Second-Order Moment Convergence Rates for Spectral Statistics of Random Matrices

Abstract

This paper considers the precise asymptotics of the spectral statistics of random matrices. Following the ideas of Gut and Spătaru (2000) and Liu and Lin (2006) on the precise asymptotics of i.i.d. random variables in the context of the complete convergence and the second-order moment convergence, respectively, we will establish the precise second-order moment convergence rates of a type of series constructed by the spectral statistics of Wigner matrices or sample covariance matrices.

1. Introduction and Main Results

This paper is concerned with the precise asymptotic behaviors of the spectral (eigenvalue) statistics of random matrices; two types of classical random matrices including Wigner matrices and sample covariance matrices will be considered. An 𝑛×𝑛 Wigner matrix is defined to be a random Hermitian matrix 𝑋_𝑛=(𝑋_𝑖𝑗)_{1≤𝑖,𝑗≤𝑛}, in which the real and imaginary parts of 𝑋_𝑖𝑗 for 𝑖<𝑗 are i.i.d. random variables with mean 0 and variance 1/2 (in that case, 𝔼𝑋²_𝑖𝑗=0), and 𝑋_𝑖𝑖, 𝑖=1,2,…,𝑛, are i.i.d. random variables with mean 0 and variance 1. Denote by 𝜆₁≤⋯≤𝜆_𝑛 the real eigenvalues of the normalized Wigner matrix 𝑀_𝑛=(1/√𝑛)𝑋_𝑛. The classical Wigner theorem states that the empirical distribution 𝐹^𝑀𝑛(𝑥)=(1/𝑛)∑^𝑛_𝑖=11_{{𝜆𝑖≤𝑥}} converges almost surely to the semicircle law with the density 𝜇_𝑠𝑐(𝑥)=(1/2𝜋)√4−𝑥²1_[−2,2](𝑥). Consequently, for any bounded continuous function 𝑓, the spectral statistics satisfy that lim_𝑛→∞1𝑛^𝑛_𝑖=1𝑓𝜆_𝑖=lim_𝑛→∞𝑓(𝑥)𝑑𝐹^𝑀𝑛(𝑥)=𝑓(𝑥)𝜇_𝑠𝑐(𝑥)𝑑𝑥a.s. (1)

The result can be viewed as an analog of the law of large number for independent random variables. As for the fluctuation of the spectral statistics, a remarkable work due to Bai et al. [1] states the following.

Lemma 1. Assume the entries of a Wigner matrix 𝑋_𝑛 satisfy that 𝐸|𝑋_𝑖𝑗|⁴=2 and 𝐸|𝑋_𝑖𝑗|⁶<∞ for all 1≤𝑖, 𝑗≤𝑛. Let Π be an open interval including the interval [−2,2] and 𝐶⁴(Π) is the space of forth-order continuous differentiable functions on Π. Denote 𝐺_𝑛𝑓∶=𝑛𝑓(𝑥)𝑑𝐹^𝑀𝑛(𝑥)−𝑓(𝑥)𝜇_𝑠𝑐(𝑥)𝑑𝑥. (2) Then the empirical process {𝐺_𝑛(𝑓)∶𝑓∈𝐶⁴(Π)} converges weakly in finite dimension to a Gaussian process {𝐺(𝑓)∶𝑓∈𝐶⁴(Π)} with mean zero and the covariance function Cov(𝑓,𝑔) given by =1Cov𝑓,𝑔4𝜋²²₋₂𝑓^′(𝑡)𝑔^′(𝑠)×log4−𝑡𝑠+4−𝑡²4−𝑠²4−𝑡𝑠−4−𝑡²4−𝑠²𝑑𝑡𝑑𝑠. (3)

In addition, Guionnet and Zeitouni [2] give a concentration inequality on the empirical spectral measure of 𝐹^𝑀𝑛(𝑥) near the semicircle law.

Lemma 2. Assume the law of the entries {Re(𝑋_𝑖𝑗),Im(𝑋_𝑖𝑗),𝑋_𝑖𝑖} satisfies the logarithmic Sobolev inequality (LSI) with constant 𝑐₀>0. Let 𝑓 be a Lipschitz function and denote 𝜂_𝑛𝑓=𝑛⁻¹𝐺_𝑛𝑓=𝑓(𝑥)𝑑𝐹^𝑀𝑛(𝑥)−𝑓(𝑥)𝜇_𝑠𝑐(𝑥)𝑑𝑥. (4) Then for any 𝛿>𝛿_𝑛(𝑓)=|𝔼∫𝑓(𝑥)𝑑𝐹^𝑀𝑛(𝑥)−∫𝑓(𝑥)𝜇_𝑠𝑐(𝑥)𝑑𝑥|, there exists 𝑐>0, such that 𝑃𝜂_𝑛𝑓≥𝛿≤2𝑒^{−𝑐𝑛2(𝛿−𝛿𝑛(𝑓))2}. (5)

Based on the above existing results on the limiting spectral properties of random matrices, we will give another type of asymptotic spectral properties, that is, the precise asymptotics of the spectral statistics of random matrices. In particular, we will consider the precise second-order moment convergence rates of a type of series constructed by the spectral statistics of random matrices. Our first result can be listed as follows.

