Necessary and Sufficient Conditions for Convergence to the Semicircle Distribution

Let $W_{n}$, for each $n\in\mathbf{N}$, be a random $n\times n$ Hermitian matrix whose upper triangular entries are independent. We call $(W_{n})_{n\in\mathbf{N}}$ a Hermitian Wigner ensemble. In case $W_{n}$ is real symmetric for all $n\in\mathbf{N}$, we call $(W_{n})_{n\in\mathbf{N}}$ a symmetric Wigner ensemble. We write $W_{n}=(w_{ij})_{i,j=1}^{n}$ throughout this paper. If $\lambda_{1}(W_{n})\geq\cdots\geq\lambda_{n}(W_{n})$ are the eigenvalues of $W_{n}$ counted with multiplicity, then the empirical spectral distribution $\mu_{W_{n}}$ of $W_{n}$ is defined by

$$\mu_{W_{n}}:=\frac{1}{n}\sum_{i=1}^{n}\delta_{\lambda_{i}(W_{n})}.$$

Since $\mu_{W_{n}}$ is a random measure, we can think of the mean measure $\operatorname{\mathbf{E}}\mu_{W_{n}}$, which is defined and treated in Appendix A.

Let us use the term semicircle law to refer to a class of theorems that state, under certain conditions, that $\mu_{W_{n}}$ converges in some sense to the semicircle distribution $\mu_{\mathrm{sc}}$ on $\mathbf{R}$ given by

$$\mu_{\mathrm{sc}}(dx):=\sqrt{(4-x^{2})_{+}}\,dx.$$

(We let $x_{+}:=\max\{x,0\}$.) Wigner initiated the spectral study of random matrices by proving the following very first version of the semicircle law in [Wig55, Wig58].

Subsequent works by [Arn71], [Pas73], and others led to the following much more general semicircle law.

Note that (1.1) for $k=3$ implies (1.2) since

$$\frac{1}{n}\sum_{i,j=1}^{n}\operatorname{\mathbf{E}}[|w_{ij}|^{2};|w_{ij}|>\epsilon]\leq\frac{1}{\epsilon n}\sum_{i,j=1}^{n}\operatorname{\mathbf{E}}|w_{ij}|^{3}\leq\frac{B_{3}}{\epsilon\sqrt{n}}\to 0.$$

Let us call (1.2) the Lindeberg condition, following the Lindeberg–Feller central limit theorem. Girko [Gir90, Theorem 9.4.1] states that the converse of Theorem 1.2 holds.

Rather surprisingly, we have the following:

A proof of this fact using a concentration-of-measure inequality is given in Appendix A. Thanks to this equivalence, we will be able to go back and forth freely between the two types of convergences throughout the paper.

Theorem 1.2 suggests an extension of the semicircle law to the case where the entries of $W_{n}$ have variances other than $1/n$. Here is one possible approach to such an extension. Assume that the underlying probability space is the product

$$(\Omega_{1}\times\Omega_{2},\mathcal{F}_{1}\times\mathcal{F}_{2},\operatorname{\mathbf{P}}_{1}\times\operatorname{\mathbf{P}}_{2})$$

of two probability spaces $(\Omega_{1},\mathcal{F}_{1},\operatorname{\mathbf{P}}_{1})$ and $(\Omega_{2},\mathcal{F}_{2},\operatorname{\mathbf{P}}_{2})$. Then let $X_{n}=(x_{ij})_{i,j=1}^{n}$ and $Y_{n}=(y_{ij})_{i,j=1}^{n}$ be random real symmetric matrices defined on $\Omega_{1}$ and $\Omega_{2}$ having i.i.d. upper triangular entries. If $x_{11}$ is standard normal and

$$\operatorname{\mathbf{P}}_{2}(y_{11}=1)=\operatorname{\mathbf{P}}_{2}(y_{11}=-1)=1/2,$$

then it is not difficult to show that $(W_{n})_{n\in\mathbf{N}}$ given by

$$w_{ij}(\omega_{1},\omega_{2}):=x_{ij}(\omega_{1})y_{ij}(\omega_{2})/\sqrt{n}$$

satisfies the conditions of Theorem 1.2.

