Berry-Esséen bound of sample quantiles for negatively associated sequence

Abstract

In this paper, we investigate the Berry-Esséen bound of the sample quantiles for the negatively associated random variables under some weak conditions. The rate of normal approximation is shown as O(n^-1/9).

2010 Mathematics Subject Classification: 62F12; 62E20; 60F05.

1 Introduction

Assume that {X _n }_n≥1is a sequence of random variables defined on a fixed probability space $\left(\Omega ,\mathcal{F},P\right)$ with a common marginal distribution function F(x) = P(X₁ ≤ x). F is a distribution function (continuous from the right, as usual). For 0 < p < 1, the p th quantile of F is defined as

$${\xi }_p=\text{inf}\left\{x:F\left(x\right)\ge p\right\}$$

and is alternately denoted by F^-1(p). The function F^-1(t), 0 < t < 1, is called the inverse function of F. It is easy to check that ξ _p possesses the following properties:

• (i) F(ξ _p -) ≤ p ≤ F(ξ _p );

• (ii) if ξ _p is the unique solution x of F (x-) ≤ p ≤ F(x), then for any ε > 0,

  $$F\left({\xi }_p-\epsilon \right)<p<F\left({\xi }_p+\epsilon \right).$$

For a sample X₁, X₂, ⋯, X _n , n ≥ 1, let F _n represent the empirical distribution function based on X₁, X₂,..., X _n , which is defined as $F_n\left(x\right)=\frac{1}{n}{\sum }_{i=1}^{n}I\left(X_i\le x\right)$, x ∈ ℝ, where I(A) denotes the indicator function of a set A and ℝ is the real line. For 0 < p < 1, we define $F_n^{-1}\left(p\right)=inf\left\{x:F_n\left(x\right)\ge p\right\}$ as the p th quantile of sample.

Recall that a finite family {X₁,..., X _n } is said to be negatively associated (NA) if for any disjoint subsets A, B ⊂ {1, 2,..., n}, and any real coordinatewise nondecreasing functions f on R^A , g on R^B ,

$$Cov\left(f\left(X_k,k\in A\right),g\left(X_k,k\in B\right)\right)\le 0.$$

A sequence of random variables {X _i }_i≥1is said to be NA if for every n ≥ 2, X₁, X₂,..., X _n are NA.

From 1960s, many authors have obtained the asymptotic results for the sample quantiles, including the well-known Bahadur representation. Bahadur [1] firstly introduced an elegant representation for the sample quantiles in terms of empirical distribution function based on independent and identically distributed (i.i.d.) random variables. Sen [2], Babu and Singh [3] and Yoshihara [4] gave the Bahadur representation for the sample quantiles under ϕ-mixing sequence and α-mixing sequence, respectively. Sun [5] established the Bahadur representation for the sample quantiles under α-mixing sequence with polynomially decaying rate. Ling [6] investigated the Bahadur representation for the sample quantiles under NA sequence. Li et al. [7] investigated the Bahadur representation of the sample quantile based on negatively orthant-dependent (NOD) sequence, which is weaker than NA sequence. Xing and Yang [8] also studied the Bahadur representation for the sample quantiles under NA sequence. Wang et al. [9] revised the results of Sun [5] and got a better bound. For more details about Bahadur representation, one can refer to Serfling [10].

For a fixed p ∈ (0, 1), let ξ _p = F^-1(p), ${\xi }_{p,n}=F_n^{-1}\left(p\right)$and Φ(t) be the distribution function of a standard normal variable. In [[10], p. 81], the Berry-Esséen bound of the sample quantiles for i.i.d. random variables is given as follows:

Theorem A Let 0 < p < 1 and {X _n }_n≥1be a sequence of i.i.d. random variables. Suppose that in a neighborhood of ξ _p , F possesses a positive continuous density f and a bounded second derivative F″. Then

$$\underset{-\infty <t<\infty }{sup}\left|P\left(\frac{n^{1∕2}\left({\xi }_{p,n}-{\xi }_p\right)}{{\left[p\left(1-p\right)\right]}^{1∕2}∕f\left({\xi }_p\right)}\le t\right)-\Phi \left(t\right)\right|=O\left(n^{-1∕2}\right),n\to \infty .$$

In this paper, we investigate the Berry-Esséen bound of the sample quantiles for NA random variables under some weak conditions. The rate of normal approximation is shown as O(n^-1/9).

Berry-Esséen theorem, which is known as the rate of convergence in the central limit theorem, can be found in many monographs such as Shiryaev [11], Petrov [12]. For the case of i.i.d. random variables, the optimal rate is $O\left(n^{-\frac{1}{2}}\right)$, and for the case of martingale, the rate is $O\left(n^{-\frac{1}{4}}logn\right)$ [[13], Chapter 3]. For other papers about Berry-Esséen bound, for example, under the association sample, Cai and Roussas [14, 15] studied the Berry-Esséen bounds for the smooth estimator of quantiles and the smooth estimator of a distribution function, respectively; Yang [16] obtained the Berry-Esséen bound of the regression weighted estimator for NA sequence; Wang and Zhang [17] provided the Berry-Esséen bound for linear negative quadrant-dependent (LNQD) sequence; Liang and Baek [18] gave the Berry-Esséen bounds for density estimates under NA sequence; Liang and Uña-Álvarez [19] studied the Berry-Esséen bound in kernel density estimation for α-mixing censored sample; Lahiri and Sun [20] obtained the Berry-Esséen bound of the sample quantiles for α-mixing random variables, etc.

Throughout the paper, C, C₁, C₂, C₃,..., d denote some positive constants not depending on n, which may be different in various places. ⌊x⌋ denotes the largest integer not exceeding x, and the second-order stationarity means that

$$\left(X_1,X_{1+k}\right)\stackrel{d}{=}\left(X_i,X_{i+k}\right),i\ge 1,k\ge 1.$$

Inspired by Serfling [10], Cai and Roussas [14, 15], Yang [16], Liang and Uña-Álvarez [19], Lahiri and Sun [20], etc., we obtain Theorem 1.1 in Section 1. Two preliminary lemmas are given in Section 2, and the proof of Theorem 1.1 is given in Section 3. Next, we give the main result as follows:

Theorem 1.1 Let 0 < p < 1 and {X _n }_n≥1be a second-order stationary NA sequence with common marginal distribution function F and EX _n = 0 for n = 1, 2, . . .. Assume that in a neighborhood of ξ _p , F possesses a positive continuous density f and a bounded second derivative F″. If there exists an ε₀> 0 such that for × ∈ [ξ _p - ε₀, ξ _p + ε₀],

$${\sum }_{j=2}^{\infty }j\left|Cov\left[I\left(X_1\le x\right),I\left(X_j\le x\right)\right]\right|<\infty ,$$

(1.1)

and

$$\text{Var}\left[I\left(X_1\le {\xi }_p\right)\right]+2{\sum }_{j=2}^{\infty }Cov\left[I\left(X_1\le {\xi }_p\right),I\left(X_j\le {\xi }_p\right)\right]:={\sigma }^2\left({\xi }_p\right)>0,$$

