Partial Linear Eigenvalue Statistics for Wigner and Sample Covariance Random Matrices

Abstract

Let $M_n$ be an $n \times n$ Wigner or sample covariance random matrix, and let $\mu _1(M_n), \mu _2(M_n), \ldots , \mu _n(M_n)$ denote the randomly ordered eigenvalues of $M_n$. We study the fluctuations of the partial linear eigenvalue statistics

$$\begin{aligned} \sum _{i=1}^{n-k} f(\mu _i(M_n)) \end{aligned}$$

as $n \rightarrow \infty$ for sufficiently nice test functions $f$. We consider both the cases when $k$ is fixed and when $\min \{k,n-k\}$ tends to infinity with $n$.

Appendix: Proof of Theorem 15

This section is devoted to the proof of Theorem 15. We will need the following version of [19, Theorem 3.6].

Theorem 19

[19] Let $M_n = \frac{1}{\sqrt{n}} W_n$ be a real symmetric Wigner matrix where $W_n = (w_{ij})_{1 \le i,j \le n}$. Suppose there exist constants $C_1,c_1>0$ and $0 < \varepsilon < 1/2$ such that

$$\begin{aligned} \sup _{1 \le i < j \le n} \mathbb{E }[w_{ij}^4] \le C_1 \quad \text{ and }\quad \sup _{1 \le i < j \le n} \mathbb{P }( |w_{ij}| > n^{1/2 - \varepsilon } ) \le e^{-n^{c_1}}. \end{aligned}$$

Then, there exist constants $c>0$ and $n_0$ (which depend only on $C_1, \varepsilon$, and $\sigma$ from Definition 1) such that the event

$$\begin{aligned} \bigcup _{j=1}^n \left\{ |\lambda _j(M_n) - \eta _j| \le (\log n)^{c \log \log n} n^{-2/3} [\min \{j,n-j+1\}]^{-1/3} \right\} \end{aligned}$$

holds with overwhelming probability for any $n > n_0$.

Proof of Theorem 15

Set $\varepsilon _n := n^{1/2 - \varepsilon }$; we remind the reader that $0 < \varepsilon < 1/2$ and hence $\varepsilon _n \rightarrow \infty$ as $n \rightarrow \infty$. We begin with a truncation. Let

$$\begin{aligned} \hat{w}_{ij} := w_{ij} \mathbf 1 _{\{|w_{ij}| \le \varepsilon _n\}} \quad \text{ for }\quad 1 \le i \le j \le n. \end{aligned}$$

We define the values $\mu _{ij} := \mathbb{E }\hat{w}_{ij}$ and $\tau ^2_{ij} := \mathbb{E }[w_{ij}^2] - \mathbb{E }[ \hat{w}_{ij}^2 ]$ for $1 \le i \le j \le n$. Then by (9), we have

$$\begin{aligned} \sup _{1 \le i < j \le n} |\mu _{ij}| \le \frac{C_1}{\varepsilon _n^3}, \quad \sup _{1 \le i \le n} |\mu _{ii}| \le \frac{\sigma ^2}{\varepsilon _n} \end{aligned}$$

(20)

$$\begin{aligned} \sup _{1 \le i < j \le n} \tau _{ij}^2 \le \frac{C_1}{\varepsilon _n^2}, \quad \sup _{1 \le i \le n} \tau _{ii}^2 \le \sigma ^2. \end{aligned}$$

(21)

For $1 \le i \le j \le n$, define the random variable $\tilde{w}_{ij}$ as a mixture of

• $\hat{w}_{ij}$ with probability $1 - \frac{|\mu _{ij}|}{\varepsilon _n} - \frac{\tau _{ij}^2}{\varepsilon _n^2}$ and

• $z_{ij}$ with probability $\frac{|\mu _{ij}|}{\varepsilon _n} + \frac{\tau _{ij}^2}{\varepsilon _n^2}$,

where $z_{ij}, 1 \le i \le j \le n$ are independent Bernoulli random variables independent of $W_n$. Set $\tilde{w}_{ji} = \tilde{w}_{ij}$ for $1 \le i < j \le n$. Let $\tilde{W}_n = (\tilde{w}_{ij})_{1 \le i,j \le n}$ and $\tilde{M}_n = \frac{1}{\sqrt{n}} \tilde{W}_n$.