Since $\mu_{W_{n}(\omega_{1},\omega_{2})}\Rightarrow\mu_{\mathrm{sc}}$ for $\operatorname{\mathbf{P}}$-a.e. $(\omega_{1},\omega_{2})$, Tonelli’s theorem implies that for $\operatorname{\mathbf{P}}_{1}$-a.e. $\omega_{1}\in\Omega_{1}$, we have $\mu_{W_{n}(\omega_{1},\cdot)}\Rightarrow\mu_{\mathrm{sc}}$ $\operatorname{\mathbf{P}}_{2}$-a.s. Note that the $(i,j)$-entry of the random matrix $W_{n}(\omega_{1},\cdot)$ defined on $(\Omega_{2},\mathcal{F}_{2},\mathcal{P}_{2})$ has variance $x_{ij}(\omega_{1})^{2}/n,$ which can deviate by any amount from $1/n$.

A problem with this approach is that we do not know for which $\omega_{1}$ we have the a.s. convergence $\mu_{W_{n}(\omega_{1},\cdot)}\Rightarrow\mu_{\mathrm{sc}}$, even though we know this happens for almost all $\omega_{1}$. For instance, the above discussion does not tell us whether $\mu_{W_{n}}\Rightarrow\mu_{\mathrm{sc}}$ a.s. is true when $(W_{n})_{n\in\mathbf{N}}$ is a symmetric Wigner ensemble such that

$$\operatorname{\mathbf{P}}(w_{ij}=\sqrt{2/n})=\operatorname{\mathbf{P}}(w_{ij}=-\sqrt{2/n})=1/2\qquad\text{if $i+j$ is even}$$

and $w_{ij}=0$ if $i+j$ is odd.

Götze, Naumov, and Tikhomirov [GNT15] covered this case by proving the following:

From our main result (Theorem 1.6) it will follow that (1.4) is not needed in Theorem 1.4, and that $(W_{n})_{n\in\mathbf{N}}$ can be assumed to be Hermitian, not necessarily real symmetric.

To illustrate that (1.3) is needed in Theorem 1.4, the authors of [GNT15] considered the random symmetric block matrix

$$W_{n}=\begin{pmatrix}A&B\\ B^{T}&D\end{pmatrix}$$

where $A$ and $D$ are of size $\lfloor{n/2}\rfloor\times\lfloor{n/2}\rfloor$ and $\lceil{n/2}\rceil\times\lceil{n/2}\rceil$, and the upper triangular entries of $W_{n}$ are independent. They let all entries of $W_{n}$ except the non-diagonal entries of $D$ be normal with mean $0$ and variance $1/n$, and simulated the spectrum of $W_{n}$ for $n=2000$ to see that $\mu_{W_{n}}$ does not look like a semicircle. Note that (1.3) does not hold.

Our main theorem will let us prove what was suggested by the simulation in [GNT15], namely that $\operatorname{\mathbf{E}}\mu_{W_{n}}\not\Rightarrow\mu_{\mathrm{sc}}$. More generally, we will prove for a large class of Hermitian Wigner ensembles $(W_{n})_{n\in\mathbf{N}}$ that $\operatorname{\mathbf{E}}\mu_{W_{n}}\Rightarrow\mu_{\mathrm{sc}}$ (or $\mu_{W_{n}}\Rightarrow\mu_{\mathrm{sc}}$ a.s., equivalently) holds if and only if (1.3) is true.

One thing we should notice is that changing $o(n)$ rows of $W_{n}$ has no effect on the limit of $\operatorname{\mathbf{E}}\mu_{W_{n}}$ due to the following:

We want to say that for certain Hermitian Wigner ensembles $(W_{n})_{n\in\mathbf{N}}$ with $\operatorname{\mathbf{E}}\mu_{W_{n}}\Rightarrow\mu_{\mathrm{sc}}$, we have (1.3). However, without further restriction on $(W_{n})_{n\in\mathbf{N}}$, we can always change $o(n)$ rows and columns of $W_{n}$ so that (1.3) becomes false, while leaving the limiting distribution of $\operatorname{\mathbf{E}}\mu_{W_{n}}$ unchanged. To avoid this problem, we assume that