(1.2)

then

$$\underset{-\infty <t<\infty }{sup}\left|P\left(\frac{n^{1∕2}\left({\xi }_{p,n}-{\xi }_p\right)}{\sigma \left({\xi }_p\right)∕f\left({\xi }_p\right)}\le t\right)-\Phi \left(t\right)\right|=O\left(n^{-1∕9}\right),n\to \infty .$$

(1.3)

Remark 1.1 Assumption (1.2) is a general condition, see for example Cai and Roussas [14]. For the stationary sequences of associated and negatively associated, Cai and Roussas [15] gave the notation $\mu (n)={\displaystyle {\sum }_{j=n}^{\infty }\mid \text{Cov}(X_1\mathrm{,}{X}_{j+1}{)\mid }^{1/3}}$ and supposed that μ(1) < ∞. In addition, they supposed that μ(n) = O(n^-α) for some α > 0 or δ(1) < ∞, where $\delta \left(i\right)={\sum }_{j=i}^{\infty }\mu \left(j\right)$, then obtained the Berry-Esséen bounds for smooth estimator of a distribution function. Under the assumptions ${\sum }_{j=n+1}^{\infty }{\left\{Cov\left(X_1,X_j\right)\right\}}^{1∕3}=O\left(n^{-\left(r-1\right)}\right)$ for some r > 1 or ${\sum }_{n=1}^{\infty }{n}^{7}Cov\left(X_1,X_n\right)<\infty$, Chaubey et al. [21] studied the smooth estimation of survival and density functions for a stationary-associated process using Poisson weights. In this paper, for x ∈ [ξ _p - ε₀, ξ _p + ε₀], the assumption (1.1) has some restriction on the covariances of Cov[I(X₁ ≤ x), I(X _j ≤ x)] in the neighborhood of ξ _p .

2 Preliminaries

Lemma 2.1 Let {X _n }_n≥1be a stationary NA sequence with EX _n = 0, |X _n | ≤ d < ∞ for n = 1, 2, . . .. There exists some β ≥ 1 such that${\sum }_{j=b_n}^{\infty }\left|Cov\left(X_1,X_j\right)\right|=O\left(b_n^{-\beta }\right)$for all 0 < b _n → ∞ as n → ∞. If

$$\underset{n\to \infty }{lim\; inf}{n}^{-1}\text{Var}\left({\sum }_{i=1}^n{X}_i\right)={\sigma }_0^2>0,$$

then

$$\underset{-\infty <t<\infty }{sup}\left|P\left(\frac{{\sum }_{i=1}^n{X}_i}{\sqrt{\text{Var}\left({\sum }_{i=1}^n{X}_i\right)}}\le t\right)-\Phi \left(t\right)\right|=O\left(n^{-1∕9}\right),n\to \infty .$$

(2.1)

Proof We employ Bernstein's big-block and small-block procedure. Partition the set {1, 2,..., n} into 2k _n + 1 subsets with large blocks of size μ = μ _n and small block of size υ = υ _n . Define

$${\mu }_n=\left[n^{2∕3}\right],{\nu }_n=\left[n^{1∕3}\right],k=k_n:=⌊\frac{n}{{\mu }_n+{\nu }_n}⌋=\left[n^{1∕3}\right],$$

(2.2)

and $Z_{n,i}=X_i∕\sqrt{\text{Var}\left({\sum }_{i=1}^n{X}_i\right)}$. Let η _j , ξ _j , ζ _j be defined as follows:

$${\eta }_j:=\sum _{i=j\left(\mu +\nu \right)+1}^{j\left(\mu +\nu \right)+\mu }{Z}_{n,i},0\le j\le k-1,$$

(2.3)

$${\xi }_j:=\sum _{i=j\left(\mu +\nu \right)+\mu +1}^{\left(j+1\right)\left(\mu +\nu \right)}{Z}_{n,i},0\le j\le k-1,$$

(2.4)

$${\zeta }_k:=\sum _{i=k\left(\mu +\nu \right)+1}^n{Z}_{n,i}.$$

(2.5)

Write

$$S_n:=\frac{{\sum }_{i=1}^n{X}_i}{\sqrt{\text{Var}\left({\sum }_{i=1}^n{X}_i\right)}}={\sum }_{j=0}^{k-1}{\eta }_j+{\sum }_{j=0}^{k-1}{\xi }_j+{\zeta }_k:=S_n^{\prime }+S_n^{″}+S_n^{‴}.$$

(2.6)

By Lemma A.3, we can see that

$$\begin{array}{c}\underset{-\infty <t<\infty }{sup}|P\left(S_n\le t\right)-\Phi \left(t\right)|\\ =\underset{-\infty <t<\infty }{sup}|P\left({S^{\prime }}_n+{S^{″}}_n+{S^{‴}}_n\le t\right)-\Phi \left(t\right)|\le \underset{-\infty <t<\infty }{sup}|P\left({S^{\prime }}_n\le t\right)-\Phi \left(t\right)|\\ +\frac{2n^{-\frac{1}{9}}}{\sqrt{2\pi }}+P\left(\left|{S^{″}}_n\right|>n^{-\frac{1}{9}}\right)+P\left(\left|{S^{‴}}_n\right|>n^{-\frac{1}{9}}\right).\end{array}$$

(2.7)

Firstly, we estimate $E{\left(S_n^{″}\right)}^2$ and $E{\left(S_n^{‴}\right)}^2$, which will be used to estimate $P\left(\left|S_n^{″}\right|>n^{-\frac{1}{9}}\right)$ and $P\left(\left|S_n^{‴}\right|>n^{-\frac{1}{9}}\right)$ in (2.7). By the conditions |X _i | ≤ d and $\underset{n\to \infty }{inf}{n}^{-1}\text{Var}\left({\sum }_{i=1}^n{X}_i\right)={\sigma }_0^2>0$, it is easy to see that $\left|Z_{n,i}\right|\le \frac{C_1}{\sqrt{n}}$. And E(ξ _j )² ≤ Cυ _n /n follows from EZ_n,i= 0 and Lemma A.1. Combining the definition of NA with the definition of ξ _j , j = 0, 1, ⋯, k - 1, we can easily prove that {ξ₀, ξ₁, ⋯, ξ_k-1} is NA. Therefore, it follows from (2.2), (2.4), (2.6) and Lemma A.1 that

$$E{\left(S_n^{″}\right)}^2\le C_1{\sum }_{j=0}^{k-1}E{\xi }_j^2\le C_2\frac{k_n{\nu }_n}{n}\le C_3\frac{n}{{\mu }_n+{\nu }_n}\frac{{\nu }_n}{n}\le C_4\frac{{\nu }_n}{{\mu }_n}=O\left(n^{-1∕3}\right).$$

(2.8)

On the other hand, we can get that

$$\begin{array}{ll}\hfill E{\left({S^{‴}}_n\right)}^2& \le \frac{C_5}{n}E{\left({\sum }_{i=k\left(\mu +\nu \right)+1}^n{X}_i\right)}^2\le \frac{C_6}{n}{\sum }_{i=k\left(\mu +\nu \right)+1}^{n}E{X}_i^2\\ \le \frac{C_7}{n}\left(n-k_n\left({\mu }_n+{\nu }_n\right)\right)\le C_8\frac{{\mu }_n+{\nu }_n}{n}=O\left(n^{-1∕3}\right)\end{array}$$

