n\) 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 \({\overline{N}}_{n}\),
converge to the solution f of (1.10) uniformly on every compact set \(K\subset {\mathbb {C}}\setminus {\mathbb {R}}\), and that
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 \(\tau _\alpha \) are bounded:
Since \({\mathcal {M}}_n-{\mathcal {M}}_n^{\alpha }=\tau _{\alpha }{Y}_{\alpha }{Y}_\alpha ^T\), the rank one perturbation formula
implies that
It follows from the spectral theorem for the real symmetric matrices that there exists a nonnegative measure \(m^{\alpha }\) such that
This yields
implying that
It also follows from (4.4) that \( A^{-1}_{\alpha }=1-\tau _{\alpha }(G Y_{\alpha },Y_{\alpha }). \) Hence,
where we use \(||G||\le |\mathfrak {I}z|^{-1}\). Let us show that
It follows from (1.2) that
Consider \({\mathbf {E}}_\alpha \{A_\alpha \}\). By the spectral theorem for the real symmetric matrices,
where \({\mathcal {N}}_{n}^{\alpha }\) is the counting measure of the eigenvalues of \({\mathcal {M}}_n^{\alpha }\). For every \(\eta \in {\mathbb {R}}\setminus \{0\}\), consider
Clearly, for \(z\in E_{\eta }\), \(|{\mathbf {E}}_\alpha \{A_{\alpha }\}|\ge 1/2\). If \(z=\mu +i\eta \notin E_{\eta }\), then
so that
This leads to (4.9) for \({\mathbf {E}}_\alpha \{A_\alpha \}\). Replacing in our argument \({\mathcal {N}}_n^\alpha \) with \({\overline{{\mathcal {N}}}}_n^\alpha \), we get (4.9) for \({\mathbf {E}}\{A_\alpha \}\).
It follows from the resolvent identity and (4.4) that
This and the identity
lead to
It follows from the Schwarz inequality that
Note that since \({\mathbf {E}}\{||Y_\alpha ||=1\}\), we have by (1.8) \({\mathbf {E}}\{||Y_\alpha ||^4\}\le C\). This and (4.8) imply that \({\mathbf {E}}\{|A^{-2}_\alpha |\}\) is uniformly bounded in \(|\tau _\alpha |\le L\) and \(z\in K\). We also have
hence
By (1.4) and (3.7) with \(p=n^k\), \(\mathbf {Var}\{g_n^\alpha \}\le Cn^{-k}|\mathfrak {I}z|^{-2}\). It follows from (1.8) and Lemma 3.3 with \({\mathcal {H}}=G^\alpha \) and \(\ell =2\) that
Thus, \({\mathbf {E}}\{|{A_\alpha ^\circ }|^2\}\le CL^2|\mathfrak {I}z|^{-2}(k\delta _n+n^{-k})\). This and (4.9) yield
uniformly in \(|\tau _\alpha |\le L\) and \(z\in K\). Hence
It follows from (4.5) and (4.7) that
This and (4.9) imply that \(|1+\tau _\alpha f_n(z)|^{-1}\) is uniformly bounded in \(|\tau _\alpha |\le L\) and \(z\in K\). Hence, in (4.15) we can replace \(f_n^\alpha \) with \(f_n\) (the corresponding error term is of the order \(O(n^{-k})\)) and pass to the limit as \(n\rightarrow \infty \). Taking into account (1.7) we get that the limit of every convergent subsequence of \(\{f_n(z)\}_n\) satisfies (1.10). This finishes the proof of the theorem under assumption (4.3).