Theorem 3. Suppose that the law of the entries {Re(𝑋_𝑖𝑗),Im(𝑋_𝑖𝑗),𝑋_𝑖𝑖} satisfies LSI with constant 𝑐₀>0 and 𝐸|𝑋_𝑖𝑗|⁴=2 for all 1≤𝑖, 𝑗≤𝑛. Denote 𝜂_𝑛(𝑓) as (4), where 𝑓∈𝐶⁴(Π) has a bounded first-order derivative. Then for any 𝑠>1/2, 𝛽>2𝑠−1, one has lim_𝜖↘0𝜖^{∞(𝛽+1)/𝑠−2}_𝑛=1𝑛^{𝛽−2𝑠+2}𝔼𝜂_𝑛𝑓²𝐼_{{|𝜂𝑛(𝑓)|≥𝜖𝑛𝑠−1}}=1𝐷𝑓𝛽−2𝑠+1^{(𝛽+1)/2𝑠}𝔼𝜉^{(𝛽+1)/𝑠}, (6) where 𝜉 is a standard Gaussian random variable and 𝐷𝑓=14𝜋²²₋₂𝑓^′(𝑡)𝑓^′(𝑠)×log4−𝑡𝑠+4−𝑡²4−𝑠²4−𝑡𝑠−4−𝑡²4−𝑠²𝑑𝑡𝑑𝑠. (7)

Now we turn to the sample covariance matrix, which is an important statistic in multivariate statistics analysis. Let 𝑌_𝑚𝑛=(𝑌_𝑖𝑗)_{1≤𝑖≤𝑚,1≤𝑗≤𝑛} be an 𝑚×𝑛 random matrix whose entries are i.i.d. complex-valued random variables with mean 0 and variance 1 and Re(𝑌_𝑖𝑗)_𝑖<𝑗 and Im(𝑌_𝑖𝑗)_𝑖<𝑗 are independent random variables with mean 0 and variance 1/2. Then 𝑊_𝑚,𝑛=(1/𝑛)𝑌_𝑚𝑛𝑌^∗_𝑚𝑛 can be viewed as the sample covariance matrix of 𝑛 samples of 𝑚-dimensional random vectors. Let 𝜆^′₁≤⋯≤𝜆^′_𝑚 be the eigenvalues of 𝑊_𝑚,𝑛, and define the empirical spectral distribution 𝐹^{𝑊𝑚,𝑛}(𝑥)=(1/𝑚)∑^𝑚_𝑖=1𝐼_{{𝜆′𝑖≤𝑥}}. When 𝑚/𝑛→𝜌∈(0,1] as 𝑛→∞, the famous Marchenko-Pastur theorem reveals that almost surely 𝐹^{𝑊𝑚,𝑛}(𝑥) converges to the Marchenko-Pastur law with the density 𝛾_𝜌(𝑥)=(1/2𝜋𝑥𝜌)(𝑏(𝜌)−𝑥)(𝑥−𝑎(𝜌))1_{[𝑎(𝜌),𝑏(𝜌)]}(𝑥), where 𝑎(𝜌)=(1−√𝜌)², 𝑏(𝜌)=(1+√𝜌)².

Denote 𝜏_𝑚,𝑛𝑓=𝑓(𝑥)𝑑𝐹^{𝑊𝑚,𝑛}(𝑥)−𝑓(𝑥)𝛾_𝑚/𝑛(𝑥)𝑑𝑥, (8) where 𝛾_𝑚/𝑛(𝑥) is the Marchenko-Pastur law 𝛾_𝜌(𝑥) with the parameter 𝜌=𝑚/𝑛. Similar to the result on Wigner matrices, we can give the result on the complex sample covariance matrices.