(2.9)

from (2.5), $\underset{n\to \infty }{lim\; inf}{n}^{-1}\text{Var}\left({\sum }_{i=1}^n{X}_i\right)={\sigma }_0^2>0$, |X _i | ≤ d and Lemma A.1. Consequently, by Markov's inequality, (2.8) and (2.9),

$$P\left(\left|{S^{″}}_n\right|>n^{-\frac{1}{9}}\right)\le n^{\frac{2}{9}}\cdot E{\left(S_n^{″}\right)}^2=O\left(n^{-1∕9}\right),$$

(2.10)

$$P\left(\left|{S^{‴}}_n\right|>n^{-\frac{1}{9}}\right)\le n^{\frac{2}{9}}\cdot E{\left(S_n^{‴}\right)}^2=O\left(n^{-1∕9}\right).$$

(2.11)

In the following, we will estimate $\underset{-\infty <t<\infty }{sup}|P\left(S_n^{\prime }\le t\right)-\Phi \left(t\right)|$. Define

$$s_n^2:={\sum }_{j=0}^{k-1}\text{Var}\left({\eta }_j\right),{\Gamma }_n:={\sum }_{0\le i<j\le k-1}Cov\left({\eta }_i,{\eta }_j\right).$$

Here, we first estimate the growth rate $|s_n^2-1|$. Since $E{S}_n^2=1$ and

$$E{\left(S_n^{\prime }\right)}^2=E{\left[S_n-\left(S_n^{″}+S_n^{‴}\right)\right]}^2=1+E{\left(S_n^{″}+S_n^{‴}\right)}^2-2E\left[S_n\left(S_n^{″}+S_n^{‴}\right)\right],$$

by (2.8) and (2.9), it has

$$\begin{array}{c}\mid E(S_n^{\prime }{)}^2-1\mid =\mid E(S_n^{\prime \prime }+S_n^{\prime \prime \prime }{)}^2-2E[S_n(S_n^{\prime \prime }+S_n^{\prime \prime \prime })]\mid \\ \le E(S_n^{\prime \prime }{)}^2+E(S_n^{\prime \prime \prime }{)}^2+2[E(S_n^{\prime \prime }{)}^2{]}^{1/2}[E(S_n^{\prime \prime \prime }{)}^2{]}^{1/2}\\ +2[E(S_n^2){]}^{1/2}[E(S_n^{\prime \prime }{)}^2{]}^{1/2}+2[E(S_n^2){]}^{1/2}[E(S_n^{\prime \prime \prime }{)}^2{]}^{1/2}\\ =O(n^{-1/3})+O(n^{-1/6})=O(n^{-1/6}).\end{array}$$

(2.12)

Notice that

$$s_n^2=E{\left(S_n^{\prime }\right)}^2-2{\Gamma }_n.$$

(2.13)

With λ _j = j(μ _n + υ _n ),

$$2{\Gamma }_n=2\sum _{0\le i<j\le k-1}\sum _{l_1=1}^{{\mu }_n}\sum _{l_2=1}^{{\mu }_n}Cov\left(Z_{n,{\lambda }_i+l_1},Z_{n,{\lambda }_j+l_2}\right),$$

but since i ≠ j, |λ _i - λ _j + l₁ - l₂| ≥ υ _n , it has that

$$\begin{array}{ll}\hfill \left|2{\Gamma }_n\right|& \le 2\sum _{\begin{array}{c}1\le i<j\le n\\ j-i\ge {\nu }_n\end{array}}\left|Cov\left(Z_{n,i},Z_{n,j}\right)\right|\le \frac{C_1}{n}\sum _{\begin{array}{c}1\le i<j\le n\\ j-i\ge {\nu }_n\end{array}}\left|Cov\left(X_i,X_j\right)\right|\\ \le C_2{\sum }_{k\ge {\nu }_n}\left|Cov\left(X_1,X_k\right)\right|=O\left(n^{-\beta ∕3}\right)=O\left(n^{-1∕3}\right)\end{array}$$

(2.14)

following from (2.2) and the conditions of stationary, $\underset{n\to \infty }{lim\; inf}{n}^{-1}\text{Var}\left({\sum }_{i=1}^n{X}_i\right)={\sigma }_0^2>0$ and ${\sum }_{j=b_n}^{\infty }\left|Cov\left(X_1,X_j\right)\right|=O\left(b_n^{-\beta }\right)$, β ≥ 1. So, by (2.12), (2.13) and (2.14), we can get that

$$|s_n^2-1|=O\left(n^{-1∕6}\right)+O\left(n^{-1∕3}\right)=O\left(n^{-1∕6}\right).$$

(2.15)

For j = 0, 1,..., k - 1, let ${\eta }_j^{\prime }$ be the independent random variables and $|s_n^2-1|=O\left(n^{-1∕6}\right)+O\left(n^{-1∕3}\right)=O\left(n^{-1∕6}\right).$ have the same distribution as η _j , j = 0, 1,..., k - 1. Define $H_n={\sum }_{j=0}^{k-1}{{\eta }^{\prime }}_j$. It can be found that

$$\begin{array}{c}\underset{-\infty <t<\infty }{sup}|P\left({S^{\prime }}_n\le t\right)-\Phi \left(t\right)|\\ \le \underset{-\infty <t<\infty }{sup}|P\left({S^{\prime }}_n\le t\right)-P\left(H_n\le t\right)|+\underset{-\infty <t<\infty }{sup}|P\left(H_n\le t\right)-\Phi \left(t∕s_n\right)|\\ +\underset{-\infty <t<\infty }{sup}|\Phi \left(t∕s_n\right)-\Phi \left(t\right)|:=D_1+D_2+D_3.\end{array}$$

(2.16)

Let ϕ(t) and ψ(t) be the characteristic functions of $S_n^{\prime }$ and H _n , respectively. By Esséen inequality [[12], Theorem 5.3], for any T > 0,

$$\begin{array}{ll}\hfill D_1& \le {\int }_{-T}^T\left|\frac{\varphi \left(t\right)-\psi \left(t\right)}{t}\right|dt+T\underset{-\infty <t<\infty }{sup}{\int }_{\left|u\right|\le \frac{C}{T}}|P\left(H_n\le u+t\right)-P\left(H_n\le t\right)|du\\ :=D_{1n}+D_{2n}.\end{array}$$

(2.17)

With λ _j = j(μ _n + υ _n ) and similar to the proof of Lemma 3.4 of Yang [16], we have that

$$\begin{array}{ll}\hfill |\phi \left(t\right)-\psi \left(t\right)|& =\left|Eexp\left(it\sum _{j=0}^{k-1}{\eta }_j\right)-\prod _{j=0}^{k-1}Eexp\left(it{\eta }_j\right)\right|\\ \le 4t^2\sum _{0\le i<j\le k-1}\sum _{l_1=1}^{{\mu }_n}\sum _{l_2=1}^{{\mu }_n}\left|Cov\left(Z_{n,{\lambda }_i+l_1},Z_{n,{\lambda }_j+l_2}\right)\right|\\ \le \frac{C_1{t}^2}{n}\sum _{\begin{array}{c}1\le i<j\le n\\ j-i\ge {\nu }_n\end{array}}\left|Cov\left(X_i,X_j\right)\right|\\ \le C_2{t}^2\sum _{j\ge {\nu }_n}\left|Cov\left(X_1,X_j\right)\right|\le C_3{t}^2{n}^{-\beta ∕3}\end{array}$$

