The Bahadur representation for sample quantiles under dependent sequence

TL;DR

This paper establishes the Bahadur representation for sample quantiles in dependent sequences, providing specific convergence rates under different mixing conditions, which enhances understanding of quantile behavior in dependent data.

Contribution

It extends the Bahadur representation to dependent sequences with new convergence rates under weaker mixing conditions.

Findings

Rate of $O(n^{-3/4}\log n)$ under $ ext{O}(n^{-3})$ mixing

Rate of $O(n^{-1/2}(\log n)^{1/2})$ under summable mixing coefficients

Results applicable to dependent data analysis and statistical inference

Abstract

On the one hand, we investigate the Bahadur representation for sample quantiles under $\varphi$-mixing sequence with $\varphi(n)=O(n^{-3})$ and obtain a rate as $O(n^{-\frac{3}{4}}\log n)$, $a.s.$. On the other hand, by relaxing the condition of mixing coefficients to $\sum\nolimits_{n=1}^\infty\varphi^{1/2}(n)<\infty$, a rate $O(n^{-1/2}(\log n)^{1/2})$, $a.s.$, is also obtained.

Equations

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

$${\xi }_{p,n}={\xi }_p-\frac{F_n({\xi }_p)-p}{f({\xi }_p)}+O\left(n^{-\frac{3}{4}}(\log n{)}^{3/4}\right),n\to \infty .$$

$$v_p=\inf \{x:F(x)\ge p\},$$

$$\phi (\mathcal{B},\mathcal{R})=\underset{A\in \mathcal{B},B\in \mathcal{R},P(A)>0}{\sup }|P(B|A)-P(B)|.$$

$$\phi (n)=\underset{k\ge 1}{\sup }\phi ({\mathcal{F}}_1^k,{\mathcal{F}}_{k+n}^{\infty }),n\ge 0.$$

$$H_{p,n}\doteq \underset{x\in {\mathcal{D}}_n}{\sup }|(F_n(x)-F(x))-(F_n({\xi }_p)-p)|.$$

$$H_{p,n}\le (d+4C_3)n^{-\frac{3}{4}}\log n,\text{for all}n\text{sufficiently large},$$

$${\xi }_{p,n}={\xi }_p-\frac{F_n({\xi }_p)-p}{f({\xi }_p)}+O\left(n^{-\frac{3}{4}}\log n\right),n\to \infty .$$

$$\underset{x\in {\mathcal{E}}_n}{\sup }|(F_n(x)-F(x))-(F_n({\xi }_p)-p)|\le (1+d){\left(\frac{(16C_3+\theta )\log n}{n}\right)}^{1/2}$$

$$\underset{x\in {\mathcal{E}}_n}{\sup }|F_n(x)-F(x)|\le (1+d){\left(\frac{(16C_3+\theta )\log n}{4n}\right)}^{1/2},\text{for all n sufficiently large}.$$

$${\xi }_{p,n}={\xi }_p-\frac{F_n({\xi }_p)-p}{f({\xi }_p)}+O\left(\frac{(\log n{)}^{1/2}}{n^{1/2}}\right),n\to \infty .$$

$$P\left(\left|\sum _{i=1}^n{X}_i\right|>\epsilon \right)\le 2e{C}_1\exp \left\{-\frac{{\epsilon }^2}{2C_2(2{\Delta }_2+n^{\beta }d\epsilon )}\right\},$$

$$|{\xi }_{p,n}-{\xi }_p|\le \frac{(2{\sqrt{C}}_3+\delta )(\log n{)}^{1/2}}{f({\xi }_p)n^{1/2}},\text{for all n sufficiently large},$$