Theorem 4. Assume the law of the entries {Re(𝑌_𝑖𝑗),Im(𝑌_𝑖𝑗)} satisfies LSI with constant 𝑐₀>0 and 𝐸|𝑌_𝑖𝑗|⁴=2 for all 1≤𝑖≤𝑚, 1≤𝑗≤𝑛, 𝑚=[𝑛𝜌](0<𝜌≤1). If one denotes Γ to be an open interval including [𝑎(𝜌),𝑏(𝜌)], 𝑓∈𝐶⁴(Γ) has a bounded first order derivative. Then for any 𝑠>1/2, 𝛽>2𝑠−1, one has lim_𝜖↘0𝜖^{∞(𝛽+1)/𝑠−2}_𝑛=1𝑛^{𝛽−2𝑠+2}𝔼𝜏_𝑚,𝑛𝑓²𝐼_{{|𝜏𝑚,𝑛(𝑓)|≥𝜖𝑛𝑠−1}}=1𝜌𝛽−2𝑠+1^{(𝛽+1)/𝑠}𝑉𝑓^{(𝛽+1)/2𝑠}𝔼𝜉^{(𝛽+1)/𝑠}, (9) where 𝜉∼𝑁(0,1), and 𝑉𝑓=12𝜋^2𝑏(𝜌)_𝑎(𝜌)𝑓^′𝑥₁𝑓^′𝑥₂×log𝑠𝑥₁−𝑠𝑥₂𝑠𝑥₁−𝑠𝑥₂𝑑𝑥₁𝑑𝑥₂, (10)𝑠(𝑥)=lim_{𝑧→𝑥+𝑖0}𝑠(𝑧) and 𝑠(𝑧) is the Stieltjes transform of 𝐹₂(𝑦)=(1−𝑦)1_[0,∞)+𝑦∫^𝑦_−∞𝛾_𝜌(𝑥)𝑑𝑥, that is, 𝑠(𝑧)=∫^+∞_−∞(1/(𝑥−𝑧))𝑑𝐹₂(𝑥).

We will give the proofs of theorems in Section 2. Below are a few words about the motivation of this paper. In a sense, our results are similar to the precise asymptotics of independent random variables in the context of complete convergence and moment convergence. Let 𝑋,𝑋₁,𝑋₂,…, be i.i.d. random variables and 𝑆_𝑛=∑^𝑛_𝑖=1𝑋_𝑖; there are a number of results on the convergence of a type of series ^∞_𝑛=1𝜑(𝑛)𝑃𝑆_𝑛≥𝜖𝑓(𝑛),𝜖>0, (11) where 𝜑(𝑥) and 𝑓(𝑥) are the positive functions defined on [0,∞), and ∑^∞_𝑛=1𝜑(𝑛)=∞. In fact, the sum (11) tends to infinity when 𝜖↘0, one of the interesting problems is to examine the precise rate at which this occurs, and this amounts to finding a suitable normalizing rate function 𝜓(𝜖) such that the sum (11) multiplied by 𝜓(𝜖) has a nontrivial limit; this kind of results is frequently called “precise asymptotics.” The first result in this direction due to Heyde [3], who proved that lim_𝜖↘0𝜖²∑^∞_𝑛=1𝑃(|𝑆_𝑛|≥𝜖𝑛)=𝔼𝑋² under the assumptions that 𝔼𝑋=0 and 𝔼𝑋²<∞. Some analogous results in more general case can be found in Gut and Spǎtaru [4, 5] and Gut and Steinebach [6]. Moreover, Chow [7] studied the convergence properties of the series ∑^∞_𝑛=1𝑛^{𝑝𝛼−2−𝛼}𝔼{max_{1≤𝑗≤𝑛}|𝑆_𝑗|−𝜖𝑛^𝛼}₊(𝜖>0) when 𝑝≥1, 𝛼>1/2, 𝑝𝛼>1. There is a remarkable result obtained by Liu and Lin [8], who considered the precise asymptotics on the second-order moment convergence, which states that lim_𝜖↘01−log𝜖^∞_𝑛=1𝑛⁻²𝔼𝑆²_𝑛𝐼_{{|𝑆𝑛|≥𝜖𝑛}}=2𝜎² (12) when 𝔼𝑋=0, 𝔼𝑋²=𝜎², and 𝔼𝑋²log⁺|𝑋|<∞. Chen and Zhang [9] got the similar results on the second-order moment convergence of empirical process. Furthermore, there are also some precise asymptotic results in other contexts, such as the self-normalized sums, martingale-difference, random fields, and renewal process. It should be mentioned that the corresponding results on random matrices and random growth model have been studied by Su [10], who presented the precise asymptotics of the largest eigenvalues of Gaussian unitary ensembles, Laguerre unitary ensembles, and the longest increasing subsequence of a random permutation. In this paper, we will study the precise asymptotics on the spectral statistics of Wigner matrices and sample covariance matrices, which is also an interesting topic in random matrix theory.