Consider now the general case and take any sequence \(\{\sigma _n\}=\{\sigma _{m(n)}\}\) satisfying (1.7). For any \(L>0\), introduce the truncated random variables
Denote \({\mathcal {M}}_n^L=\sum _{\alpha =1}^{m}\tau _{\alpha }^{L}{Y}_{\alpha }{Y}_{\alpha }^T.\) Then
Take any sequence \(\{L_i\}_i\) which does not contain atoms of \(\sigma \) and tends to infinity as \(i\rightarrow \infty \). If \(N_{n}^{L_i}\) is the NCM of the eigenvalues of \( {\mathcal {M}}_n^{L_i}\) and \({\overline{N}}_{n}^{L_i}\) is its expectation, then the mini-max principle implies that for any interval \(\Delta \subset {\mathbb {R}}\):
We have
where by (1.7) the first term on the r.h.s. tends to zero as \(n\rightarrow \infty \). Hence,
Thus if f and \(f^{L_{i}}\) are the Stieltjes transforms of \({\overline{N}}\) and \(\lim _{n\rightarrow \infty }{\overline{N}}_{n}^{L_{i}}\), then
uniformly on K. It follows from the first part of the proof that
where \(c_{L_{i}}=c\sigma [-L_i,L_{i}]\rightarrow c\) as \(L_{i}\rightarrow \infty \). Since \( N({\mathbb {R}})=1\), there exists \(C>0\), such that
Hence we have for all sufficiently big \(L_{i}\):
Thus \(|\tau /(1+\tau f^{L_{i}}(z))|\le |\mathfrak {I}f^{L_{i}}(z)|^{-1}\le 2/C<\infty ,\;z\in K\). This allows us to pass to the limit \(L_i\rightarrow \infty \) in (4.17) and to obtain (1.10) for f, which completes the proof of the theorem. \(\square \)
Remark 4.1
It follows from the proof that in the model we can take k depending on n such that
as \(n\rightarrow \infty \), and the theorem remains valid (see 4.14).
5 Variance of Bilinear Forms
Lemma 5.1
Let \({Y}\) be defined in (1.1–1.2), where \({\mathbf {y}}\) has an unconditional distribution and satisfies (1.19). Then for every symmetric \(n^k\times n^k\) matrix H which does not depend on \({\mathbf {y}}\) and whose operator norm is uniformly bounded in n, there is an absolute constant C such that
If additionally \({\mathbf {y}}\) satisfies (1.13–1.14), then we have
where \({\mathbf {j}}(p_i)=\{j_1,\ldots ,j_{i-1},p_i,j_{i+1},\ldots ,j_k\}\).
Proof
Since \({\mathbf {y}}\) has an unconditional distribution, we have
Hence,
where
For \(W\subset [k]\), \(W^c=[k]\setminus W\), denote
For every fixed \(W,{\mathbf {j}},{\mathbf {s}}\), we have
Indeed, the number of pairs for which \(\Lambda (W,{\mathbf {j}},{\mathbf {s}},{\mathbf {p}},{\mathbf {q}})\ne 0\) does not exceed \(2^{|W|}n^{k-|W|}\) (the number of choices of indices \(p_i=q_i\) for \(i\notin W\) equals to \(n^{k-|W|}\); all other indices \(p_\ell ,\, q_\ell \) (\(\ell \in W\)) must satisfy \(\{p_\ell ,\, q_\ell \}=\{j_\ell ,\, s_\ell \}\) and, therefore, can be chosen in at most two ways each). Since \(a_{2,2}\), \(w_i=O(n^{-2})\), (5.4) follows.
For every fixed W,
Since by (1.2) \( {\mathbf {E}}\{(H{Y},{Y})\}=n^{-k}{{\mathrm{Tr}}}H, \) we have
By (1.19), the term corresponding to \(W=\emptyset \), \(W^c=[k]\), has the form
This and (1.19) imply that
and by (1.13),
The term corresponding to \(\sum _{|W|=1}\) (i.e., \(W=\{1\},\ldots ,W=\{k\}\)), has the form
and by (1.13)
Also it follows from (5.5) that the terms corresponding to W: \(|W|\ge 2\) are less than \(Cn^{-k-2}||H||_{HS}^2\). Summarizing (5.6–5.8), we get (5.1) and (5.2) and complete the proof of the lemma. \(\square \)
6 Proof of Lemma 1.6
Lemma 6.1
