Central Limit Theorem for linear eigenvalue statistics of the Wigner and sample covariance random matrices

Introduction

The Wigner Ensembles for real symmetric matrices is a family of $n\times n$ real symmetric matrices $M$ of the form

Here and below we denote $\mathbf{E}\{.\}$ the averaging with respect to all random parameters of the problem. Let $\{\lambda_{j}^{(n)}\}_{i=1}^{n}$ – be eigenvalues of $M$. Since the pioneer work of Wigner it is known that if we consider the linear eigenvalue statistic corresponding to any continuous test function $\varphi$:

then $n^{-1}\mathcal{N}_{n}[\varphi]$ converges in probability to the limit

where $\rho_{sc}(\lambda)$ is the famous semicircle density

The result of this type, which is the analog of the Law of Large Numbers of the classical probability theory, normally is the first step in studies of the eigenvalue distribution for any ensemble of random matrices. For the Wigner ensemble this result, obtained initially in for Gaussian W=\big{\{}w_{jk}^{(n)}\big{\}}_{j,k=1}^{n}, was improved in , where the convergence of $N_{n}(\lambda)$ to the semicircle law was shown under the minimal conditions on the distribution of W=\big{\{}w_{jk}^{(n)}\big{\}}_{j,k=1}^{n} (the Lindeberg type conditions).

The second classical ensemble which we consider in the paper is a sample covariance matrix of the form

where $X$ is a $n\times m$ matrix whose entries \big{\{}X_{jk}^{(n)}\big{\}}_{j=1,.,n,k=1,.,m} are independent random variables, satisfying the conditions

Corresponding results on the convergence of normalized linear eigenvalue statistics to integrals with the Marchenko-Pastur distribution were obtained in .

Central Limit Theorem (CLT) for fluctuations of linear eigenvalue statistics is a natural second step in studies of the eigenvalue distribution of any ensemble of random matrices. That is why there are a lot of papers, devoted to the proofs of CLT for different ensembles of random matrices (see ). CLT for the traces of resolvents for the classical Wigner and sample covariance matrices was proved by Girko in 1975 (see and references therein), but the expression for the variance found by him was rather complicated. A simple expression for the covariance of the resolvent traces for the Wigner matrix in the case $E\{(w_{ii}^{(n)})^{2}\}=2$ was found in . CLT for polynomial test functions for some generalizations of the Wigner and sample covariance matrices was proved in by using moment methods. CLT for real analytic test functions for the Wigner and sample covariance matrices was established in under additional assumptions that $E\{(w_{ii}^{(n)})^{2}\}=2$, $E\{(w_{jk}^{(n)})^{4}\}=3E^{2}\{(w_{jk}^{(n)})^{2}\}=3$ (or $E\{(X_{jk}^{(n)})^{4}\}=3E^{2}\{(X_{jk}^{(n)})^{2}\}$ for the model (1.5)). In the recent paper CLT for the linear eigenvalue statistics of the Wigner and sample covariance matrix ensemble was proved under assumptions that $E\{(w_{ii}^{(n)})^{2}\}=2$, the third and the forth moments of all entries are the same, but $E\{(w_{jk}^{(n)})^{4}\}$ is not necessary 3. Moreover, the test functions, studied in , are not supposed to be real analytic. It was assumed that the Fourier transform $\widehat{\varphi}$ of the test function $\varphi$ satisfies the inequality

which means that $\varphi$ has more than 5 bounded derivatives.

In the present paper we prove CLT for the Wigner ensemble (1.1) under the following assumptions on the matrix entries

We consider the test functions from the space $\mathcal{H}_{s}$, possessing the norm (cf (1.7))

Consider the Wigner model with entries satisfying condition (1.8). Let the real valued test function $\varphi$ satisfy condition $||\varphi||_{3/2+\varepsilon}<\infty$ ($\varepsilon>0$). Then $\mathcal{N}^{\circ}_{n}[\varphi]$ converges in distribution to the Gaussian random variable with zero mean and the variance

Let us note that similarly to the result of it is easy to check that Theorem 1 remains valid if the second condition of (1.8) is replaced by the Lindeberg type condition for the fourth moments of entries of $W$

The proof will be the same as for Theorem 1, but everywhere below $n^{-\varepsilon_{1}/2}$ will be replaced by $L_{n}(\tau)/\tau^{\gamma}$, with some positive $\gamma$.