In the rest of the paper, we will always write 𝑁(𝜖,𝑀)=[𝑀𝜖^−1/𝑠], where 𝜖>0 and 𝑀 is an arbitrary positive real number. ‖𝑓‖=sup_𝑥∈𝑅|𝑓(𝑥)|, and 𝜉 stands for a random variable observing the standard Gaussian distribution. We also denote 𝐶 to be the absolutely positive constant whose value can be different from one place to another.

2. The Proofs

Before the main proof of Theorem 3, we will first give four propositions.

Proposition 5. Under the assumptions of Theorem 3, one has lim_𝜖↘0𝜖^{∞(𝛽+1)/𝑠}_𝑛=1𝑛^𝛽𝑃𝜉≥𝜖𝑛^𝑠𝐷𝑓=1𝐷𝑓𝛽+1^{(𝛽+1)/2𝑠}𝐸𝜉^{(𝛽+1)/𝑠}. (13)

Proof. We calculate that lim_𝜖↘0𝜖^{∞(𝛽+1)/𝑠}_𝑛=1𝑛^𝛽𝑃𝜉≥𝜖𝑛^𝑠𝐷𝑓=lim_𝜖↘0𝜖^{(𝛽+1)/𝑠∞}₁𝑦^𝛽𝑃𝜉≥𝜖𝑦^𝑠𝐷𝑓=1𝑑𝑦𝑠𝐷𝑓^{(𝛽+1)/2𝑠∞}₀𝑡^{(𝛽+1)/𝑠−1}𝑃𝜉=1≥𝑡𝑑𝑡𝐷𝑓𝛽+1^{(𝛽+1)/2𝑠}𝐸𝜉^{(𝛽+1)/𝑠}. (14)

Proposition 6. Under the assumptions of Theorem 3, one has lim_𝜖↘0𝜖^{∞(𝛽+1)/𝑠}_𝑛=1𝑛^𝛽𝑃𝑛𝜂_𝑛𝑓≥𝜖𝑛^𝑠−𝑃𝜉≥𝜖𝑛^𝑠𝐷𝑓=0. (15)

Proof. We can write ^∞_𝑛=1𝑛^𝛽𝑃𝑛𝜂_𝑛𝑓≥𝜖𝑛^𝑠−𝑃𝜉≥𝜖𝑛^𝑠𝐷𝑓≤_{()𝑛≤𝑁𝜖,𝑀}𝑛^𝛽𝑃𝑛𝜂_𝑛𝑓≥𝜖𝑛^𝑠−𝑃𝜉≥𝜖𝑛^𝑠𝐷𝑓+_{()𝑛>𝑁𝜖,𝑀}𝑛^𝛽𝑃𝑛𝜂_𝑛𝑓≥𝜖𝑛^𝑠+_{()𝑛>𝑁𝜖,𝑀}𝑛^𝛽𝑃𝜉≥𝜖𝑛^𝑠𝐷𝑓∶=𝐼₁+𝐼₂+𝐼₃. (16)
Under the assumptions of Theorem 3, the law of each entry 𝑋_𝑖𝑗 also satisify LSI. By Lemma 5 of Vershynin [11], for each fixed positive integer 𝑘, 𝔼|𝑋_𝑖𝑗|^𝑘<+∞. According to Lemma 1, we can see that 𝑛𝜂_𝑛𝑓𝐷𝑓^𝑑⟶𝑁(0,1). (17) Thus, as 𝑛→∞, Δ_𝑛=∶sup_𝑥∈ℝ𝑃𝑛𝜂_𝑛𝑓𝜉≥𝑥≥𝑥−𝑃𝐷𝑓⟶0. (18) Using Toeplitz's lemma, we can deduce that lim_𝜖↘0𝜖^{(𝛽+1)/𝑠}𝐼₁=0. (19)
For the term 𝐼₂, if 𝑓∈𝐶⁴(Π) has a bounded first-order derivative, then 𝑓 is a Lipschitz function.
By Theorem 8.2 of Bai and Silverstein [12], there exists a real number 𝐶₁>0, such that 𝛿_𝑛𝑓=𝔼𝑓(𝑥)𝑑𝐹^𝑀𝑛(𝑥)−𝑓(𝑥)𝜇_𝑠𝑐(𝑥)𝑑𝑥≤𝐶₁𝑓^′𝑛^−1/2. (20) By Lemma 2, for any 𝛿₀>𝑛𝛿_𝑛(𝑓), there exists 𝑐>0 such that 𝑃𝑛𝜂_𝑛𝑓𝐷𝑓≥𝛿₀≤2𝑒^{−𝑐(𝛿0−𝑛𝛿𝑛(𝑓))2}. (21)
For each fixed 𝜖>0, the assumption 𝑠>1/2 reveals that, when 𝑛>𝑁(𝜖,𝑀) with a sufficient large number 𝑀, we have 𝜖𝑛^𝑠>𝑛𝛿_𝑛(𝑓). Thus there exists 𝑐₁>0, such that lim_𝑀→∞𝐼₂≤lim_𝑀→∞𝐶^∞_{𝑁(𝜖,𝑀)}𝑥^𝛽𝑒^{−𝑐1𝑥2𝑠}𝑑𝑥=0. (22)
For the term 𝐼₃, as the fact that 𝜉∼𝑁(0,1), the well-known tail probability estimation states that for any 𝑥∈ℝ, 𝑃()≤1𝜉≥𝑥√𝑒2𝜋𝑥^−𝑥2/2. (23) Hence, for any fixed 𝜖>0, we can get lim_𝑀→∞𝐼₃≤lim_𝑀→∞𝐶^∞_{𝑁(𝜖,𝑀)}𝑥^𝛽−1𝑒^−𝑥2/2𝑑𝑥=0. (24) By letting 𝜖↘0 and then 𝑀→∞, a combination of (22) and (24) can get that lim_𝜖↘0𝜖^{(𝛽+1)/𝑠}(𝐼₂+𝐼₃)=0. Then combined with the relations (16) and (19), we can complete the proof.