(2.18)

by (2.2) and the conditions of stationary, $\underset{n\to \infty }{lim\; inf}{n}^{-1}\text{Var}\left({\sum }_{i=1}^n{X}_i\right)={\sigma }_0^2>0$ and ${\sum }_{j=b_n}^{\infty }\left|Cov\left(X_1,X_j\right)\right|=O\left(b_n^{-\beta }\right)$. Set T = n^{(3β - 1)/18}for β ≥ 1, we have by (2.18) that

$$D_{1n}={\int }_{-T}^T\left|\frac{\phi \left(t\right)-\psi \left(t\right)}{t}\right|dt\le C{n}^{-\beta ∕3}\cdot T^2=O\left(n^{-1∕9}\right).$$

(2.19)

It follows from the Berry-Esséen inequality [[12], Theorem 5.7], that

$$\underset{-\infty <t<\infty }{sup}|P\left(H_n∕s_n\le t\right)-\Phi \left(t\right)|\le \frac{C}{s_n^3}{\sum }_{j=0}^{k-1}E|{{\eta }^{\prime }}_j{|}^3=\frac{C}{s_n^3}{\sum }_{j=0}^{k-1}E|{\eta }_j{|}^3.$$

(2.20)

By (2.3) and Lemma A.1,

$$\begin{array}{l}{\displaystyle {\sum }_{j=0}^{k-1}E\mid {\eta }_j{\mid }^3}={\displaystyle {\sum }_{j=0}^{k-1}E{\left|{\displaystyle {\sum }_{i=j(\mu +\nu )+1}^{j(\mu +\nu )+\mu }{Z}_{n\mathrm{,}i}}\right|}^3}\\ \le \frac{C_1}{n^{3/2}}{\displaystyle {\sum }_{j=0}^{k-1}E}{\left|{\displaystyle {\sum }_{i=j(\mu +\nu )+1}^{j(\mu +\nu )+\mu }{X}_i}\right|}^3\\ \le \frac{C_2}{n^{3/2}}{\displaystyle {\sum }_{j=0}^{k-1}\left\{{\displaystyle {\sum }_{i=j(\mu +\nu )+1}^{j(\mu +\nu )+\mu }E\mid X_i{\mid }^3}+({\displaystyle {\sum }_{i=j(\mu +\nu )+1}^{j(\mu +\nu )+\mu }E\mid X_i{\mid }^2{)}^{3/2}}\right\}}\\ \le \frac{C_3}{n^{3/2}}{\displaystyle {\sum }_{j=0}^{k-1}(\mu +{\mu }^{3/2})}\le \frac{C_{4}k{\mu }^{3/2}}{n^{3/2}}=O(n^{-1/6}).\end{array}$$

(2.21)

Combining (2.20) with (2.21), we obtain that

$$\underset{-\infty <t<\infty }{sup}|P\left(\frac{H_n}{s_n}\le t\right)-\Phi \left(t\right)|=O\left(n^{-1∕6}\right),$$

(2.22)

since s _n → 1 as n → ∞ by (2.15). It follows from (2.22) that

$$\begin{array}{c}\underset{-\infty <t<\infty }{sup}|P\left(H_n\le u+t\right)-P\left(H_n\le t\right)|\\ \le \underset{-\infty <t<\infty }{sup}\left|P\left(\frac{H_n}{s_n}\le \frac{u+t}{s_n}\right)-\Phi \left(\frac{u+t}{s_n}\right)\right|\\ +\underset{-\infty <t<\infty }{sup}\left|P\left(\frac{H_n}{s_n}\le \frac{t}{s_n}\right)-\Phi \left(\frac{t}{s_n}\right)\right|+\underset{-\infty <t<\infty }{sup}\left|\Phi \left(\frac{u+t}{s_n}\right)-\Phi \left(\frac{t}{s_n}\right)\right|\\ \le 2\underset{-\infty <t<\infty }{sup}\left|P\left(\frac{H_n}{s_n}\le t\right)-\Phi \left(t\right)|+\underset{-\infty <t<\infty }{sup}\right|\Phi \left(\frac{u+t}{s_n}\right)-\Phi \left(\frac{t}{s_n}\right)|\\ =O\left(n^{-1∕6}\right)+O\left(\frac{\left|u\right|}{s_n}\right),\end{array}$$

which implies that

$$\begin{array}{ll}\hfill D_{2n}& =T\underset{-\infty <t<\infty }{sup}{\int }_{\left|u\right|\le C∕T}|P\left(H_n\le u+t\right)-P\left(H_n\le t\right)|du\\ \le \frac{C_1}{n^{1∕6}}+\frac{C_2}{T}=O\left(n^{-1∕6}\right)+O\left(n^{-1∕9}\right)=O\left(n^{-1∕9}\right),\end{array}$$

(2.23)

where T = n^{(3β - 1)/18}. It is known that [[12], Lemma 5.2],

$$\underset{-\infty <x<\infty }{sup}|\Phi \left(px\right)-\Phi \left(x\right)|\le \frac{\left(p-1\right)I\left(p\ge 1\right)}{{\left(2\pi e\right)}^{1∕2}}+\frac{\left(p^{-1}-1\right)I\left(0<p<1\right)}{{\left(2\pi e\right)}^{1∕2}}.$$

Thus, by (2.15),

$$\begin{array}{ll}\hfill D_3& =\underset{-\infty <t<\infty }{sup}|\Phi \left(t∕s_n\right)-\Phi \left(t\right)|\\ \le {\left(2\pi e\right)}^{-1∕2}\left(s_n-1\right)I\left(s_n\ge 1\right)+{\left(2\pi e\right)}^{-1∕2}\left(s_n^{-1}-1\right)I\left(0<s_n<1\right)\\ \le {\left(2\pi e\right)}^{-1∕2}max\left(|s_n-1|,|s_n-1|∕s_n\right)\\ \le C_{1}max\left(|s_n-1|,|s_n-1|∕s_n\right)\cdot \left(s_n+1\right)\left(\text{note that}{s}_n\to 1\right)\\ \le C_2|s_n^2-1|=O\left(n^{-1∕6}\right),\end{array}$$

(2.24)

and by (2.22),

$$D_2=\underset{-\infty <t<\infty }{sup}\left|P\left(\frac{H_n}{s_n}\le \frac{t}{s_n}\right)-\Phi \left(\frac{t}{s_n}\right)\right|=O\left(n^{-1∕6}\right).$$

(2.25)

Therefore, it follows from (2.16), (2.17), (2.19), (2.23), (2.24) and (2.25) that

$$\underset{-\infty <t<\infty }{sup}|P\left(S_n^{\prime }\le t\right)-\Phi \left(t\right)|=O\left(n^{-1∕9}\right)+O\left(n^{-1∕6}\right)=O\left(n^{-1∕9}\right).$$