The proof of Theorem 1 is based on some combination of the resolvent approach with martingal bounds for the variance of the resolvent traces, used before by many authors, in particularly, by Girko (see and references therein). An important advantage of our approach is that it is shown by the marginal difference method that (see Proposition 2 below)

The proposition allows one to transform the bounds for the variances of the resolvent traces into the bounds for the variances of linear eigenvalue statistics of $\varphi\in\mathcal{H}_{s}$, where the value of $s$ depends on the exponent of $|\Im z|$ in the r.h.s. of (1.13). It is important, that Proposition 1 has a rather general form and therefore it is applicable to any ensemble of random matrices for which the bounds of the type (1.13) (may be with a different exponent of $|\Im z|$) are found. This makes Proposition 1 an important tool of the proof of CLT for linear eigenvalue statistics for different random matrices. The idea of Proposition 1 was taken from the paper , where a similar argument was used to study the first order correction terms of $n^{-1}\mathbf{E}\{\mathcal{N}_{n}[\varphi]\}$ for the matrix models. Having in mind Proposition 1, one can prove CLT for any dense in $\mathcal{H}_{s}$ set of the test functions, and then extend this result to the whole $\mathcal{H}_{s}$ by the standard procedure (see Proposition 3). In the present paper for this aim we use a set of convolutions of integrable functions with the Poisson kernel (see (2.32) and (2.3)). This choice simplifies considerably the argument in the proof of CLT and makes the proof more short than that in the previous papers .

The result for sample covariance matrices is very similar. We assume that the moments of the entries of $X$ from (1.5) satisfy the bounds

Consider a random matrix (1.5) – (1.6) with entries of $X$, satisfying the condition (1.15). Let the real valued test function $\varphi$ satisfy condition $||\varphi||_{3/2+\varepsilon}<\infty$ ($\varepsilon>0$). Then $\mathcal{N}^{\circ}_{n}[\varphi]$ in the limit $m,n\to\infty$, $m/n\to c\geq 1$ converges in distribution to the Gaussian random variable with zero mean and the variance

where $\dfrac{\Delta\varphi}{\Delta\lambda}=\dfrac{\varphi(\lambda_{1})-\varphi(\lambda_{2})}{\lambda_{1}-\lambda_{2}}$, $\kappa_{4}=X_{4}-3$ is the fourth cumulant of entries of $X$, $a_{\pm}=(1\pm\sqrt{c})^{2}$, and $a_{m}=\frac{1}{2}(a_{+}+a_{-})$.

Proofs

Proof of Proposition 1. Consider the operator $\mathcal{D}_{s}$

where $\Pi_{\varphi}$ is the projection on the vector $\varphi$

where $||.||_{0}$ means the norm (1.9) with $s=0$. We can write

where the symbol $*$ means the convolution of functions, and $P_{y}$ is the Poisson kernel

This relation combined with (2.2) proves (1.14).$\square$

In what wallows we need to estimate \mathbf{E}\big{\{}|w_{jk}^{(n)}|^{8}\big{\}} (see the proof of Proposition 2). Hence, if $\varepsilon_{1}<4$, then it is convenient to consider the truncated matrix

Let $\widetilde{\mathcal{N}}_{n}[\varphi]=\emph{Tr }\varphi(\widetilde{M}^{(\tau)\circ})$ be the linear eigenvalue statistic of the matrix $\widetilde{M}^{(\tau)\circ}$, corresponding to the test function $\varphi$ with bounded first derivative. Then

Proof. Consider the matrix $M(t)=\widetilde{M}+t(M-\widetilde{M})$. Let $\{\lambda_{i}(t)\}$ be eigenvalues of $M(t)$ and $\{\psi_{i}(t)\}$ be corresponding eigenvectors. Then

where $M^{\prime}(t)=\frac{d}{dt}M(t)=(M-\widetilde{M})$, $U=\{u_{ik}\}$ is the unitary matrix such that $M-\widetilde{M}=U^{*}\Lambda U$, where $\Lambda$ is a diagonal matrix and $|M-\widetilde{M}|=U^{*}|\Lambda|U$. Hence,

It follows from Lemma 1 that for our purposes it suffices to prove CLT for $\widetilde{\mathcal{N}}^{\circ}_{n}[\varphi]$. Hence, starting from this point we will assume that $M$ is replaced by $\widetilde{M}^{(\tau)o}$, but to simplify notations we will write $M$ instead of $\widetilde{M}^{(\tau)o}$ just assuming below that the matrix entries of $W$ satisfy conditions

Here and below we omit also the super index $(n)$ of matrix entries $w_{jk}^{(n)}$ and $X_{jk}^{(n)}$.