Proposition 7. Under the assumptions of Theorem 3, one has lim_𝜖↘0𝜖^{∞(𝛽+1)/𝑠−2}_𝑛=1𝑛^{𝛽−2𝑠∞}_𝜖𝑛𝑠𝜉≥𝑦𝑦𝑃𝐷𝑓=𝑠𝑑𝑦𝐷𝑓𝛽+1𝛽−2𝑠+1^{(𝛽+1)/2𝑠}𝔼𝜉^{(𝛽+1)/𝑠}. (25)

Proof. By the similar argument in the proof of Proposition 5, we can show that lim_𝜖↘0𝜖^{∞(𝛽+1)/𝑠−2}_𝑛=1𝑛^{𝛽−2𝑠∞}_𝜖𝑛𝑠𝜉≥𝑦𝑦𝑃𝐷𝑓𝑑𝑦=lim_𝜖↘0𝜖^{(𝛽+1)/𝑠−2}𝐷𝑓^∞₁𝑥^{𝛽−2𝑠∞}_{𝜖𝑥𝑠/𝐷𝑓}𝑦𝑃𝜉≥𝑦𝑑𝑦=lim_𝜖↘02𝑠𝐷𝑓^1/2𝑠∞_{𝜖/𝐷𝑓}𝑡^{1/𝑠−3∞}_𝑡𝜉=2𝑦𝑃≥𝑦𝑑𝑦𝑠𝐷𝑓^{(𝛽+1)/2𝑠∞}₀𝑦𝑃𝜉≥𝑦^𝑦₀𝑡^{(𝛽+1)/𝑠−3}=𝑠𝑑𝑡𝑑𝑦𝐷𝑓𝛽+1𝛽−2𝑠+1^{(𝛽+1)/2𝑠}𝔼𝜉^{(𝛽+1)/𝑠}. (26)

Proposition 8. Under the assumptions of Theorem 3, one has lim_𝜖↘0𝜖^{∞(𝛽+1)/𝑠−2}_𝑛=1𝑛^{𝛽−2𝑠∞}_𝜖𝑛𝑠𝑦𝑃𝑛𝜂_𝑛𝑓−≥𝑦𝑑𝑦_{∞𝜖𝑛𝑠}𝜉≥𝑦𝑦𝑃𝐷𝑓𝑑𝑦=0. (27)