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

$$P({\xi }_{p,n}>{\xi }_p+{\epsilon }_n)$$

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

$$P({\xi }_{p,n}>{\xi }_p+{\epsilon }_n)\le 2e{C}_1\exp \left\{-\frac{n{\delta }_{n1}^2}{2C_3(2+n^{\beta }{\delta }_{n1})}\right\},$$

$$P({\xi }_{p,n}<{\xi }_p-{\epsilon }_n)\le 2e{C}_1\exp \left\{-\frac{n{\delta }_{n2}^2}{2C_3(2+n^{\beta }{\delta }_{n2})}\right\},$$

$$P(|{\xi }_{p,n}-{\xi }_p|>{\epsilon }_n)\le 4e{C}_1\exp \left\{-\frac{n[\min ({\delta }_{n1},{\delta }_{n2}){]}^2}{2C_3(2+n^{\beta }[\max ({\delta }_{n1},{\delta }_{n2})])}\right\}.$$

$${\delta }_{n1}=F({\xi }_p+{\epsilon }_n)-p=f({\xi }_p){\epsilon }_n+o({\epsilon }_n)=\frac{(2{\sqrt{C}}_3+\delta )(\log n{)}^{1/2}}{n^{1/2}}+o({\epsilon }_n),$$

$${\delta }_{n2}=p-F({\xi }_p-{\epsilon }_n)=f({\xi }_p){\epsilon }_n+o({\epsilon }_n)=\frac{(2{\sqrt{C}}_3+\delta )(\log n{)}^{1/2}}{n^{1/2}}+o({\epsilon }_n).$$

$$[\min ({\delta }_{n1},{\delta }_{n2}){]}^2\ge (2{\sqrt{C}}_3+\delta /2{)}^2\cdot \frac{\log n}{n},\text{for all}n\text{sufficiently large}.$$

$$n^{\beta }[\max ({\delta }_{n1},{\delta }_{n2})]\le {\delta }_1,\text{and}{\delta }_2\doteq \frac{(2{\sqrt{C}}_3+\delta /2{)}^2}{2C_3(2+{\delta }_1)}>1$$

$$P(|{\xi }_{p,n}-{\xi }_p|>{\epsilon }_n)\le 4e{C}_1\exp \left\{-\frac{(2{\sqrt{C}}_3+\delta /2{)}^2}{2C_3(2+{\delta }_1)}\log n\right\}=\frac{4e{C}_1}{n^{{\delta }_2}}$$

$$C_1=\exp \{2e{n}^{1-\beta }\phi (m)\}\le C\exp \{2e{n}^{1-\beta }{n}^{-2\beta }\}\le C\exp \{2e\},$$

$$\sum _{n=1}^{\infty }P(|{\xi }_{p,n}-{\xi }_p|>{\epsilon }_n)$$

$$p\le F_n({\xi }_{p,n})<p+\frac{1}{n},a.s.$$

$${\eta }_{r,n}={\xi }_p+a_n{b}_n^{-1}r,{\alpha }_{r,n}=F({\eta }_{r+1,n})-F({\eta }_{r,n}),$$

$$G_{r,n}=F_n({\eta }_{r,n})-F_n({\xi }_p)-F({\eta }_{r,n})+p\doteq \frac{1}{n}\sum _{i=1}^n(Y_i-E{Y}_i),$$

$$g(x)=F_n(x)-F(x)-F_n({\xi }_p)+p,x\in {\mathcal{D}}_n.$$

Full text

The Bahadur representation for sample quantiles under dependent sequence

Wenzhi Yang    Shuhe Hu    Xuejun Wang

School of Mathematical Science, Anhui University, Hefei, Anhui 230039, China Corresponding author. E-mail: hushuhe@263.net (S.H. Hu); wzyang827@163.com (W.Z. Yang); wxjahdx2000@126.com (X.J. Wang)

Abstract:  On the one hand, we investigate the Bahadur representation for sample quantiles under $\varphi$-mixing sequence with $\varphi(n)=O(n^{-3})$ and obtain a rate as $O(n^{-\frac{3}{4}}\log n)$, $a.s.$. On the other hand, by relaxing the condition of mixing coefficients to $\sum\nolimits_{n=1}^{\infty}\varphi^{1/2}(n)<\infty$, a rate $O(n^{-1/2}(\log n)^{1/2})$, $a.s.$, is also obtained.