Let \(\{\tau _{\alpha }\}_{\alpha }\) be a collection of real numbers satisfying (1.7), (1.18), and let \({\mathbf {y}}\) be a normalized isotropic vector having an unconditional distribution and satisfying (1.19). Consider the corresponding matrix \({\mathcal {M}}_n\) (1.3) and the trace of its resolvent \(\gamma _{n}(z)={{\mathrm{Tr}}}({\mathcal {M}}_n-zI)^{-1}\). We have
If additionally \({\mathbf {y}}\) satisfies (1.15) and \(\tau _\alpha \) are uniformly bounded in \(\alpha \) and m, then
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
Applying (4.5), (4.7), and (4.9) we get
Here by (5.1)
and
This and (6.3–6.4) lead to (6.1). Also it follows from (1.15) and Lemma 3.3 that
which leads to (6.2). \(\square \)
Proof of Lemma 1.6
The proof of (1.20) is based on the following inequality obtained in [25]: for \(\varphi \in {\mathcal {H}}_s\) (see 1.17),
Let \(z=\mu +i\eta \), \(\eta >0\). It follows from (6.3) – (6.6) that
By the spectral theorem for the real symmetric matrices,
where \(\overline{{\mathcal {N}}_{n}^{\alpha }}\) is the expectation of the counting measure of the eigenvalues of \({\mathcal {M}}_n^{\alpha }\). We have
Summarizing, we get
provided that \(s>5/2\). This finishes the proof of Lemma 1.6. \(\square \)
7 Case \(k=2\): Some Preliminary Results
From now on we fix \(k=2\) and consider matrices \({\mathcal {M}}_n={\mathcal {M}}_{n,m,2}\). For every \({\mathbf {j}}=\{j_1,j_2\}=j_1j_2\),
In this section we establish some asymptotic properties of \(A_\alpha \), \((G^\alpha Y_\alpha ,Y_\alpha )\), and their central moments. We start with
Lemma 7.1
Under conditions of Theorem 1.9,
and
Proof
Since \((A_{\alpha })_\alpha ^\circ =\tau _\alpha (G^{\alpha }Y_\alpha ,Y_\alpha )_\alpha ^\circ \), Lemma 3.3 and (1.22) imply that
and by the H\(\ddot{\text {o}}\)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)
which together with (4.13) and (7.1) leads to (7.2). \(\square \)
Let
It follows from (5.2) with \(k=2\) that
Consider an \(n\times n\) matrix of the form
Since \({\mathcal {G}}=\sum _j {\mathcal {G}}^{(j)}\), where for every j, \({\mathcal {G}}^{(j)}=\{H_{js,jp}\}_{s,p}\) is a block of \(G^\alpha \), we have
We define functions
Similarly, we introduce the matrix
and define functions
It follows from (7.3) that
We have:
Lemma 7.2
Under conditions of Theorem 1.9, we have for \(i=1,2\):
where f is the solution of (1.10).
Proof
We prove the lemma for \(g_n^{(1)}\); the cases of \({\widetilde{g}}_n^{(2)}\), \(g_n^{(2)}\), and \({\widetilde{g}}_n^{(2)}\) can be treated similarly. Without loss of generality we can assume that in the definitions of \({\mathcal {G}}\) and \(g_n^{(1)}\), \(H=G\). It follows from (3.2) that
We have
Hence
and to get (7.7), it is enough to show that
Consider \(S_n^{(1)}\). It follows from (4.4) that
Since for \(x,\xi \in {\mathbb {R}}^n\) and an \(n\times n\) matrix D
taking into account \(||H||\le 1/|\mathfrak {I}z|\), (4.8), and (7.4) we get
This and following from (1.2) and (1.22) bound
imply (7.9) for \(j=1\). The case \(j=2\) can be treated similarly. So we get (7.7) for \(g_n^{(1)}\).
Let us prove (7.8) for \(g_n^{(1)}\). Let \(f_n^{(1)}={\mathbf {E}}\{g_n^{(1)}\}\). For a convergent subsequence \(\{f_{n_i}^{(1)}\}\), put \(f^{(1)}:=\lim _{n_i\rightarrow \infty }f_{n_i}^{(1)}\). It follows from (4.4) that
This and the resolvent identity yield
Hence,
By the H\(\ddot{\text {o}}\)lder 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 \(f_n^\alpha \) and \(H^\alpha \) 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. \(\square \)
It follows from Lemmas 5.1 and 7.2 that under conditions of Theorem 1.9
where f is the solution of (1.10).