Proof. Similar to the argument in the proof of Proposition 6, we can write ^∞_𝑛=1𝑛^{𝛽−2𝑠∞}_𝜖𝑛𝑠𝑦𝑃𝑛𝜂_𝑛𝑓−≥𝑦𝑑𝑦_{∞𝜖𝑛𝑠}𝜉≥𝑦𝑦𝑃𝐷𝑓=𝑑𝑦_{𝑁(𝜖,𝑀)𝑛=1}𝑛^{𝛽−2𝑠∞}_𝜖𝑛𝑠𝑦𝑃𝑛𝜂_𝑛𝑓≥𝑦𝑑𝑦−^∞_𝜖𝑛𝑠𝜉≥𝑦𝑦𝑃𝐷𝑓𝑑𝑦+_{()𝑛>𝑁𝜖,𝑀}𝑛^{𝛽−2𝑠∞}_𝜖𝑛𝑠𝑦𝑃𝑛𝜂_𝑛𝑓−≥𝑦𝑑𝑦_{∞𝜖𝑛𝑠}𝜉≥𝑦𝑦𝑃𝐷𝑓𝑑𝑦∶=𝐽₁+𝐽₂. (28)
For the term 𝐽₁, we have 𝐽₁≤^{𝑁(𝜖,𝑀)}_𝑛=1𝑛^𝛽−2𝑠×^∞_𝜖𝑛𝑠𝑦𝑃𝑛𝜂_𝑛𝑓𝜉≥𝑦≥𝑦−𝑃𝐷𝑓≤𝑑𝑦_{𝑁(𝜖,𝑀)𝑛=1}𝑛^{𝛽𝑛−𝑠Δ𝑛−1/4}₀()×𝑃𝑥+𝜖𝑛𝜂_𝑛𝑓≥(𝑥+𝜖)𝑛^𝑠𝜉≥()𝑛−𝑃𝑥+𝜖^𝑠𝐷𝑓+𝑑𝑥_{𝑁(𝜖,𝑀)𝑛=1}𝑛^𝛽×^∞_{𝑛−𝑠Δ𝑛−1/4}(𝑥+𝜖)𝑃𝑛𝜂_𝑛𝑓≥(𝑥+𝜖)𝑛^𝑠+𝑑𝑥_{𝑁(𝜖,𝑀)𝑛=1}𝑛^𝛽×^∞_{𝑛−𝑠Δ𝑛−1/4}()𝑃𝜉≥()𝑛𝑥+𝜖𝑥+𝜖^𝑠𝐷𝑓𝑑𝑥∶=^{𝑁(𝜖,𝑀)}_𝑛=1𝑛^𝛽𝐽₁₁+𝐽₁₂+𝐽₁₃. (29) As 𝑁(𝜖,𝑀)=[𝑀𝜖^−1/𝑠], for 𝑛≤𝑁(𝜖,𝑀), we have 𝜖≤𝑀^𝑠𝑛^−𝑠. Thus 𝑛^2𝑠𝐽₁₁≤𝑛^{2𝑠𝑛−𝑠Δ𝑛−1/4}₀(𝑥+𝜖)Δ_𝑛𝑑𝑥≤𝐶𝑛^2𝑠Δ_𝑛𝑛^−𝑠Δ_𝑛^−1/4+𝑀^𝑠𝑛^−𝑠2=𝐶Δ_𝑛^1/4+𝑀^𝑠Δ_𝑛^1/22⟶0as𝑛⟶∞. (30)
Noticing that when 𝜖>0 is fixed, 𝜖𝑛^𝑠>𝑛𝛿_𝑛(𝑓) for sufficient large 𝑛, the relation (21) tells us that there exists 𝑐₁>0, such that lim_𝑛→∞𝑛^2𝑠𝐽₁₂=lim_𝑛→∞𝑛^2𝑠∞_{𝑛−𝑠Δ𝑛−1/4}(𝑥+𝜖)𝑃𝑛𝜂_𝑛𝑓≥(𝑥+𝜖)𝑛^𝑠𝑑𝑥=lim_𝑛→∞^∞_{𝜖𝑛𝑠+Δ𝑛−1/4}𝑦𝑃𝑛𝜂_𝑛𝑓≥𝑦𝑑𝑦≤lim_𝑛→∞𝐶^∞_{𝜖𝑛𝑠+Δ𝑛−1/4}𝑦𝑒^{−𝑐1𝑦2}𝑑𝑦≤lim_𝑛→∞𝐶^∞_Δ𝑛−1/4𝑦𝑒^{−𝑐1𝑦2}𝑑𝑦≤lim_𝑛→∞𝐶𝑒^{−𝑐1Δ𝑛−1/2}=0. (31)
By the same argument as for 𝐽₁₂, using the relation (23), we can easily prove that lim_𝑛→∞𝑛^2𝑠𝐽₁₃=0. (32)