Keywords: Bahadur representation; sample quantiles; mixing sequence

AMS 2000 Mathematical Subject Classification:  62F12

1 Introduction

Assume that $\{X_{n},n\geq 1\}$ is a sequence of random variables defined on a fixed probability space $(\Omega,\mathcal{F},P)$ with a common marginal distribution function $F(x)=P(X_{1}\leq x)$. Let $F$ be a distribution function (continuous from the right, as usual). For $0<p<1$, the $p$th quantile of $F$ is defined as

[TABLE]

and is alternately denoted by $F^{-1}(p)$. The function $F^{-1}(t)$, $0<t<1$, is called the inverse function of $F$.

For a sample $X_{1},X_{2},\cdots,X_{n}$, $n\geq 1$, let $F_{n}$ represent the empirical distribution function based on $X_{1},X_{2},\cdots,X_{n}$, which is defined as $F_{n}(x)=\frac{1}{n}\sum\nolimits_{i=1}^{n}I(X_{i}\leq x)$, $x\in R$. Here $I(A)$ denotes the indicator function of the set $A$ and $R$ is the real line. For $0<p<1$, we define $F_{n}^{-1}(p)=\inf\{x:F_{n}(x)\geq p\}$ as the sample $p$th quantile.

Let $\xi_{p,n}=F_{n}^{-1}(p)$. Bahadur [2] firstly introduced an elegant representation for sample quantile in terms of empirical distribution function based on independent and identically distributed ($i.i.d.$) random variables and obtained the following result (or see Serfling [9, Theorem 2.5.1])

Theorem 1.1. Let $0<p<1$ and and $\{X_{n},n\geq 1\}$ be a sequence of $i.i.d.$ random variables. Suppose that $F$ is twice differentiable at $\xi_{p}$, with $F^{\prime}(\xi_{p})=f(\xi_{p})>0$. Then with probability 1,

[TABLE]

At present, many researchers have extended Bahadur representation for $i.i.d.$ random variables to the dependent cases. One can refer to Sen [8] and Babu and Singh [1] for $\varphi$-mixing sequence, Yoshihara [17] for $\alpha$-mixing sequence, Zhou and Zhu [19] for the smooth quantile estimator, Sun [10] for $\alpha$-mixing sequence, Cheng and Gooijer [4] for $M$-estimator under $\alpha$-mixing sequence, etc. Ling [7] extended the results of Sun [10] to the case of NA sequence and obtained a rate $O(\tau_{n})$ $a.s.$, where $\tau_{n}\rightarrow 0$ and $\sqrt{n}\tau_{n}^{2}/\log n\rightarrow 0$ as $n\rightarrow\infty$. Li et al. [6] extended the results of Ling [7] to the case of NOD sequence, which is weaker than NA sequence, and they got a better rate $O(n^{-1/2}(\log n)^{1/2})$ $a.s$. For more works on Bahadur representation, one can refer to [11, 12, 13, 16, 18], etc. Meanwhile, for the Berry-Esseen bounds of sample quantiles, one can refer to Serfling [9, Theorem 2.3.3 C], Lahiri and Sun [5], Yang et al. [14] and the references therein.

One of the applications of the quantile function is in finance where many financial returns can be modeled as time series data. Value-at-risk(VaR) is a popular measure of market risk associated with an asset or a portfolio of assets. It has been chosen by the Basel Committee on Banking Supervision as a benchmark risk measure and has been used by financial institutions for asset management and minimization of risk. Let $\{X_{t}\}_{t=1}^{n}$ be the market value of an asset over $n$ periods of a time unit, and let $Y_{t}=\log(X_{t}/X_{t-1})$ be the log-returns. Suppose $\{Y_{t}\}_{t=1}^{n}$ is a strictly stationary dependent process with marginal distribution function $F$. Given a positive value $p$ close to zero, the $1-p$ level VaR is

[TABLE]

which specifies the smallest amount of loss such that the probability of the loss in market value being large than $v_{p}$ is less than $p$. So, the study of VaR is a well application of $p$th quantile. Chen and Tang [3] considered the nonparametric estimation of VaR and associated standard error estimation for dependent financial returns. Theoretical properties of the kernel VaR estimator were investigated in the context of dependent. For more details, one can refer to Chen and Tang [3] and the references therein.