(1.5)

$$\frac{1}{n}\sum_{i\in J_{n}}\sum_{j=1}^{n}\operatorname{\mathbf{E}}|w_{ij}|^{2}\to 0\qquad\text{for any $J_{n}\subset\{1,\ldots,n\}$ with $|J_{n}|/n\to 0$.}$$

Notice that this condition is weaker than (1.4). If $W_{n}$ satisfies (1.3) and (1.5), and we change $o(n)$ rows and columns of it to obtain Hermitian $W_{n}^{\prime}$ which also satisfies (1.5), then $W_{n}^{\prime}$ also satisfies (1.3). The following is our first main theorem:

We can actually go beyond Theorem 1.6 and allow the entries of $W_{n}$ to have infinite variances, for example when $w_{ij}=c_{ij}x_{ij}/\sqrt{n\log n}$ where $x_{ij}$ has a density

$$f(x)=\begin{cases}1/|x|^{3}&\text{if $|x|>1$}\\ 0&\text{if $|x|\leq 1$}\end{cases}$$

and $c_{ij}$ is a real number close to $1$. To achieve this, instead of $\operatorname{\mathbf{E}}w_{ij}=0$ and the Lindeberg condition (1.2), we assume

(1.6)

$$\frac{1}{n}\sum_{i,j=1}^{n}\bigl{(}\operatorname{\mathbf{E}}[w_{ij};|w_{ij}|\leq 1]\bigr{)}^{2}\to 0$$

and

(1.7)

$$\frac{1}{n}\sum_{i,j=1}^{n}\operatorname{\mathbf{P}}(|w_{ij}|>\epsilon)\to 0\qquad\text{for all $\epsilon>0$.}$$

If $\operatorname{\mathbf{E}}w_{ij}=0$ and (1.2) hold, then (1.6) follows due to

$$\bigl{(}\operatorname{\mathbf{E}}[w_{ij};|w_{ij}|\leq 1]\bigr{)}^{2}=\bigl{(}\operatorname{\mathbf{E}}[w_{ij};|w_{ij}|>1]\bigr{)}^{2}\leq\operatorname{\mathbf{E}}[|w_{ij}|^{2};|w_{ij}|>1],$$

and (1.7) follows by

$$\operatorname{\mathbf{P}}(|w_{ij}|>\epsilon)\leq\epsilon^{-2}\operatorname{\mathbf{E}}[|w_{ij}|^{2};|w_{ij}|>\epsilon].$$

Finally, (1.5) is replaced by

(1.8)

$$\frac{1}{n}\sum_{i\in J_{n}}\sum_{j=1}^{n}\operatorname{\mathbf{E}}[|w_{ij}|^{2};|w_{ij}|\leq 1]\to 0\\ \text{for any $J_{n}\subset\{1,\ldots,n\}$ with $|J_{n}|/n\to 0$.}$$

The following is our second main theorem:

The following corollary relates $\operatorname{\mathbf{E}}\mu_{W_{n}}\Rightarrow\mu_{\mathrm{sc}}$ to the convergence in distribution of the sum of a row of $W_{n}$ to the standard normal random variable. The sufficiency direction when the entries of $W_{n}$ are identically distributed was covered by [Jun18]. We denote the Lévy metric by $L$.

The rest of the paper is organized as follows. Section 2 is a short section that introduces Theorem 2.1. This theorem is a reduction of Theorems 1.6 and 1.8, and will ultimately imply them as shown in Appendix B. This section also proves the sufficiency part of Theorem 2.1 in the case when the entries of $W_{n}$ are real.

Section 3 is the essence of the proof of the necessity part of Theorem 2.1. We add an assumption that the sixth moment of $\operatorname{\mathbf{E}}\mu_{W_{n}}$ is bounded, but as a return we obtain a clean proof that is right to the point. The idea is to express the second and the fourth moments of $\operatorname{\mathbf{E}}\mu_{W_{n}}$ in terms of the variances of the entries of $W_{n}$.

Section 4 shows that we can remove the additional assumption on the sixth moments, assuming that a certain lemma (Lemma 4.1) holds. The condition (1.5) is used in this section.