Hence, we have proved that lim_𝑛→∞Ξ_𝑛(𝜖)∶=lim_𝑛→∞𝑛^2𝑠(𝐽₁₁+𝐽₁₂+𝐽₁₃)=0 uniformly for 𝜖>0. By Toeplitz's lemma again, we can further deduce that lim_𝜖↘0𝜖^{(𝛽+1)/𝑠−2}𝐽₁=lim_𝜖↘0𝜖^{(𝛽+1)/𝑠−2𝑁(𝜖,𝑀)}_𝑛=1𝑛^𝛽𝐽₁₁+𝐽₁₂+𝐽₁₃≤lim_𝜖↘0𝐶[𝑁(𝜖,𝑀)]^{2𝑠−𝛽−1𝑁(𝜖,𝑀)}_𝑛=1𝑛^𝛽−2𝑠Ξ_𝑛(𝜖)=0. (33)
For the term 𝐽₂, we can write 𝐽₂≤_{()𝑛>𝑁𝜖,𝑀}𝑛^𝛽−2𝑠×^∞_𝜖𝑛𝑠𝑦𝑃𝑛𝜂_𝑛𝑓𝜉≥𝑦≥𝑦+𝑃𝐷𝑓𝑑𝑦∶=𝐽₂₁+𝐽₂₂. (34)
By using the relation (21) again, for any fixed 𝜖>0, when 𝑀→∞, there exists 𝑐₁>0, such that 𝜖^{(𝛽+1)/𝑠−2}𝐽₂₁≤𝐶𝜖^{(𝛽+1)/𝑠−2}_{()𝑛>𝑁𝜖,𝑀}𝑛^{𝛽−2𝑠∞}_𝑛𝜖²𝑥^2𝑠−1𝑃𝑛𝜂_𝑛𝑓≥𝜖𝑥^𝑠𝑑𝑥≤𝐶𝜖^{(𝛽+1)/𝑠}_{𝑛>𝑁(𝜖,𝑀)}𝑛^{𝛽−2𝑠∞}_𝑛𝑥^2𝑠−1𝑒^{−𝑐1𝜖2𝑥2𝑠}𝑑𝑥≤𝐶𝜖^{(𝛽+1)/𝑠∞}_{𝑁()𝜖,𝑀}𝑦^{𝛽−2𝑠∞}_𝑦𝑥^2𝑠−1𝑒^{−𝑐1𝜖2𝑥2𝑠}𝑑𝑥𝑑𝑦≤𝐶𝜖^{(𝛽+1)/𝑠∞}_{𝑁(𝜖,𝑀)}𝑥^{𝛽−2𝑠+1}𝑥^2𝑠−1𝑒^{−𝑐1𝜖2𝑥2𝑠}𝑑𝑥≤𝐶^∞_𝑀𝑠𝑦^{(𝛽+1)/𝑠−1}𝑒^{−𝑐1𝑦2}𝑑𝑦⟶0. (35) Hence, by letting 𝜖↘0 firstly and then taking 𝑀→∞, we get lim_𝜖↘0𝜖^{(𝛽+1)/𝑠−2}𝐽₂₁=0.
Following the proof of 𝐽₂₁ and using the relation (23), we can easily prove that 𝜖^{(𝛽+1)/𝑠−2}𝐽₂₂ vanishes as 𝜖↘0, which yields that lim_𝜖↘0𝜖^{(𝛽+1)/𝑠−2}𝐽₂=0. (36)
By combining (33) and (36), we conclude that the proof is completed.