Before stating our works, we need to recall the definition of $\varphi$-mixing. Let $n$ and $m$ be positive integers. Write $\mathcal{F}_{n}^{m}=\sigma(X_{i},n\leq i\leq m)$. Given $\sigma$-algebras $\mathcal{B},\mathcal{R}$ in $\mathcal{F}$, let

[TABLE]

Define the $\varphi$-mixing coefficients by

[TABLE]

Definition 1.1. A random variable sequence  $\{X_{n},n\geq 1\}$ is said to be a $\varphi$-mixing random variable sequence if $\varphi(n)\downarrow 0$ as $n\rightarrow\infty$.

In this paper, we go on investigating the Bahadur representation for sample quantiles under $\varphi$-mixing sequence. Under some mild conditions such as $\varphi(n)=O(n^{-3})$ and $P(X_{i}=X_{j})=0$, $\forall~{}i\neq j$, the rate is established as $O(n^{-\frac{3}{4}}\log n)$ $a.s.$, which is close to the rate $O(n^{-\frac{3}{4}}(\log n)^{3/4})$ $a.s.$ in Theorem 1.1 for $i.i.d.$ random variables. By relaxing the mixing coefficients to $\sum\nolimits_{n=1}^{\infty}\varphi^{1/2}(n)<\infty$ and removing the condition $P(X_{i}=X_{j})=0$, $\forall~{}i\neq j$, we get the rate as $O(n^{-1/2}(\log n)^{1/2})$ $a.s.$. The Bahadur representation for sample quantiles under $\varphi$-mixing sequence has been studied by Sen [8], Babu and Singh [1] and Yoshihara [17], etc. Comparing our Theorem 2.1 in Section 2 with Theorem 3.1 of Sen [8], under some conditions, Sen obtained the rate $O(n^{-\frac{3}{4}}\log n)$ $a.s.$, where the mixing coefficients satisfy that $\sum\nolimits_{n=1}^{\infty}n^{k}\varphi^{1/2}(n)<\infty$ for some $k\geq 1$. The mixing coefficients condition $\varphi(n)=O(n^{-3})$ in our Theorem 2.1 is relatively weaker.

Under the mixing coefficients condition $\alpha(n)=O(n^{-\beta})$, $\beta>3$, Zhang et al. [18] studied the Bahadur representation for sample quantiles under $\alpha$-mixing sequence and obtained the rate $O(n^{-\frac{1}{2}}(\log\log n\cdot\log n)^{\frac{1}{2}})$, $a.s.$ ( see Theorems 2.3 of Zhang et al. [18]). For any $\delta>0$ and $\alpha(n)=O(n^{-\beta})$, $\beta>\max\{3+\frac{5}{1+\delta},1+\frac{2}{\delta}\}$, they also obtained the rate $O(n^{-\frac{3}{4}+\frac{\delta}{4(2+\delta)}}(\log\log n\cdot\log n)^{\frac{1}{2}})$, $a.s.$ (see Theorem 2.5 of Zhang et al. [18]).

It is a fact that $\varphi$-mixing random variables are $\alpha$-mixing random variables and $\alpha(n)\leq\varphi(n)$. In this paper, we investigate the Bahadur representation for sample quantiles under $\varphi$-mixing sequence. By taking $\varphi(n)=O(n^{-3})$ in our Theorem 2.2, we get a better rate $O\left(n^{-\frac{3}{4}}\log n\right)$, $a.s.$ than $O(n^{-\frac{3}{4}+\frac{\delta}{4(2+\delta)}}(\log\log n\cdot\log n)^{\frac{1}{2}})$, $a.s.$ obtained by Theorem 2.5 of Zhang et al. [18]. Similarly, by taking $\sum_{n=1}^{\infty}\varphi^{1/2}(n)<\infty$ in our Theorem 2.5, we also obtain a better rate $O(n^{-1/2}(\log n)^{1/2})$, $a.s.$ than $O(n^{-\frac{1}{2}}(\log\log n\cdot\log n)^{\frac{1}{2}})$, $a.s.$ obtained by Theorems 2.3 of Zhang et al. [18]. Although $\varphi$-mixing random variables are $\alpha$-mixing random variables, the bounds in our Theorems 2.1-2.5 are better than the ones obtained by Theorem 2.4, Theorem 2.5 and Theorems 2.1-2.3 of Zhang et al. [18], respectively.