We now show that there exist Bernoulli random variables $z_{ij}$ such that $\tilde{M}_n$ is a real symmetric Wigner matrix that satisfies

$$\begin{aligned} \sup _{1 \le i \le j \le n} |\tilde{w}_{ij}| \le n^{1/2 - \varepsilon /2} \quad \text{ almost } \text{ surely } \end{aligned}$$

(22)

and

$$\begin{aligned} \sup _{1 \le i < j \le n} \mathbb{E }[ \tilde{w}_{ij}^4] \le 513 C_1. \end{aligned}$$

(23)

In particular, we will construct $z_{ij}$ to be a Bernoulli random variable, symmetric about its mean, such that its mean and second moment satisfy

$$\begin{aligned} 0&= \mu _{ij} \left( 1 - \frac{|\mu _{ij}|}{\varepsilon _n} - \frac{\tau _{ij}^2}{\varepsilon _n^2} \right) + \mathbb{E }[z_{ij}] \left( \frac{|\mu _{ij}|}{\varepsilon _n} + \frac{\tau _{ij}^2}{\varepsilon _n^2} \right) \end{aligned}$$

(24)

$$\begin{aligned} \mathbb{E }\left[ w_{ij}^2\right]&= \left( \mathbb{E }\left[ w_{ij}^2\right] - \tau _{ij}^2\right) \left( 1 - \frac{|\mu _{ij}|}{\varepsilon _n} - \frac{\tau _{ij}^2}{\varepsilon _n^2} \right) + \mathbb{E }[z_{ij}^2] \left( \frac{|\mu _{ij}|}{\varepsilon _n} + \frac{\tau _{ij}^2}{\varepsilon _n^2} \right) \end{aligned}$$

(25)

for $1 \le i \le j \le n$. We first note that, by definition of $\tilde{w}_{ij}$, we only need to consider the case when $\frac{|\mu _{ij}|}{\varepsilon _n} + \frac{\tau _{ij}^2}{\varepsilon _n^2} > 0$. Suppose $a_{ij},b_{ij}$ are real numbers that satisfy

$$\begin{aligned} 0&= \mu _{ij} \left( 1 - \frac{|\mu _{ij}|}{\varepsilon _n} - \frac{\tau _{ij}^2}{\varepsilon _n^2} \right) + a_{ij} \left( \frac{|\mu _{ij}|}{\varepsilon _n} + \frac{\tau _{ij}^2}{\varepsilon _n^2} \right) \\ \mathbb{E }\left[ w_{ij}^2\right]&= \left( \mathbb{E }\left[ w_{ij}^2\right] - \tau _{ij}^2\right) \left( 1 - \frac{|\mu _{ij}|}{\varepsilon _n} - \frac{\tau _{ij}^2}{\varepsilon _n^2} \right) + b_{ij}^2 \left( \frac{|\mu _{ij}|}{\varepsilon _n} + \frac{\tau _{ij}^2}{\varepsilon _n^2} \right) . \end{aligned}$$

From the first equation, we obtain

$$\begin{aligned} |a_{ij}| \left( \frac{|\mu _{ij}|}{\varepsilon _n} + \frac{\tau _{ij}^2}{\varepsilon _n^2} \right) \le |\mu _{ij}|. \end{aligned}$$

(26)

From the second equation, we have

$$\begin{aligned} b_{ij}^2 \left( \frac{|\mu _{ij}|}{\varepsilon _n} + \frac{\tau _{ij}^2}{\varepsilon _n^2} \right) = (\mathbb{E }[w_{ij}^2] - \tau _{ij}^2) \left( \frac{|\mu _{ij}|}{\varepsilon _n} + \frac{\tau _{ij}^2}{\varepsilon _n^2} \right) + \tau _{ij}^2 \ge \tau _{ij}^2. \end{aligned}$$

(27)

We now note that $\tau _{ij}^2 \ge \varepsilon _n |\mu _{ij}|$ by definition of $\hat{w}_{ij}$ and hence

$$\begin{aligned} \tau _{ij}^4 + \tau _{ij}^2|\mu _{ij}| \varepsilon _n - |\mu _{ij}|^2 \varepsilon _n^2 \ge 0. \end{aligned}$$

It then follows that

$$\begin{aligned} \tau _{ij}^2 \ge \frac{|\mu _{ij}|^2}{\left( \frac{|\mu _{ij}|}{\varepsilon _n} + \frac{\tau _{ij}^2}{\varepsilon _n^2} \right) }. \end{aligned}$$