Section 5 proves Lemma 4.1 by a systematic computation of the moments of $\operatorname{\mathbf{E}}\mu_{W_{n}}$. The computation is a variant of Wigner’s original moment method, but it can handle the case when the entries have non-identical variances.

In Section 6, we prove the sufficiency part of Theorem 2.1 using the results of Section 5. The classical argument involving Dyck paths is discussed for completeness.

Finally in Section 7, we derive Corollary 1.10 from Theorem 1.8 by an elementary argument involving the Lindeberg–Feller central limit theorem.

Appendix A defines the mean of a random probability measure, and proves some of their properties that we need. Appendix B contains the fairly standard proof that Theorem 1.8 implies Theorem 1.6, and that Theorem 2.1 implies Theorem 1.8.

It is enough to prove the following in order to prove Theorems 1.6 and 1.8. The justification for the reduction is fairly standard, and is covered by Lemmas B.1 and B.4 in the appendix.

The following shows that we can assume (1.4) in the proof of the sufficiency part of Theorem 2.1.

If $W_{n}$ is real symmetric, then the sufficiency part of Theorem 2.1 follows from Lemma 2.2 and Theorem 1.4. In Section 6, we will present a direct proof of the sufficiency that applies to Hermitian Wigner ensembles and does not depend on the result of [GNT15].

Assume (2.1) throughout this section. In this section, we present a relatively simple proof of necessity in Theorem 2.1 under the following additional assumption:

(3.1)

$$\sup_{n\in\mathbf{N}}\int_{\mathbf{R}}x^{6}\,\operatorname{\mathbf{E}}\mu_{W_{n}}<\infty.$$

The number $6$ comes out just because it is an even number greater than $4$. Our proof is based on an examination of the second and the fourth moments of $\operatorname{\mathbf{E}}\mu_{W_{n}}$. If $\lambda_{1},\ldots,\lambda_{n}$ are the eigenvalues of $W_{n}$, then for each $k\in\mathbf{N}$ we have

$$\int_{\mathbf{R}}x^{k}\,\mu_{W_{n}}(dx)=\frac{1}{n}\sum_{i=1}^{n}\lambda_{i}^{k}=\frac{1}{n}\operatorname{tr}W_{n}^{k},$$

and thus

(3.2)

$$\int_{\mathbf{R}}x^{k}\,\operatorname{\mathbf{E}}\mu_{W_{n}}(dx)=\frac{1}{n}\operatorname{\mathbf{E}}\operatorname{tr}W_{n}^{k}.$$

(See Lemma A.1.)

The second moment of $\operatorname{\mathbf{E}}\mu_{W_{n}}$ can be easily expressed in terms of the variances of $w_{ij}$.

Computing the fourth moment requires more effort, but is still tractable.

We remark that (1.5) implies (3.3). The proof for that is similar to that of Lemma 4.2 below.

Now we are ready to prove the necessity part of Theorem 2.1 assuming (3.1). Assume $\operatorname{\mathbf{E}}\mu_{W_{n}}\Rightarrow\mu_{\mathrm{sc}}$. By Skorokhod’s representation theorem [Bil12, Theorem 25.6], we can take real-valued random variables $X,X_{1},X_{2},\ldots$ on a common probability space such that $\operatorname{\mathbf{E}}\mu_{W_{n}}$ is the distribution of $X_{n}$, $\mu_{\mathrm{sc}}$ is the distribution of $X$, and $X_{n}\to X$ a.s.

Since

(3.4)

$$\sup_{n\in\mathbf{N}}\operatorname{\mathbf{E}}X_{n}^{6}=\sup_{n\in\mathbf{N}}\int_{\mathbf{R}}x^{6}\,\operatorname{\mathbf{E}}\mu_{W_{n}}(dx)<\infty,$$