Proof of Theorem 3. According to the fact that for any random variable 𝜁 and 𝑎∈ℝ, 𝔼𝜁𝐼_{{𝜁≥𝑎}}=𝑎𝑃(𝜁≥𝑎)+^∞_𝑎𝑃(𝜁≥𝑡)𝑑𝑡, (37) we can see that 𝔼𝜂_𝑛𝑓²𝐼_{{|𝜂𝑛(𝑓)|≥𝜖𝑛𝑠−1}}=𝜖²𝑛^2𝑠−2𝑃𝜂_𝑛𝑓≥𝜖𝑛^𝑠−1+2𝑛^−2∞_𝜖𝑛𝑠𝑦𝑃𝑛𝜂_𝑛𝑓≥𝑦𝑑𝑦. (38)
By Propositions 5–8, we can get Theorem 3 easily.

Proof of Theorem 4. The proof is essentially the same as for Theorem 3, which mainly depends on the central limit theorem of the spectral statistics and the concentration inequality of the empirical spectral measure. We will only list some key tools in the proof, and the details are omitted here.

Lemma 9 (see [13]). Consider the sample covariance matrix 𝑊_𝑚,𝑛=(1/𝑛)𝑌_𝑚𝑛𝑌^∗_𝑚𝑛 as above, where 𝐸|𝑌_𝑖𝑗|⁴=2 and 𝐸|𝑌_𝑖𝑗|⁸<∞ for all 1≤𝑖≤𝑚, 1≤𝑗≤𝑛, 𝑚=[𝑛𝜌](0<𝜌≤1). Let 𝐺_𝑚,𝑛𝑓=𝑚𝑓(𝑥)𝑑𝐹^{𝑊𝑚,𝑛}(𝑥)−𝑓(𝑥)𝛾_𝑚/𝑛(𝑥)𝑑𝑥 (39) and denote Γ to be an open interval including [𝑎(𝜌),𝑏(𝜌)]. Then the empirical processes {𝐺_𝑚,𝑛(𝑓)∶𝑓∈𝐶⁴(Γ)} converge weakly in finite dimension to a Gaussian process {𝐺(𝑓)∶𝑓∈𝐶⁴(Γ)} with mean zero and the covariance function Cov(𝑓,𝑔) given by =1Cov𝑓,𝑔2𝜋²_𝑎𝜌^𝑏(𝜌)𝑓^′𝑥₁𝑔^′𝑥₂×log𝑠𝑥₁−𝑠𝑥₂𝑠𝑥₁−𝑠𝑥₂𝑑𝑥₁𝑑𝑥₂,𝑓,𝑔∈𝐶⁴(Γ), (40) where 𝑠(𝑥) is defined in Theorem 4.

Lemma 10. Assume 𝑓 is a Lipschitz function and 𝜏_𝑚,𝑛(𝑓) is defined by (8). Denote 𝛿_𝑚,𝑛(𝑓)=|𝔼∫𝑓(𝑥)𝑑𝐹^{𝑊𝑚,𝑛}(𝑥)−∫𝑓(𝑥)𝛾_𝑚/𝑛(𝑥)𝑑𝑥|. Then under the assumptions of Theorem 4, for any 𝛿>𝛿_𝑚,𝑛(𝑓), there exists 𝑐₂>0, such that 𝑃𝜏_𝑚,𝑛𝑓≥𝛿≤2𝑒^{−𝑐2𝑚2(𝛿−}𝛿_{𝑚,𝑛(𝑓))}². (41)

Proof. If we denote 𝜏^∘_𝑚,𝑛𝑓=𝑓(𝑥)𝑑𝐹^{𝑊𝑚,𝑛}(𝑥)−𝐸𝑓(𝑥)𝑑𝐹^{𝑊𝑚,𝑛}(𝑥), (42) then by the Corollary 1.8(b) in Guionnet and Zeitouni [2], for any 𝛿>0, there exists 𝑐₂>0 such that 𝑃𝜏^∘_𝑚,𝑛𝑓≥𝛿≤2𝑒^{−𝑐2𝛿2𝑚2}. (43)
As we can write 𝑃𝜏_𝑚,𝑛𝑓𝜏≥𝛿≤𝑃^∘_𝑚,𝑛𝑓+𝔼𝑓(𝑥)𝑑𝐹^{𝑊𝑚,𝑛}(𝑥)−𝑓(𝑥)𝛾_𝑚/𝑛(𝑥)𝜏𝑑𝑥≥𝛿≤𝑃^∘_𝑚,𝑛𝑓≥𝛿−𝛿_𝑚,𝑛𝑓, (44) for each 𝛿>𝛿_𝑚,𝑛(𝑓), we have 𝑃𝜏_𝑚,𝑛𝑓≥𝛿≤2𝑒^{−𝑐2𝑚2(𝛿−}𝛿_{𝑚,𝑛(𝑓))}². (45)

Just like the proof of Proposition 6, in order to use Lemma 10, we will need to estimate the order of 𝛿_𝑚,𝑛(𝑓), and the following remark is needed.

Remark 11. For the estimation of 𝛿_𝑚,𝑛(𝑓), by the result of Bai et al. [14], we can see that sup_𝑥∈ℝ𝔼𝐹₂^(𝑚,𝑛)(𝑥)−𝐹₂^(𝑚,𝑛)(𝑥)≤𝐶𝑚^−1/2, (46) where 𝐹₂^(𝑚,𝑛)(𝑥)=∫^𝑥₀𝑑𝐹^{𝑊𝑚,𝑛}(𝑡) is the empirical spectral distribution of 𝑊_𝑚,𝑛. As a direct consequence, when 𝑓 is differentiable and 𝑓^′ is bounded, there exists a real number 𝐶₂>0, such that 𝛿_𝑚,𝑛(𝑓)≤𝐶₂‖𝑓^′‖𝑚^−1/2.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant nos. 11326173 and 11301146) and the Foundation of Henan Educational Committee in China (Grant no. 13A110087).