(2.26)

Finally, by (2.7), (2.10), (2.11) and (2.26), (2.1) holds true. □

Lemma 2.2 Let {X _n }_n≥1be a second-order stationary NA sequence with common marginal distribution function and EX _n = 0, |X _n | ≤ d< ∞, n = 1,2,.... We give an assumption such that${\sum }_{j=2}^{\infty }j\left|Cov\left(X_1,X_j\right)\right|<\infty$. If$\text{Var}\left(X_1\right)+2{\sum }_{j=2}^{\infty }Cov\left(X_1,X_j\right)={\sigma }_1^2>0$, then

$$\underset{-\infty <t<\infty }{sup}\left|P\left(\frac{{\sum }_{i=1}^n{X}_i}{\sqrt{n}{\sigma }_1}\le t\right)-\Phi \left(t\right)\right|=O\left(n^{-1∕9}\right),n\to \infty .$$

(2.27)

Proof Define ${\sigma }_n^2=\text{Var}\left({\sum }_{i=1}^n{X}_i\right)$, ${\sigma }^2\left(n,{\sigma }_1^2\right)=n{\sigma }_1^2$ and γ(k) = Cov (X_i+k, X _i ) for k = 0, 1, 2,.... For the second-order stationarity process {X _n }_n≥1with common marginal distribution function, it can be found by the condition ${\sum }_{j=1}^{\infty }j\left|\gamma \left(j\right)\right|<\infty$ that

$$\begin{array}{ll}\hfill |{\sigma }_n^2-{\sigma }^2\left(n,{\sigma }_1^2\right)|& =\left|n\gamma \left(0\right)+2n{\sum }_{j=1}^{n-1}\left(1-\frac{j}{n}\right)\gamma \left(j\right)-n\gamma \left(0\right)-2n{\sum }_{j=1}^{\infty }\gamma \left(j\right)\right|\\ =\left|2n{\sum }_{j=1}^{n-1}\frac{j}{n}\gamma \left(j\right)-2n{\sum }_{j=n}^{\infty }\gamma \left(j\right)\right|\\ \le 2{\sum }_{j=1}^{\infty }j\left|\gamma \left(j\right)\right|+2n{\sum }_{j=n}^{\infty }\left|\gamma \left(j\right)\right|\\ \le 4{\sum }_{j=1}^{\infty }j\left|\gamma \left(j\right)\right|=O\left(1\right).\end{array}$$

(2.28)

On the other hand,

$$\begin{array}{c}\underset{-\infty <t<\infty }{sup}\left|P\left(\frac{{\sum }_{i=1}^n{X}_i}{\sigma \left(n,{\sigma }_1^2\right)}\le t\right)-\Phi \left(t\right)\right|\\ \le \underset{-\infty <t<\infty }{sup}\left|P\left(\frac{{\sum }_{i=1}^n{X}_i}{{\sigma }_n}\le \frac{\sigma \left(n,{\sigma }_1^2\right)}{{\sigma }_n}t\right)-\Phi \left(\frac{\sigma \left(n,{\sigma }_1^2\right)}{{\sigma }_n}t\right)\right|\\ +\underset{-\infty <t<\infty }{sup}\left|\Phi \left(\frac{\sigma \left(n,{\sigma }_1^2\right)}{{\sigma }_n}t\right)-\Phi \left(t\right)\right|\\ :=D_1+D_2.\end{array}$$

(2.29)

Obviously, if b _n → ∞ as n → ∞, then it follows from ${\sum }_{j=2}^{\infty }j\left|Cov\left(X_1,X_j\right)\right|<\infty$ that

$${\sum }_{j=b_n}^{\infty }\left|Cov\left(X_1,X_j\right)\right|\le \frac{1}{b_n}{\sum }_{j=b_n}^{\infty }j\left|Cov\left(X_1,X_j\right)\right|=o\left(b_n^{-1}\right).$$

(2.28) and the fact ${\sigma }^2\left(n,{\sigma }_1^2\right)=n{\sigma }_1^2\to \infty$ yield that $\underset{n\to \infty }{lim}{\sigma }_n^2∕{\sigma }^2\left(n,{\sigma }_1^2\right)=1$. Thus, by Lemma 2.1,

$$D_1=O\left(n^{-1∕9}\right).$$

(2.30)

By (2.28) again and similar to the proof of (2.24), it follows

$$D_2\le C\left|\frac{{\sigma }_n^2}{{\sigma }^2\left(n,{\sigma }_1^2\right)}-1\right|=\frac{C}{{\sigma }^2\left(n,{\sigma }_1^2\right)}\left|{\sigma }_n^2-{\sigma }^2\left(n,{\sigma }_1^2\right)\right|=O\left(n^{-1}\right).$$

(2.31)

Finally, by (2.29), (2.30) and (2.31), (2.27) holds true. □

Remark 2.1 Under the conditions of Lemma 2.2, we have (27). Furthermore, by the proof of Lemma 2.2, we can obtain that

$$\underset{-\infty <t<\infty }{sup}\left|P\left(\frac{{\sum }_{i=1}^n{X}_i}{\sqrt{n}{\sigma }_1}\le t\right)-\Phi \left(t\right)\right|\le C\left({\sigma }_1^2\right)n^{-1∕9},n\to \infty ,$$

(2.32)

where $C\left({\sigma }_1^2\right)$ is a positive constant depending only on ${\sigma }_1^2$.

3 Proof of the main result

Proof of Theorem 1.1 The proof is inspired by the proofs of Theorem A and Theorem C of Serfling [[10], pp. 77-84]. Denote A = σ (ξ _p ) / f (ξ _p ) and

$$G_n\left(t\right)=P\left(n^{1∕2}\left({\xi }_{p,n}-{\xi }_p\right)∕A\le t\right).$$

Let L _n = (log n log log n)^1/2, we have

$$\begin{array}{ll}\hfill \underset{\left|t\right|>L_n}{sup}|G_n\left(t\right)-\Phi \left(t\right)|& =max\left\{\underset{t<-L_n}{sup}|G_n\left(t\right)-\Phi \left(t\right)|,\underset{t>L_n}{sup}|G_n\left(t\right)-\Phi \left(t\right)|\right\}\\ \le max\left\{G_n\left(-L_n\right)+\Phi \left(-L_n\right),1-G_n\left(L_n\right)+1-\Phi \left(L_n\right)\right\}\\ \le G_n\left(-L_n\right)+1-G_n\left(L_n\right)+1-\Phi \left(L_n\right)\\ \le P\left(|{\xi }_{p,n}-{\xi }_p|\ge A{L}_n{n}^{-1∕2}\right)+1-\Phi \left(L_n\right).\end{array}$$

(3.1)

Since $1-\Phi \left(x\right)\le \frac{{\left(2\pi \right)}^{-1∕2}}{x}{e}^{-x^2∕2}$, x > 0 it follows

$$1-\Phi \left(L_n\right)\le \frac{{\left(2\pi \right)}^{-1∕2}}{L_n}{e}^{-lognloglogn∕2}=O\left(n^{-1}\right).$$

(3.2)