$(X_{n}^{2})_{n\in\mathbf{N}}$ and $(X_{n}^{4})_{n\in\mathbf{N}}$ are uniformly integrable. Thus, $X_{n}\to X$ a.s. implies $\operatorname{\mathbf{E}}X_{n}^{2}\to\operatorname{\mathbf{E}}X^{2}$ and $\operatorname{\mathbf{E}}X_{n}^{4}\to\operatorname{\mathbf{E}}X^{4}$. By Lemma 3.1 and Lemma 3.2, we have

$$\frac{1}{n}\sum_{i,j=1}^{n}\operatorname{\mathbf{E}}|w_{ij}|^{2}\to\operatorname{\mathbf{E}}X^{2}=\int_{\mathbf{R}}x^{2}\,\mu_{\mathrm{sc}}(dx)=1$$

and

$$\frac{2}{n}\sum_{i=1}^{n}\biggl{(}\sum_{j=1}^{n}\operatorname{\mathbf{E}}|w_{ij}|^{2}\biggr{)}^{2}\to\operatorname{\mathbf{E}}X^{4}=\int_{\mathbf{R}}x^{4}\,\mu_{\mathrm{sc}}(dx)=2.$$

See [AGZ10, 2.1.1] for a computation of the moments of $\mu_{\mathrm{sc}}$.

Here is the punchline: using $(\operatorname{\mathbf{E}}Y)^{2}\leq\operatorname{\mathbf{E}}Y^{2}$ in the first line, and then applying the two convergences we have just established, we have

$$\begin{split}\biggl{(}\frac{1}{n}\sum_{i=1}^{n}\Bigl{|}\sum_{j=1}^{n}\operatorname{\mathbf{E}}|w_{ij}|^{2}-1\Bigr{|}\biggr{)}^{2}&\leq\frac{1}{n}\sum_{i=1}^{n}\biggl{(}\sum_{j=1}^{n}\operatorname{\mathbf{E}}|w_{ij}|^{2}-1\biggr{)}^{2}\\ &=\frac{1}{n}\sum_{i=1}^{n}\biggl{(}\sum_{j=1}^{n}\operatorname{\mathbf{E}}|w_{ij}|^{2}\biggr{)}^{2}-\frac{2}{n}\sum_{i,j=1}^{n}\operatorname{\mathbf{E}}|w_{ij}|^{2}+1\\ &\to 1-2+1=0.\end{split}$$

Thus, the necessity in Theorem 2.1 is proved under the assumption (3.1).

In this section, we prove the necessity part of Theorem 2.1 without assuming the bounded sixth moment condition (3.1). We rely on the following lemma, which will be proved in the next section.

The number $8$ is here just because it is even and greater than $6$. In fact, our proof easily extends to any even natural number. Given $K_{n}\subset\{1,\ldots,n\}$ for all $n\in\mathbf{N}$, let $W_{n}^{K}$ be the matrix obtained from $W_{n}$ by replacing $w_{ij}$ with $0$ for all $(i,j)\not\in K_{n}\times K_{n}$.

Given $K_{n}\subset\{1,\ldots,n\}$ for each $n\in\mathbf{N}$, let $W_{n}^{K}$ be the matrix obtained from $W_{n}$ by replacing $w_{ij}$ with $0$ for all $(i,j)\not\in K_{n}\times K_{n}$.

We are ready to prove the necessity part of Theorem 2.1. Assume (1.5), (2.1), and $\operatorname{\mathbf{E}}\mu_{W_{n}}\Rightarrow\mu_{\mathrm{sc}}$. By Lemma 1.3, we have $\mu_{W_{n}}\Rightarrow\mu_{\mathrm{sc}}$ a.s. If $K_{1},K_{2},\ldots$ are as in Lemma 4.3, then

$$\frac{\operatorname{rank}(W_{n}-W_{n}^{K})}{n}\leq\frac{2(n-|K_{n}|)}{n}\to 0,$$

and thus $\mu_{W_{n}^{K}}\Rightarrow\mu_{\mathrm{sc}}$ a.s. by Lemma 1.5. By another application of Lemma 1.3, $\operatorname{\mathbf{E}}\mu_{W_{n}^{K}}\Rightarrow\mu_{\mathrm{sc}}$. As we have (4.1), the previous section tells us that

$$\frac{1}{n}\sum_{i\in K_{n}}\Bigl{|}\sum_{j\in K_{n}}\operatorname{\mathbf{E}}|w_{ij}|^{2}-1\Bigr{|}\to 0.$$

Since $|K_{n}^{c}|/n\to 0$, the assumption (1.5) implies (1.3). Thus, the necessity part of Theorem 2.1 is proved assuming that Lemma 4.1 holds.