Lemma 7.3
Under conditions of Theorem 1.9
Proof
Since \(\tau _\alpha \), \(\alpha \in [m]\), are uniformly bounded in \(\alpha \) and n, then to get the desired bounds it is enough to consider the case \(\tau _\alpha =1\), \( \alpha \in [m]\). By (4.13), we have
where by (6.2) \({\mathbf {E}}\{|(g_n^\alpha )^{\circ }|^{2p}\}=O(n^{-6})\), \(p=2,3\), and by (7.1) and (6.1)
Hence,
It also follows from (7.6) and Lemmas 6.1 and 7.2 that
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 (3.8) with \(h_1=g_n^\alpha \), \(h_2=n{\mathbf {E}}_\alpha \{(H Y_\alpha ,Y_\alpha )_\alpha ^{\circ 2}\}\) that
Hence,
and to get (7.17) for \(p=3\) it is enough to show that
We have
where
and by (1.21)
Also, due to the unconditionality of the distribution, \(\Lambda \) contains only even moments. Thus in the index pairs \({\mathbf {i}},{\mathbf {j}},{\mathbf {p}},{\mathbf {q}},{\mathbf {s}},{\mathbf {t}}\in [n]^2\), 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: \(\le 3\) on the first positions (call them i, j, k) and \(\le 3\) on the second positions (call them u, v, w). For every fixed set of independent indices, consider maps \(\Phi \) from this set to the sets of index pairs \(\{{\mathbf {i}},{\mathbf {j}},{\mathbf {p}},{\mathbf {q}},{\mathbf {s}},{\mathbf {t}}\}\). We call such maps the index schemes. Let \(|\Phi |\) 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
where the last sum is taken over the set of independent indices of cardinality \(\ell \), \(\Phi \) is an index scheme constructing pairs \(\{{\mathbf {i}},{\mathbf {j}},{\mathbf {p}},{\mathbf {q}},{\mathbf {s}},{\mathbf {t}}\}\) from this set, and \(\Lambda '(\Phi )\) is a certain expression, depending on \(\Phi \), \(a_{2,2,2}\), \(a_{2,4}\), 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 \(\Lambda '(\Phi )\) in (7.21), the only thing we need to know is that
and that in the particular case of
we have by (1.21)
and the corresponding term in \(S_6\) has the form \(a_{2,2,2}^2({{\mathrm{Tr}}}H)^3\).
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
and on the observation that the more the mixing of matrix entries we have the lower order of sums we get. Let \(V\subset {\mathbb {R}}^n\) be the set of vectors of the form
and let W be the set of \(n\times n\) matrices of the form
It follows from (7.24) that
Hence,
In particular, by (7.24) and (7.25), we have for \(S_5'\)
so that \(\mathbf {Var}\{S_5'\}=O(n^{-4})\). Consider \(S_5''\). Note that if in \(S_5''\) we have a single “external” pairing with respect to at least one index on the second positions, then similar to \(S_5'\), the variance of this term is of the order \(O(n^{-4})\). So we are left with the terms of the form
It follows from (7.5) that
Now (3.8), (6.1), and (7.7) imply that
Summarizing we get \(\mathbf {Var}\{S_5\}=O(n^{-4})\).
Consider \(S_6\) and show that \(\mathbf {Var}\{S_6-g_n^{\alpha 3}\}=O(n^{-4})\). 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
It follows from (7.26) that \(S_6'=O(n^{-2})\); hence \( \mathbf {Var}\{S_6'\}=O(n^{-4}). \) Consider \(S_6''\)
If the second indices in \(H_{k\cdot ,\,k\cdot }\) are not equal, then we get the expression of the form
It follows from (7.26) that \(S_6'''=O(n^{-2})\); hence \( \mathbf {Var}\{S_6'''\}=O(n^{-4}). \) If the second indices in \(H_{k\cdot ,\,k\cdot }\) in (7.27) are equal, then we get the expressions of three types:
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, \(\mathbf {Var}\{S_6'''\}=O(n^{-4})\). It remains to consider the term without external pairing, which corresponds to
(see (7.22)). Summarizing we get
where we used (1.21) and (6.1). This leads to (7.23) and completes the proof of the lemma. \(\square \)
8 Covariance of the Resolvent Traces
Lemma 8.1
Suppose that the conditions of Theorem 1.9 are fulfilled. Let
Then \(\{C_n(z_1,z_2)\}_n\) converges uniformly in \(z_{1,2}\in K\) to
Proof
For a convergent subsequence \(\{C_{n_i}\}\), denote
We will show that for every converging subsequence, its limit satisfies (8.1). Applying the resolvent identity, we get (see (4.11))
Consider \(T_{n}^{(1)}\). Iterating (4.12) four times, we get
It follows from (4.9), (6.1), and (7.15) that \(S_n^{(i)}=O(n^{-1/2})\), \(i=2,3\). Also, by (4.8) we have