If the conditions (2.6) are satisfied, then for any $1>\delta>0$

If the conditions of (2.7) are also satisfied, then

Proof. Denote $\mathbf{E}_{\leq k}$ the averaging with respect to $\{w_{ij}\}_{1\leq i\leq j\leq k}$. Then, according to the standard martingal method (see ), we have

Denote $\mathbf{E}_{k}$ the averaging with respect to $\{w_{ki}\}_{1\leq i\leq n}$. Then, using the Schwarz inequality, we obtain that

Let us estimate the first summand (with $k=1$) of the above sum. The other ones can be estimated similarly. Denote $M^{(1)}$ the $(n-1)\times(n-1)$ matrix which is the main bottom $(n-1)\times(n-1)$ minor of $M$

The first identity of (2.13) yields that it suffices to estimate $\mathbf{E}\{|BA^{-1}-\mathbf{E}_{1}\{BA^{-1}\}|^{2}\}$ and $\mathbf{E}\{|A^{-1}-\mathbf{E}_{1}\{A^{-1}\}|^{2}\}$. We will estimate the first expression. The second one can be estimated similarly. Denote $\xi^{\circ}_{1}=\xi-\mathbf{E}_{1}\{\xi\}$ for any random variable $\xi$ and note that for any $a$ independent of $\{w_{1i}$ we have

Let us use also the identities that follow from the spectral theorem

where $|G^{(1)}|=(G^{(1)}G^{(1)*})^{1/2}$. The first relation yields, in particular, that $|{B}/{A}|\leq|\Im z|^{-1}$. Moreover, using the second identity of (2.14), we have

we get by (2.15) and the second identity of (2.14):

Then, using the Jensen inequality $|\mathbf{E}_{1}\{A\}|^{-1}\leq\mathbf{E}_{1}\{|A|^{-1}\}$, and the second identity of (2.13), we conclude that

To prove (2.9) we use the inequality similar to (2.11) (see )

Thus, in view of (2.13), it is enough to check that

The first relation here evidently follow from (2.16), if we take the forth degree of the r.h.s., average with respect to $\{w_{1i}\}$, and take into account (2.7). The second relation can be obtained similarly. $\square$

Proposition 2 gives the bound for the variance of the linear eigenvalue statistics for the functions $\varphi(\lambda)=(\lambda-z)^{-1}$. We are going to extend the bound for a wider class of test functions.

If $||\varphi||_{3/2+\epsilon}\leq\infty$, with any $\epsilon>0$, then

Proof. In view of Proposition 1 we need to estimate

Take in (2.8) $\delta=\varepsilon/2$. Then we need to estimate

We do this for $j=1$. For other $j$ the estimates are the same. The spectral representation

Taking $s=3/2+\varepsilon$ in (1.14), we get

To simplify formulas we will assume below that $\{w_{jk}\}_{1\leq j<k\leq n}$ are i.i.d. and $\{w_{jj}\}_{1\leq j\leq n}$ are i.i.d. Note that this assumption does not change the proof seriously, it just allows us to write the bounds only for $G_{11}$ instead of all $G_{ii}$.

The next lemma collects relations which we need to prove CLT.

Using notations of (2.13) we have uniformly in $z_{1},z_{2}:\Im z_{1,2}>a$ with any $a>0$:

Proof. Note that since $\Im z\Im(G^{(1)}m,m)\geq 0$, we can use the bound

Relations (2.21) follow from the representations

combined with (2.19), and (2.9), applied to $\gamma_{n}^{(1)}$. Relations (2.22) and (2.23) follow from (2.16) and (2.29), if we take the products of the r.h.s. of (2.16) with different $z$ and average with respect to $\{w_{1i}\}$. Relation (2.25) follows from (2.13), (2.21), and (2.28). The first relation of (2.26) is the analog of the relation

if in the latter we replace the matrix $M$ by $M^{(1)}$. But since $G_{11}(z_{1})=-A^{-1}(z_{1})$, (2.30) follows from (2.21) and (2.28). The second relation of (2.26) follows from (2.13).

The first relations of (2.27) follows from the above bound for $n^{-1}\mathbf{E}\{\gamma_{n}-\gamma_{n}^{(1)}\}$ and the well known estimate (see e.g. )

The second one of (2.27) is the corollary of the above estimate and of the relation

Finally we obtain the first bound of (2.24) from (2.22), (2.26), (2.25), and the identity

The second bound of (2.24) follows from the first one, (2.23), and the Cauchy theorem. $\square$

Proof of Theorem 1. We prove first Theorem 1 for the functions $\varphi_{\eta}$ of the form

where $P_{\eta}$ is the Poisson kernel defined in (2.3). One can see easily that

On the other hand, using the symmetry of the problem and the notations of (2.13), we have

Since $e_{1}(x)$ does not depend on $\{w_{1i}\}$, using that $\mathbf{E}\{...\}=\mathbf{E}\{\mathbf{E}_{1}\{...\}\}$, we obtain in view of (2.37) and (2.21)