Let ε _n = (A - ε₀) (log n log log n)^1/2n^-1/2, where 0 < ε₀ < A. Seeing that

$$P\left(|{\xi }_{p,n}-{\xi }_p|\ge A{\left(lognloglogn\right)}^{1∕2}{n}^{-1∕2}\right)\le P\left(|{\xi }_{p,n}-{\xi }_p|>{\epsilon }_n\right)$$

and

$$P\left(|{\xi }_{p,n}-{\xi }_p|>{\epsilon }_n\right)=P\left({\xi }_{p,n}>{\xi }_p+{\epsilon }_n\right)+P\left({\xi }_{p,n}<{\xi }_p-{\epsilon }_n\right),$$

by Lemma A.4 (iii), we obtain

$$\begin{array}{ll}\hfill P\left({\xi }_{p,n}>{\xi }_p+{\epsilon }_n\right)& =P\left(p>F_n\left({\xi }_p+{\epsilon }_n\right)\right)=P\left(1-F_n\left({\xi }_p+{\epsilon }_n\right)>1-p\right)\\ =P\left({\sum }_{i=1}^{n}I\left(X_i>{\xi }_p+{\epsilon }_n\right)>n\left(1-p\right)\right)\\ =P\left({\sum }_{i=1}^n\left(V_i-E{V}_i\right)>n{\delta }_{n1}\right),\end{array}$$

where V _i = I (X _i > ξ _p + ξ _n ) and δ_{n 1}= F(ξ _p + ε _n ) - p. Likewise,

$$P\left({\xi }_{p,n}<{\xi }_p-{\epsilon }_n\right)\le P\left(p\le F_n\left({\xi }_p-{\epsilon }_n\right)\right)=P\left({\sum }_{i=1}^n\left(W_i-E{W}_i\right)\ge n{\delta }_{n2}\right),$$

where W _i = I (X _i > ξ _p - ξ_n) and δ_{n 2}= p - F(ξ _p - ε _n ). It is easy to see that {V _i - EV _i }_1≤i≤n. and {W _i - EV _i }_1≤i≤nare still NA sequences. Obviously, |V _i - EV _i | ≤ 1, ${\sum }_{i=1}^{n}E{\left(V_i-E{V}_i\right)}^2\le n$, |W _i - EW _i | ≤ 1, ${\sum }_{i=1}^{n}E{\left(W_i-E{W}_i\right)}^2\le n$. By Lemma A.2, we have that

$$\begin{array}{c}P\left({\xi }_{p,n}>{\xi }_p+{\epsilon }_n\right)\le 2exp\left\{-\frac{n{\delta }_{n1}^2}{2\left(2+{\delta }_{n1}\right)}\right\},\\ P\left({\xi }_{p,n}<{\xi }_p-{\epsilon }_n\right)\le 2exp\left\{-\frac{n{\delta }_{n2}^2}{2\left(2+{\delta }_{n2}\right)}\right\}.\end{array}$$

Consequently,

$$P\left(|{\xi }_{p,n}-{\xi }_p|>{\epsilon }_n\right)\le 4exp\left\{-\frac{n{\left[min\left({\delta }_{n1},{\delta }_{n2}\right)\right]}^2}{2\left(2+max\left({\delta }_{n1},{\delta }_{n2}\right)\right)}\right\}.$$

(3.3)

Since F (x) is continuous at ξ _p with F' (ξ _p ) > 0, ξ _p is the unique solution of F (x-) ≤ p ≤ F (x) and F (ξ _p ) = p. By the assumption on f'(x) and Taylor's expansion,

$$\begin{array}{c}F\left({\xi }_p+{\epsilon }_n\right)-p=F\left({\xi }_p+{\epsilon }_n\right)-F\left({\xi }_p\right)=f\left({\xi }_p\right){\epsilon }_n+o\left({\epsilon }_n\right),\\ p-F\left({\xi }_p-{\epsilon }_n\right)=F\left({\xi }_p\right)-F\left({\xi }_p-{\epsilon }_n\right)=f\left({\xi }_p\right){\epsilon }_n+o\left({\epsilon }_n\right).\end{array}$$

Therefore, we can get that for n large enough,

$$\begin{array}{l}\frac{f({\xi }_p){\epsilon }_n}{2}=\frac{f({\xi }_p)(A-{\epsilon }_0)(lognloglogn{)}^{1/2}}{2n^{1/2}}\le F({\xi }_p+{\epsilon }_n)-p\mathrm{,}\\ \frac{f({\xi }_p){\epsilon }_n}{2}=\frac{f({\xi }_p)(A-{\epsilon }_0)(lognloglogn{)}^{1/2}}{2n^{1/2}}\le p-F({\xi }_p-{\epsilon }_n).\end{array}$$

Note that max(δ_{n 1}, δ_{n 2}) → 0. as n → ∞. So with (3), for n large enough,

$$P(\mid {\xi }_{p\mathrm{,}n}-{\xi }_p\mid >{\epsilon }_n)\le 4exp\left\{-\frac{f^2({\xi }_p)(A-{\epsilon }_0{)}^{2}lognloglogn}{8(2+max({\delta }_{n1}\mathrm{,}{\delta }_{n2}))}\right\}=O(n^{-1}).$$

(3.4)

Next, we define

$${\sigma }^2\left(n,t\right)=\text{Var}\left(Z_1\right)+2{\sum }_{j=2}^{\infty }Cov\left(Z_1,Z_j\right),$$

where Z _i = I [X _i ≤ ξ _p + tAn^-1/2] - EI [X _i ≤ ξ _p + tAn^-1/2]. Seeing that

$${\sigma }^2\left({\xi }_p\right)=\text{Var}\left[I\left(X_1\le {\xi }_p\right)\right]+2{\sum }_{j=2}^{\infty }Cov\left[I\left(X_1\le {\xi }_p\right),I\left(X_j\le {\xi }_p\right)\right],$$

we will estimate the convergence rate of |σ² (n, t) - σ² (ξ _p )|. By the condition (1.1), we can see that σ² (ξ _p ) < ∞. Since that F possesses a positive continuous density f and a bounded second derivative F', for |t| ≤ L _n = (log n log log n)^1/2, we will obtain by Taylor's expansion that

$$\begin{array}{l}\mid \text{Var}(Z_1)-\text{Var}[I(X_1\le {\xi }_p)]\mid \\ =\mid \text{Var}[I(X_1\le {\xi }_p+tA{n}^{-1/2})]-\text{Var}[I(X_1\le {\xi }_p)]\mid \\ =\mid F({\xi }_p+tA{n}^{-1/2})-F({\xi }_p)+[F^2({\xi }_p)-F^2({\xi }_p+tA{n}^{-1/2})]\mid \\ \le f({\xi }_p)\cdot \mid t\mid A{n}^{-1/2}+o(\mid t\mid A{n}^{-1/2})\\ +\mid F({\xi }_p)+F({\xi }_p+tA{n}^{-1/2})\mid \cdot [f({\xi }_p)\cdot \mid t\mid A{n}^{-1/2}+o(\mid t\mid A{n}^{-1/2})]\\ =O((lognloglogn{)}^{1/2}{n}^{-1/2}).\end{array}$$