The organization of this paper is as follows. The main results are presented in Section 2, some preliminary lemmas are given in Section 3 and the proofs of theorems are provided in Section 4. Throughout the paper, $C$ and $C_{0}$ denote positive constants which may be different in various places. $\lceil x\rceil$ denotes the largest integer not exceeding $x$. Denote $c_{n}\sim d_{n}$, which means $c_{n}d_{n}^{-1}\rightarrow 1$ as $n\rightarrow\infty$.

2 Main results

For a fixed $p\in(0,1)$, let $\xi_{p}=F^{-1}(p)$, $\xi_{p,n}=F_{n}^{-1}(p)$.

Theorem 2.1. Let $\{X_{n},n\geq 1\}$ be a sequence of $\varphi$-mixing random variables with the mixing coefficients $\varphi(n)=O(n^{-3})$. Assume that the common marginal distribution function $F(x)$ possesses a positive continuous density $f(x)$ in a neighborhood $\mathscr{N}_{p}$ of $\xi_{p}$ such that $0<d=\sup\{f(x):x\in\mathscr{N}_{p}\}<\infty$. Let $a_{n}\sim C_{0}n^{-\frac{1}{2}}(\log n)^{\frac{3}{4}}$, $\mathscr{D}_{n}=[\xi_{p}-a_{n},\xi_{p}+a_{n}]$ and

[TABLE]

Then with probability 1,

[TABLE]

*where $C_{3}=4[1+4\sum\nolimits_{n=1}^{\infty}\varphi^{1/2}(n)]$. *

Theorem 2.2. Let the conditions of Theorem 2.1 be satisfied and $P(X_{i}=X_{j})=0$, $\forall~{}i\neq j$. Assume that $f^{\prime}(x)$ is bounded in some neighborhood of $\xi_{p}$, say $\mathscr{N}_{p}$. Then with probability 1,

[TABLE]

Theorem 2.3. Let $\{X_{n},n\geq 1\}$ be a sequence of $\varphi$-mixing random variables with $\sum_{n=1}^{\infty}\varphi^{1/2}(n)<\infty$. Assume that the common marginal distribution function $F(x)$ possesses a positive continuous density $f(x)$ in a neighborhood $\mathscr{N}_{p}$ of $\xi_{p}$ such that $0<d=\sup\{f(x):x\in\mathscr{N}_{p}\}<\infty$. For any $\theta>0$, put $\tau_{n}=\frac{\sqrt{16C_{3}+\theta}(\log n)^{3/2}}{n^{1/2}(\log\log n)^{1/2}}$, $\mathscr{E}_{n}=[\xi_{p}-\tau_{n},\xi_{p}+\tau_{n}]$, where $C_{3}=4[1+4\sum\nolimits_{n=1}^{\infty}\varphi^{1/2}(n)]$. Then with probability 1

[TABLE]

for all $n$ sufficiently large.

Theorem 2.4. Let the conditions of Theorem 2.3 hold. Then with probability 1

[TABLE]

Theorem 2.5. Let the conditions of Theorem 2.3 be satisfied and $f^{\prime}(x)$ be bounded in some neighborhood of $\xi_{p}$. Then with probability 1

[TABLE]

3 Preliminary lemmas

Lemma 3.1. (Yang et al. [15, Corollary A.1]) Let $\{X_{n},n\geq 1\}$ be a $\varphi$-mixing sequence with $EX_{i}=0$, $|X_{i}|\leq d<\infty$, a.s. $i=1,2,\cdots$, $0<\beta<1$, $m=\lceil n^{\beta}\rceil$, $\Delta_{2}=\sum\nolimits_{i=1}^{n}EX_{i}^{2}$. Then for any $\varepsilon>0$ and $n\geq 2$, it follows

[TABLE]

where $C_{1}=\exp\{2en^{1-\beta}\varphi(m)\}$, $C_{2}=4[1+4\sum\nolimits_{i=1}^{2m}\varphi^{1/2}(i)]$ and $e$ is the base of natural logarithm.