But the Schwarz inequality and (2.25) yield

Using the Schwartz inequality, (2.13), (2.25), and (2.22), we conclude that the term $O((\gamma_{n}^{\circ}-(\gamma^{(1)}_{n})^{\circ})^{2})$ gives the contribution $O(n^{-\varepsilon_{1}/4})$. Then, since $e_{1}(x)$ does not depend on $\{w_{1i}\}$, we average first with respect to $\{w_{1i}\}$ and obtain in view of (2.25)

Using (2.37) and (2.21), we conclude that only linear terms with respect to $B^{\circ}$ and $A^{\circ}$ give non vanishing contribution, hence in view of (2.23) and (2.24) we obtain

where we used also (2.27) to replace $\mathbf{E}^{-1}\{A(z)\}$ by $f(z)$ and $\mathbf{E}\{B(z_{\mu})\}$ by $f^{\prime}(z_{\mu})$. The identity (2.31) yields

In addition, similarly to (2.38), we have

we can transform $C_{n}(z,z_{\mu})$ to the form

Now, taking into account (2.35), (2.41), and (2.42), we obtain the equation

and since $\widetilde{Z}_{n}(0)=Z_{n}(0)=1$, we obtain uniformly in $|x|\leq C$

Thus, we have proved CLT for the functions of the form (2.32). To extend CLT to a wider class of functions we use

be the variance of $\mathcal{N}_{n}[\varphi]$. Assume that

(a) there exists a vector space $\mathcal{L}$ endowed with a norm $||...||$ and such that $V_{n}$ is defined on $\mathcal{L}$ and admits the bound

Then $V$ admits a continuous extension to $\mathcal{L}$ and CLT is valid for all $\mathcal{N}_{n}[\varphi]$, $\varphi\in\mathcal{L}$.

Proof. Let $\{\varphi_{k}\}$ be a sequence of elements of $\mathcal{L}_{1}$ converging to $\varphi\in\mathcal{L}$. We have then in view of the inequality $|e^{ia}-e^{ib}|\leq|a-b|$, the linearity of $\overset{\circ}{\mathcal{N}}_{n}[\varphi]$ in $\varphi$, the Schwarz inequality, and (2.45):

Now, passing first to the limit $n\rightarrow\infty$ and then $k\rightarrow\infty$, we obtain the assertion. $\square$

The proposition and Lemma 2 allow us to complete the proof of Theorem 1.$\square$

Proof of Theorem 2 The proof of Theorem 2 can be performed by the same way as that for Theorem 1. We start from the proposition which is the analog of Proposition 2.

Taking into account Proposition 4, on the basis of Proposition 1 and Lemma 2 we obtain immediately the bound (2.20) for the variance of linear eigenvalue statistics of sample covariance matrices. Then one can use the same method as in the proof of Theorem 1 to prove CLT for $\varphi_{\eta}$ of (2.32) or just use the result of for the functions, satisfying conditions (1.7). Then Proposition 3 implies immediately the assertion of Theorem 2.

Thus, to complete the proof of Theorem 2 we are left to prove Proposition 4.

Proof of Proposition 4. Similarly to the proof of Proposition 2 we use the identity (2.10) where this time $\mathbf{E}_{\leq k}$ means the averaging with respect to $\{X_{jl}\}_{l=1,.,m,j\leq k}$. Then we obtain (2.11) with $\mathbf{E}_{k}$ meaning the averaging with respect to $\{X_{kl}\}_{l=1,.,m}$.

Denote $M^{(1)}=X^{(1)}X^{(1)*}$, where the $(n-1)\times m$ matrix $X^{(1)}$ is made from the lines $X$, from the second to the last one. Then denote

and use (2.13) with these $G^{(1)}$ and $m^{(1)}$.

To obtain the estimate for \mathbf{E}_{1}\Big{\{}|\gamma_{n}-\mathbf{E}_{1}\{\gamma_{n}\}|^{2}\Big{\}} we need (as in the proof of Proposition 2) to estimate $\mathbf{E}_{1}\{|A^{\circ}_{1}|^{2}\}/(\Im z\mathbf{E}_{1}\{A\})^{2}$ and $\mathbf{E}_{1}\{|B^{\circ}_{1}|^{2}\}/(\mathbf{E}_{1}\{A\})^{2}$. Since $G^{(1)}$ does not depend on $\{X_{1i}\}_{i=1,.,m}$, averaging with respect to $\{X_{1i}\}_{i=1,.,m}$, using the Jensen inequality and (2.47), we get

But it is known (see and references therein) that for any fixed $\delta>0$

Combining this bound with the above inequality and repeating the argument of Proposition 2, we obtain the bound (2.17). The bound for $\mathbf{E}_{1}\{|B^{\circ}_{1}|^{2}\}/\mathbf{E}_{1}^{2}\{A\}$ can be obtained similarly.

References