(3.5)

Similarly, for j ≥ 2 and |t| ≤ L _n ,

$$\begin{array}{l}\mid E[I(X_1\le {\xi }_p+tA{n}^{-1/2})I(X_j\le {\xi }_p+tA{n}^{-1/2})]-E[I(X_1\le {\xi }_p+tA{n}^{-1/2})I(X_j\le {\xi }_p)]\mid \\ \le E\mid I(X_j\le {\xi }_p+tA{n}^{-1/2})-I(X_j\le {\xi }_p)\mid \\ =[F({\xi }_p+tA{n}^{-1/2})-F({\xi }_p)]I(t\ge 0)+[F({\xi }_p)-F({\xi }_p+tA{n}^{-1/2})]I(t<0)\\ =O((lognloglogn{)}^{1/2}{n}^{-1/2}),\end{array}$$

Therefore, by a similar argument, for j ≥ 2 and |t| ≤ L _n ,

$$\begin{array}{l}\mid \text{Cov}(Z_1\mathrm{,}{Z}_j)-\text{Cov}[I(X_1\le {\xi }_p),I(X_j\le {\xi }_p)]\mid \\ \le \mid E[I(X_1\le {\xi }_p+tA{n}^{-1/2})I(X_j\le {\xi }_p+tA{n}^{-1/2})]-E[I(X_1\le {\xi }_p)I(X_j\le {\xi }_p)]\mid \\ +\mid E[I(X_1\le {\xi }_p+tA{n}^{-1/2})]E[I(X_j\le {\xi }_p+tA{n}^{-1/2})]-E[I(X_1\le {\xi }_p)]E[I(X_j\le {\xi }_p)]\mid \\ \le \mid E[I(X_1\le {\xi }_p+tA{n}^{-1/2})I(X_j\le {\xi }_p+tA{n}^{-1/2})]\\ -E[I(X_1\le {\xi }_p+tA{n}^{-1/2})I(X_j\le {\xi }_p)]\mid \\ +\mid E[I(X_1\le {\xi }_p+tA{n}^{-1/2})I(X_j\le {\xi }_p)]-E[I(X_1\le {\xi }_p)I(X_j\le {\xi }_p)]\mid \\ +\mid E[I(X_1\le {\xi }_p+tA{n}^{-1/2})]E[I(X_j\le {\xi }_p+tA{n}^{-1/2})]\\ -E[I(X_1\le {\xi }_p+tA{n}^{-1/2})]E[I(X_j\le {\xi }_p)]\mid \\ +\mid E[I(X_1\le {\xi }_p+tA{n}^{-1/2})]E[I(X_j\le {\xi }_p)]-E[I(X_1\le {\xi }_p)]E[I(X_j\le {\xi }_p)]\mid \\ =O((lognloglogn{)}^{1/2}{n}^{-1/2}).\end{array}$$

(3.6)

Consequently, by the conditions (1.1) and (3.5), (3.6), for |t| ≤ L _n ,

$$\begin{array}{c}|{\sigma }^2\left(n,t\right)-{\sigma }^2\left({\xi }_p\right)|\\ \le |\text{Var}\left(Z_1\right)-\text{Var}\left[I\left(X_1\le {\xi }_p\right)\right]|\\ +2{\sum }_{j=2}^{\left[n^{1∕5}\right]}|Cov\left(Z_1,Z_j\right)-Cov\left[I\left(X_1\le {\xi }_p\right),I\left(X_j\le {\xi }_p\right)\right]|\\ +2{\sum }_{j=\left[n^{1∕5}\right]+1}^{\infty }\left|Cov\left(Z_1,Z_j\right)\right|+2{\sum }_{\left[j=n^{1∕5}\right]+1}^{\infty }\left|Cov\left[I\left(X_1\le {\xi }_p\right),I\left(X_j\le {\xi }_p\right)\right]\right|\\ \le C_1{\left(lognloglogn\right)}^{1∕2}{n}^{-1∕2}+C_2{n}^{1∕5}{\left(lognloglogn\right)}^{1∕2}{n}^{-1∕2}+o\left(n^{-1∕5}\right)\\ =o\left(n^{-1∕5}\right).\end{array}$$

(3.7)

By Lemma A.4 (iii) again, it has

$$\begin{array}{ll}\hfill G_n\left(t\right)& =P\left({\xi }_{p,n}\le {\xi }_p+tA{n}^{-1∕2}\right)=P\left[p\le F_n\left({\xi }_p+tA{n}^{-1∕2}\right)\right]\\ =P\left[np\le {\sum }_{i=1}^{n}I\left(X_i\le {\xi }_p+tA{n}^{-1∕2}\right)\right]\\ =P\left[\frac{n^{1∕2}\left(p-F\left({\xi }_p+tA{n}^{-1∕2}\right)\right)}{\sigma \left(n,t\right)}\le \frac{{\sum }_{i=1}^n{Z}_i}{\sqrt{n}\sigma \left(n,t\right)}\right].\end{array}$$

Thus,

$$G_n\left(t\right)=P\left[\frac{{\sum }_{i=1}^n{Z}_i}{\sqrt{n}\sigma \left(n,t\right)}\ge -c_{nt}\right]=1-P\left[\frac{{\sum }_{i=1}^n{Z}_i}{\sqrt{n}\sigma \left(n,t\right)}<-c_{nt}\right],$$

where

$$c_{nt}=\frac{n^{1∕2}\left(F\left({\xi }_p+tA{n}^{-1∕2}\right)-p\right)}{\sigma \left(n,t\right)}.$$

It is easy to check that

$$\begin{array}{ll}\hfill \Phi \left(t\right)-G_n\left(t\right)& =\Phi \left(t\right)-1+P\left[\frac{{\sum }_{i=1}^n{Z}_i}{\sqrt{n}\sigma \left(n,t\right)}<-c_{nt}\right]\\ =P\left[\frac{{\sum }_{i=1}^n{Z}_i}{\sqrt{n}\sigma \left(n,t\right)}<-c_{nt}\right]-\left[1-\Phi \left(t\right)\right]\\ =P\left[\frac{{\sum }_{i=1}^n{Z}_i}{\sqrt{n}\sigma \left(n,t\right)}<-c_{nt}\right]-\Phi \left(-c_{nt}\right)+\Phi \left(t\right)-\Phi \left(c_{nt}\right).\end{array}$$

(3.8)