Lemma 3.2. (Serfling [9, 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)   $F^{-1}(F(x))\leq x,~{}-\infty<x<\infty$;

(ii)  $F(F^{-1}(t))\geq t,~{}0<t<1$;

(iii) $F(x)\geq t$ if and only if $x\geq F^{-1}(t)$.

Inspired by Serfling [9, Theorem 2.3.2 and Lemma 2.5.4 B], we obtain the following result.

Lemma 3.3. Let $p\in(0,1)$ and $\{X_{n},n\geq 1\}$ be a sequence of $\varphi$-mixing random variables with $\sum_{n=1}^{\infty}\varphi^{1/2}(n)<\infty$. Assume that the common marginal distribution function $F(x)$ is differentiable at $\xi_{p}$, with $F^{{}^{\prime}}(\xi_{p})=f(\xi_{p})>0$. Suppose that $f^{{}^{\prime}}(x)$ is bounded in a neighborhood of $\xi_{p}$, say $\mathscr{N}_{p}$. Then for any $\delta>0$, with probability 1

[TABLE]

*where $C_{3}=4[1+4\sum\nolimits_{n=1}^{\infty}\varphi^{1/2}(n)]$. *

Proof.  Let $\varepsilon_{n}=\frac{(2\sqrt{C}_{3}+\delta)(\log n)^{1/2}}{f(\xi_{p})n^{1/2}}$, $n>1$, $\frac{1}{3}\leq\beta<\frac{1}{2}$. Write

[TABLE]

By Lemma 3.2 $(iii)$,

[TABLE]

where $V_{i}=I(X_{i}>\xi_{p}+\varepsilon_{n})$ and $\delta_{n1}=F(\xi_{p}+\varepsilon_{n})-p>0$. Likewise, by Lemma 3.2 $(iii)$,

[TABLE]

where $W_{i}=I(X_{i}\leq\xi_{p}-\varepsilon_{n})$ and $\delta_{n2}=p-F(\xi_{p}-\varepsilon_{n})>0$. It is easy to see that $\{V_{i}-EV_{i},1\leq i\leq n\}$ and $\{W_{i}-EW_{i},1\leq i\leq n\}$ are still $\varphi$-mixing random variables with mean zero and same mixing coefficients. Since $|V_{i}-EV_{i}|\leq 1$, $\sum\nolimits_{i=1}^{n}E(V_{i}-EV_{i})^{2}\leq n$, $|W_{i}-EW_{i}|\leq 1$, $\sum\nolimits_{i=1}^{n}E(W_{i}-EW_{i})^{2}\leq n$, it follows from Lemma 3.1 that

[TABLE]

[TABLE]

where $C_{1}=\exp\{2en^{1-\beta}\varphi(m)\}$, $m=\lceil n^{\beta}\rceil$ and $C_{3}=4[1+4\sum\nolimits_{n=1}^{\infty}\varphi^{1/2}(n)]$. Consequently,

[TABLE]

Since $F(x)$ is continuous at $\xi_{p}$ with $F^{{}^{\prime}}(\xi_{p})>0$, $\xi_{p}$ is the unique solution of $F(x-)\leq p\leq F(x)$ and $F(\xi_{p})=p$. By the assumption on $f^{\prime}(x)$ and Taylor’s expansion,

[TABLE]

and

[TABLE]

So

[TABLE]

Since $\frac{1}{3}\leq\beta<\frac{1}{2}$, it has $n^{\beta}[\max(\delta_{n1},\delta_{n2})]\rightarrow 0$ as $n\rightarrow\infty$. So we can choose $\delta_{1}>0$ such that

[TABLE]

for all $n$ sufficiently large. Hence,

[TABLE]

for all $n$ sufficiently large.