where by the Schwarz inequality, (6.2), (7.1), and (7.13)
Hence \(S_n^{(4)}=O(n^{-1/2})\), and we are left with \(S_n^{(1)}\). We have
It follows from (4.5) and (4.7) that \(|\gamma _n(z)-\gamma ^\alpha _n(z)|\le 1/|\mathfrak {I}z|\). This and (6.1) yield
Hence,
and we have
Summarizing, we get
Consider now \(T_n^{(2)}\) of (8.2). By (4.5),
For shortness let for the moment \(A_i=A_\alpha (z_i)\), \(i=1,2\), \(B_2=B_\alpha (z_2)\). 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 \(|B_2/A_2|\), \(|A_i|^{-1}\), \(|{\mathbf {E}}\{A_i\}|^{-1}\), \(i=1,2\), one can show that the terms containing at least three centered factors \(A_1^\circ \), \(A_2^\circ \), \(B_2^\circ \) are of the order \(O(n^{-3/2})\). This implies that
Returning to the original notations and taking into account that
we get
Denote for the moment
It follows from (7.14) and (8.4–8.5) that
Note that by (1.10),
Hence
which completes the proof of the lemma. \(\square \)
9 Proof of Theorem 1.9
The proof essentially repeats the proofs of Theorem 1 of [25] and Theorem 1.8 of [13]; the technical details are provided by the calculations of the proof of Lemma 8.1. It suffices to show that if
then we have uniformly in \(|x|\le C\)
with \(V[\varphi ]\) of (1.23). Define for every test functions \(\varphi \in {\mathcal {H}}_{s}\), \(s>5/2\),
where \(P_\eta \) is the Poisson kernel
and “\(*\)” denotes the convolution. We have
Denote for the moment the characteristic function (9.1) by \( Z_n[\varphi ]\), to make explicit its dependence on the test function. Take any converging subsequence \(\{Z_{n_j}[\varphi ]\}_{j=1}^\infty \) Without loss of generality assume that the whole sequence \(\{Z_{n_j}[\varphi _{\eta }]\}\) converges as \(n_j\rightarrow \infty \). By (1.20), we have
hence
This and the equality \(Z_{n_j}[\varphi ]=(Z_{n_j}[\varphi ]-Z_{n_j}[\varphi _{\eta }])+Z_{n_j}[\varphi _{\eta }]\) imply that
Thus it suffices to find the limit of
as \({n} \rightarrow \infty \). It follows from (9.2) – (9.3) that
This allows to write
where
Since \(|{\mathcal {Y}}_n(z,x)|\le 2n^{-1/2}\mathbf {Var}\{\gamma _n(z)\}^{1/2}\), it follows from the proof of Lemma 1.6 that for every \(\eta >0\) the integrals of \(|{\mathcal {Y}}_n(z,x)|\) over \(\mu \) are uniformly bounded in n. This and the fact that \(\varphi \in L^2\) 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 \({\mathcal {Y}}_n(\mu +i\eta ,x)\). We have
where \(e_{\eta n}^{\alpha }(x)=\exp \{ix{\mathcal {N}}_n^{\alpha \circ }[\varphi _\eta ]/\sqrt{n}\}\) and \({\mathcal {N}}_n^{\alpha }[\varphi _\eta ]={{\mathrm{Tr}}}\varphi _\eta (M^\alpha )\). By (9.6),
so that
where \(z_j=\lambda _j+i\eta \), \(j=1,2\), and
Using the argument of the proof of the Lemma 8.1, it can be shown that \(R_n= O(n^{-5/2})\). Hence,
Treating the r.h.s. similarly to \(T_n^{(1)}\) and \(T_n^{(2)}\) of (8.2), we get
where \(C(z,z_1)\) is defined in (8.1). It follows from (9.7) and (9.8) that
(see (1.23)) and finally
Taking into account (9.5), we pass to the limit \(\eta \downarrow 0\) and complete the proof of the theorem. \(\square \)
It remains to prove the following lemma.
Lemma 9.1
Let \(g\in L^2({\mathbb {R}})\) and let \(\{h_n\}\subset L^2({\mathbb {R}})\) be a sequence of complex-valued functions such that
Then
Proof
According to the convergence theorem of Vitali (see, e.g., [24]), if \((X,{\mathcal {F}},\mu )\) is a positive measure space and
then \(F\in L^1(\mu )\) and \(\lim _{n\rightarrow \infty }\int _X |F_n-F|\hbox {d}\mu =0\). Without loss of generality assume that \(g(x)\ne 0\), \(x\in {\mathbb {R}}\), and take
Then
Hence, the conditions of Vitali’s theorem are fulfilled and we get
which completes the proof of the lemma. \(\square \)
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Acknowledgements
The author would like to thank Leonid Pastur for an introduction to the problem and for fruitful discussions.
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Lytova, A. Central Limit Theorem for Linear Eigenvalue Statistics for a Tensor Product Version of Sample Covariance Matrices. J Theor Probab 31, 1024–1057 (2018). https://doi.org/10.1007/s10959-017-0741-9
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DOI: https://doi.org/10.1007/s10959-017-0741-9