(28)

Combining (26), (27), and (28), we obtain

$$\begin{aligned} b_{ij}^2 \left( \frac{|\mu _{ij}|}{\varepsilon _n} + \frac{\tau _{ij}^2}{\varepsilon _n^2} \right) \ge \tau _{ij}^2 \ge \frac{\mu _{ij}^2}{\left( \frac{|\mu _{ij}|}{\varepsilon _n} + \frac{\tau _{ij}^2}{\varepsilon _n^2} \right) } \ge a_{ij}^2 \left( \frac{|\mu _{ij}|}{\varepsilon _n} + \frac{\tau _{ij}^2}{\varepsilon _n^2} \right) \end{aligned}$$

and hence $b_{ij}^2 \ge a_{ij}^2$. We can now define

$$\begin{aligned} z_{ij} := \left\{ \begin{array}{lr} a_{ij} + \sqrt{b_{ij}^2 - a_{ij}^2} &{} \text{ with } \text{ probability } 1/2 \\ a_{ij} - \sqrt{b_{ij}^2 - a_{ij}^2} &{} \text{ with } \text{ probability } 1/2 \end{array} \right. . \end{aligned}$$

It is straightforward to verify that $z_{ij}$ has mean $a_{ij}$ and second moment $b_{ij}^2$.

By construction, $\tilde{M}_n$ is a real symmetric Wigner matrix. We now verify (22) and (23). By solving equations (24) and (25) for $\mathbb{E }[z_{ij}]$ and $\mathbb{E }[z_{ij}^2]$ and applying the bounds (20) and (21), it follows that $|z_{ij}| \le 4 \varepsilon _n$. Thus, we conclude that (22) holds for $n$ sufficiently large.

We also have for $1 \le i < j \le n$

$$\begin{aligned} \mathbb{E }[\tilde{w}_{ij}^4]&= \mathbb{E }[\hat{w}_{ij}^4] \left( 1 - \frac{|\mu _{ij}|}{\varepsilon _n} - \frac{\tau _{ij}^2}{\varepsilon _n^2} \right) + \mathbb{E }[z_{ij}^4] \left( \frac{|\mu _{ij}|}{\varepsilon _n} + \frac{\tau _{ij}^2}{\varepsilon _n^2} \right) \\&\le C_1 + (4 \varepsilon _n)^4 2 \frac{C_1}{\varepsilon _n^4} \\&\le 513 C_1 \end{aligned}$$

by (20) and (21). This verifies (23), and hence, $\tilde{M}_n$ satisfies the conditions of Theorem 19.

By Theorem 19, there exists a constant $c>0$ such that the event

$$\begin{aligned} \Omega _n := \bigcup _{j=1}^n \left\{ |\lambda _j(\tilde{M}_n) - \eta _j| \le (\log n)^{c \log \log n} n^{-2/3} [\min \{j,n-j+1\}]^{-1/3} \right\} \end{aligned}$$

holds with overwhelming probability. Thus, we obtain

$$\begin{aligned} \mathbb{P }&\left( \exists j : |\lambda _j(M_n) - \eta _j| \ge (\log n)^{c \log \log n} n^{-2/3} [\min \{j,n-j+1\}]^{-1/3} \right) \\&\qquad \qquad \le \mathbb{P }(\Omega _n^C) + \mathbb{P }(M_n \ne \tilde{M}_n). \end{aligned}$$

The proof of Theorem 15 is now complete by noting that

$$\begin{aligned} \mathbb{P }( M_n \ne \tilde{M}_n)&\le \sum _{i,j=1}^n \mathbb{P }(|w_{ij}| > \varepsilon _n) + \sum _{i,j=1}^n \left( \frac{|\mu _{ij}|}{\varepsilon _n} + \frac{\tau _{ij}^2}{\varepsilon _n^2} \right) \\&\le \frac{2}{\varepsilon _n^4} \sum _{1 \le i < j \le n} \mathbb{E }\left[ w_{ij}^4 \mathbf 1 _{\{|w_{ij}| > \varepsilon _n\}}\right] + \frac{2}{\varepsilon _n^2} \sum _{i =1}^n \mathbb{E }\left[ w_{ii}^2 \mathbf 1 _{\{|w_{ii}| > \varepsilon _n\}}\right] . \end{aligned}$$

$\square$