Since $\varphi(n)\downarrow 0$ as $n\rightarrow\infty$ and $\sum\nolimits_{n=1}^{\infty}\varphi^{1/2}(n)<\infty$, it follows that $\varphi^{1/2}(n)=o(\frac{1}{n})$. Therefore,

[TABLE]

and

[TABLE]

which implies that with probability 1 ($wp1$), the relations $|\xi_{p,n}-\xi_{p}|>\varepsilon_{n}$ hold for only finitely many $n$ by Borel-Cantelli Lemma. Thus (3.1) holds. $\sharp$

Lemma 3.4. (Wang et al. [11, Lemma 3.4]) Let $p\in(0,1)$ and $\xi_{p,n}=F_{n}^{-1}(p)=\inf\{x:F_{n}(x)\geq p\}$. For any $i\neq j$, we assume that $P(X_{i}=X_{j})=0$. Then

[TABLE]

4 Proofs of main results

Proof of Theorem 2.1. The proof is inspired by Serfling [9, Lemma 2.5.4 E]. Let $\{b_{n},n\geq 1\}$ be a sequence of positive integers such that $b_{n}\sim C_{0}n^{\frac{1}{4}}\log n$ as $n\rightarrow\infty$. For integers $r=-b_{n}\cdots,b_{n}$, put

[TABLE]

and

[TABLE]

where $Y_{i}=I(X_{i}\leq\eta_{r,n})-I(X_{i}\leq\xi_{p})$. Denote

[TABLE]

Then for all $x\in[\eta_{r,n},\eta_{r+1,n}]$,

[TABLE]

and

[TABLE]

So it has

[TABLE]

where

[TABLE]

By the fact $\eta_{r+1,n}-\eta_{r,n}=a_{n}b_{n}^{-1}=n^{-\frac{3}{4}}(\log n)^{-\frac{1}{4}}$, $-b_{n}\leq r\leq b_{n}-1$, we have by the Mean Value Theorem that

[TABLE]

thus

[TABLE]

Taking $\gamma_{n}=4C_{3}n^{-\frac{3}{4}}\log n$, we can check that $\{Y_{i}-EY_{i},1\leq i\leq n\}$ are still $\varphi$-mixing random variables with $|Y_{i}-EY_{i}|\leq 1$, $i=1,2,\cdots,n$. Applying Lemma 3.1 to $\{Y_{i}-EY_{i},1\leq i\leq n\}$ and $\varepsilon=n\gamma_{n}$, we obtain that

[TABLE]

where $z_{r,n}=|F(\eta_{r,n})-F(\xi_{p})|$. Since $\varphi(n)=O(n^{-3})$, let $\beta=\frac{1}{4}$ we have that $C_{1}=\exp\{2en^{1-\beta}\varphi(m)\}\leq C\exp\{2en^{1-\beta}n^{-3\beta}\}=C\exp\{2e\}$, and

[TABLE]

Let $C_{4}$ be some positive constant such that $f(\xi_{p})<C_{4}$. Then there exists $N\in\mathcal{N^{+}}$ such that

[TABLE]

and

[TABLE]

for all $n>N$. Thus

[TABLE]

By (4.3) and (4.4), it follows that

[TABLE]

for all $n$ sufficiently large. Therefore,

[TABLE]

Together with Borel–Cantelli Lemma, it follows that with probability 1 ($wp1$), the relations $K_{n}>\gamma_{n}$ hold for only finitely many $n$. Hence $wp1$, $K_{n}\leq\gamma_{n}$,  for all $n$ sufficiently large, i.e., $wp1$,

[TABLE]

Finally, (4.1), (4.2) and (4.5) yield (2.1). $\sharp$

Proof of Theorem 2.2.  By Lemma 3.3, we can see that $wp1$,

[TABLE]

which implies that $wp1$, $\xi_{p,n}\in\mathscr{D}_{n}$, for all $n$ sufficiently large. It follows from Theorem 2.1 that $wp1$,

[TABLE]

which implies that $wp1$,

[TABLE]