By (3.7), it has that $lim\frac{{\sigma }^2\left(n,t\right)}{{\sigma }^2\left({\xi }_p\right)}\to 1$ as n → ∞, which implies that 0 < σ² (n, t) for |t| ≤ L _n and n large enough. Obviously, {Z _i } is a second-order stationary NA sequence. Thus, for a fixed t, |t| ≤ L _n , by the Lemma 2.2, (2.32) in Remark 2.1 and (3.7), it has for n large enough that

$$\begin{array}{c}\left|P\left[\frac{{\sum }_{i=1}^n{Z}_i}{\sqrt{n}\sigma \left(n,t\right)}<-c_{nt}\right]-\Phi \left(-c_{nt}\right)\right|\\ \le \underset{-\infty <x<\infty }{sup}\left|P\left[\frac{{\sum }_{i=1}^n{Z}_i}{\sqrt{n}\sigma \left(n,t\right)}<x\right]-\Phi \left(x\right)\right|\\ \le C\left({\sigma }^2\left(n,t\right)\right)n^{-1∕9}=C\left({\sigma }^2\left({\xi }_p\right)+o\left(n^{-1∕5}\right)\right)n^{-1∕9}\\ \le C_1{n}^{-1∕9},\end{array}$$

where C₁ does not depend on t for |t| ≤ L _n . Therefore, for n large enough, we have

$$\underset{\left|t\right|\le L_n}{sup}\left|P\left[\frac{{\sum }_{i=1}^n{Z}_i}{\sqrt{n}\sigma \left(n,t\right)}<-c_{nt}\right]-\Phi \left(-c_{nt}\right)\right|\le C_1{n}^{-1∕9}.$$

By (3.8) and the inequality above, we can get that for n large enough,

$$\begin{array}{c}\underset{\left|t\right|\le L_n}{sup}|G_n\left(t\right)-\Phi \left(t\right)|\\ \le \underset{\left|t\right|\le L_n}{sup}\left|P\left[\frac{{\sum }_{i=1}^n{Z}_i}{\sqrt{n}\sigma \left(n,t\right)}<-c_{nt}\right]-\Phi \left(-c_{nt}\right)\right|+\underset{\left|t\right|\le L_n}{sup}|\Phi \left(t\right)-\Phi \left(c_{nt}\right)|\\ \le C{n}^{-1∕9}+\underset{\left|t\right|\le L_n}{sup}|\Phi \left(t\right)-\Phi \left(c_{nt}\right)|.\end{array}$$

(3.9)

On the other hand,

$$\begin{array}{ll}\hfill \underset{\left|t\right|\le L_n}{sup}|\Phi \left(t\right)-\Phi \left(c_{nt}\right)|& \le \underset{-\infty <t<\infty }{sup}\left|\Phi \left(t\right)-\Phi \left(\frac{\sigma \left({\xi }_p\right)}{\sigma \left(n,t\right)}t\right)\right|\\ +\underset{\left|t\right|\le L_n}{sup}\left|\Phi \left(\frac{\sigma \left({\xi }_p\right)}{\sigma \left(n,t\right)}t\right)-\Phi \left(c_{nt}\right)\right|:=H_1+H_2.\end{array}$$

(3.10)

By (3.7) again and similar to the proof of (2.31), we have

$$H_1\le C\left|\frac{{\sigma }^2\left(n,t\right)}{{\sigma }^2\left({\xi }_p\right)}-1\right|=C{\sigma }^{-2}\left({\xi }_p\right)|{\sigma }^2\left(n,t\right)-{\sigma }^2\left({\xi }_p\right)|=o\left(n^{-1∕5}\right).$$

(3.11)

By Taylor's expansion again, we obtain that

$$\begin{array}{c}{c}_{nt}=t\cdot \frac{A}{\sigma (n\mathrm{,}t)}\cdot \frac{F({\xi }_p+tA{n}^{-1/2})-F({\xi }_p)}{tA{n}^{-1/2}}\\ =t\cdot \frac{A}{\sigma (n\mathrm{,}t)}\cdot \frac{F^{\prime }({\xi }_p)tA{n}^{-1/2}+\frac{1}{2}{F}^{″}({\xi }_{p\mathrm{,}t})(tA{n}^{-1/2}{)}^2}{tA{n}^{-1/2}}\\ =t\frac{\sigma ({\xi }_p)}{\sigma (n\mathrm{,}t)}+t^2\frac{A^2{F}^{″}({\xi }_{p\mathrm{,}t})}{2\sigma (n\mathrm{,}t)}{n}^{-1/2}\mathrm{,}\end{array}$$

(3.12)

where ξ_p,tlies between ξ _p and ξ _p + Atn^-1/2. It is known that [[12], Lemma 5.2],

$$\underset{x}{sup}|\Phi \left(x+q\right)-\Phi \left(x\right)|\le \left|q\right|\cdot {\left(2\pi \right)}^{-1∕2},foreveryq.$$

(3.13)

Therefore, by (3.12), (3.13) and the condition that F' is bounded in a neighborhood of ξ _p , we get for n large enough that

$$H_2=\underset{\left|t\right|\le L_n}{sup}\left|\Phi \left(\frac{\sigma \left({\xi }_p\right)}{\sigma \left(n,t\right)}t\right)-\Phi \left(c_{nt}\right)\right|\le C{L}_n^2{n}^{-1∕2}=O\left(logn\cdot loglogn\cdot n^{-1∕2}\right),$$

(3.14)

since σ² (ξ _p ) < ∞ and $\underset{n\to \infty }{lim}\frac{{\sigma }^2\left(n,t\right)}{{\sigma }^2\left({\xi }_p\right)}=1$ for |t| ≤ L _n . Therefore, it follows from (3.9), (3.10), (3.11) and (3.14) that

$$\underset{\left|t\right|\le L_n}{sup}|G_n\left(t\right)-\Phi \left(t\right)|=O\left(n^{-1∕9}\right).$$

(3.15)

Finally, the desired result (1.3) follows from (3.1), (3.2), (3.4) and (3.15) immediately. □

Appendix

Lemma A.1 [[22], Theorem 1] Let {X _n }_n≥1be NA random variables, EX _i = 0, E|X _i | ^p < ∞, where i = 1, 2,..., n and p ≥ 2. Then, there exists some constant c _p depending only on p such that

$$E|{\sum }_{i=1}^n{X}_i{|}^p\le c_p\left\{{\sum }_{i=1}^{n}E|X_i{|}^p+{\left({\sum }_{i=1}^{n}E{X}_i^2\right)}^{p∕2}\right\}.$$

Lemma A.2 [[16], Lemma 3.5] Let {X _n }_n≥1be a NA sequence with EX _i = 0, |X _i | ≤ b, a.s. i = 1, 2,..., Denote${\Delta }_n={\sum }_{i=1}^{n}E{X}_i^2$. Then for ∀ ε > 0,

$$P\left(\left|{\sum }_{i=1}^n{X}_i\right|>\epsilon \right)\le 2exp\left\{-\frac{{\epsilon }^2}{2\left(2{\Delta }_n+b\epsilon \right)}\right\}.$$

Lemma A.3 [[23], Lemma 2] Let × and Y be random variables, then for any a > 0,

$$\underset{t}{sup}|P\left(X+Y\le t\right)-\Phi \left(t\right)|\le \underset{t}{sup}|P\left(X\le t\right)-\Phi \left(t\right)|+\frac{a}{\sqrt{2\pi }}+P\left(\left|Y\right|>a\right).$$

Lemma A.4 [[10], Lemma 1.1.4] Let F(x) be a right-continuous distribution function. The inverse function F^-1(t), 0 < t < 1, is nondecreasing and left-continuous, and satisfies

• (i) F ^-1 (F(x)) ≤ x, - ∞ < x < ∞;

• (ii) F (F ^-1 (t)) ≥ t, 0 < t < 1;

• (ii) F (x) ≥ t if and only if × ≥ F ^-1 (t).