By (4.6), (4.7) and Lemma 3.4, we can obtain that $wp1$,

[TABLE]

where $w_{n}$ is a random variable between $\xi_{p,n}$ and $\xi_{p}$. By reorganizing the above equality, $wp1$, (2.2) holds. $\sharp$

Proof of Theorem 2.3. For $n>2$, let

[TABLE]

for $r=0,\pm 1,\pm 2,\cdots,\pm\lceil b_{n}\rceil$ and $b_{n}=\frac{\log n}{(\log\log n)^{1/2}}$, $\frac{1}{3}\leq\beta<\frac{1}{2}$. Denote

[TABLE]

Then for all $x\in[\eta_{r,n},\eta_{r+1,n}]$, it has

[TABLE]

and

[TABLE]

which imply that

[TABLE]

By the notations above, we can see that

[TABLE]

Denote $\xi_{i}=I(X_{i}\leq\eta_{r,n})-EI(X_{i}\leq\eta_{r,n}),~{}i=1,2,\cdots,n.$ Applying Lemma 3.1 to $\xi_{1},\xi_{2},\cdots,\xi_{n}$ and $\varepsilon=nt_{n}/2$, we have

[TABLE]

Here, $C_{1}=\exp\{2en^{1-\beta}\varphi(m)\}\leq C\exp\{2e\}$. It is easy to check that $n^{\beta}t_{n}\rightarrow 0$ as $n\rightarrow\infty$, so we can choose $\theta_{1}>0$ such that

[TABLE]

for all $n$ sufficiently large. Hence,

[TABLE]

Likewise, $I_{n}^{(2)}=P\left(|F_{n}(\xi_{p})-p|>\frac{t_{n}}{2}\right)$ satisfies a similar relation. So,

[TABLE]

By Borel-Cantelli Lemma, it follows that $wp1$, the relations $\max\limits_{0\leq|r|\leq\lceil b_{n}\rceil}|\Delta_{r,n}|>t_{n}$ hold for only finitely many $n$. Hence $wp1$

[TABLE]

for all $n$ sufficiently large. The proof of (2.3) is completed. $\sharp$

Proof of Theorem 2.4.  For $n>2$, denote

[TABLE]

for $r=0,\pm 1,\pm 2,\cdots,\pm\lceil b_{n}\rceil$ and $b_{n}=\frac{2\log n}{(\log\log n)^{1/2}}$, $\frac{1}{3}\leq\beta<\frac{1}{2}$. Then for any $x\in[\xi_{p}+rt_{n},\xi_{p}+(r+1)t_{n}]$,

[TABLE]

Hence

[TABLE]

Denote $\eta_{i}=I(X_{i}\leq\xi_{p}+rt_{n})-EI(X_{i}\leq\xi_{p}+rt_{n})$, $1\leq i\leq n$. Similar to the proof of (4.8), by Lemma 3.1, we have

[TABLE]

where $C_{1}=\exp\{2en^{1-\beta}\varphi(m)\}<\infty$ and $\theta_{2}\doteq\frac{16C_{3}+\theta}{16C_{3}+\theta_{1}}>1$. Therefore,

[TABLE]

By Borel-Cantelli Lemma, we obtain that $wp1$, the relations $\max\limits_{0\leq|r|\leq\lceil b_{n}\rceil}|d_{r,n}|>t_{n}$ hold for only finitely many $n$. Together with (4.9), we can get (2.4) immediately. $\sharp$

Proof of Theorem 2.5. Lemma 3.3 implies that $wp1$,

[TABLE]

$i.e.$ $wp1$, $\xi_{p,n}\in\mathscr{E}_{n}$ for all $n$ sufficiently large. It follows from Theorem 2.3 that $wp1$,

[TABLE]

By (4.10), assumption on $f^{\prime}(x)$, Taylor’s expansion and Theorem 2.4, we have that $wp1$,

[TABLE]

Together with (4.11), we obtain that $wp1$,

[TABLE]

where $w_{n}$ is a random variable between $\xi_{p,n}$ and $\xi_{p}$. Reorganizing the above equality, we obtain that $wp1$,

[TABLE]

The proof of (2.5) is completed. $\sharp$