Assume thatwith a common marginal distribution function . Let be a distribution function (continuous from the right, as usual). For , the th quantile of is defined as
and is alternately denoted by . The function , , is called the inverse function of .
For a sample , , let represent the empirical distribution function based on , which is defined as , . Here denotes the indicator function of the set and is the real line. For , we define as the sample th quantile.
Let . Bahadur  firstly introduced an elegant representation for sample quantile in terms of empirical distribution function based on independent and identically distributed () random variables and obtained the following result (or see Serfling [9, Theorem 2.5.1])
Theorem 1.1. Let and and be a sequence of random variables. Suppose that is twice differentiable at , with . Then with probability 1,
At present, many researchers have extended Bahadur representation for random variables to the dependent cases. One can refer to Sen  and Babu and Singh  for -mixing sequence, Yoshihara  for -mixing sequence, Zhou and Zhu 
for the smooth quantile estimator, Sun for -mixing sequence, Cheng and Gooijer  for -estimator under -mixing sequence, etc. Ling  extended the results of Sun  to the case of NA sequence and obtained a rate , where and as . Li et al.  extended the results of Ling  to the case of NOD sequence, which is weaker than NA sequence, and they got a better rate . 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 , Yang et al.  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 be the market value of an asset over periods of a time unit, and let be the log-returns. Suppose is a strictly stationary dependent process with marginal distribution function . Given a positive value close to zero, the level VaR is
which specifies the smallest amount of loss such that the probability of the loss in market value being large than is less than . So, the study of VaR is a well application of th quantile. Chen and Tang 
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 and the references therein.
Before stating our works, we need to recall the definition of -mixing. Let and be positive integers. Write . Given -algebras in , let
Define the -mixing coefficients by
Definition 1.1. A random variable sequence is said to be a -mixing random variable sequence if as .
In this paper, we go on investigating the Bahadur representation for sample quantiles under -mixing sequence. Under some mild conditions such as and , , the rate is established as , which is close to the rate in Theorem 1.1 for random variables. By relaxing the mixing coefficients to and removing the condition , , we get the rate as . The Bahadur representation for sample quantiles under -mixing sequence has been studied by Sen , Babu and Singh  and Yoshihara , etc. Comparing our Theorem 2.1 in Section 2 with Theorem 3.1 of Sen , under some conditions, Sen obtained the rate , where the mixing coefficients satisfy that for some . The mixing coefficients condition in our Theorem 2.1 is relatively weaker.
Under the mixing coefficients condition , , Zhang et al.  studied the Bahadur representation for sample quantiles under -mixing sequence and obtained the rate , ( see Theorems 2.3 of Zhang et al. ). For any and , , they also obtained the rate , (see Theorem 2.5 of Zhang et al. ).
It is a fact that -mixing random variables are -mixing random variables and . In this paper, we investigate the Bahadur representation for sample quantiles under -mixing sequence. By taking in our Theorem 2.2, we get a better rate , than , obtained by Theorem 2.5 of Zhang et al. . Similarly, by taking in our Theorem 2.5, we also obtain a better rate , than , obtained by Theorems 2.3 of Zhang et al. . Although -mixing random variables are -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. , 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, and denote positive constants which may be different in various places. denotes the largest integer not exceeding . Denote , which means as .
2 Main results
For a fixed , let , .
Theorem 2.1. Let be a sequence of -mixing random variables with the mixing coefficients . Assume that the common marginal distribution function possesses a positive continuous density in a neighborhood of such that . Let , and
Then with probability 1,
Theorem 2.2. Let the conditions of Theorem 2.1 be satisfied and , . Assume that is bounded in some neighborhood of , say . Then with probability 1,
Theorem 2.3. Let be a sequence of -mixing random variables with . Assume that the common marginal distribution function possesses a positive continuous density in a neighborhood of such that . For any , put , , where . Then with probability 1
for all sufficiently large.
Theorem 2.4. Let the conditions of Theorem 2.3 hold. Then with probability 1
Theorem 2.5. Let the conditions of Theorem 2.3 be satisfied and be bounded in some neighborhood of . Then with probability 1
3 Preliminary lemmas
Lemma 3.1. (Yang et al. [15, Corollary A.1]) Let be a -mixing sequence with , , a.s. , , , . Then for any and , it follows
where , and is the base of natural logarithm.
Lemma 3.2. (Serfling [9, Lemma 1.1.4]) Let be a right-continuous distribution function. The inverse function , , is nondecreasing and left-continuous, and satisfies
(iii) if and only if .
Inspired by Serfling [9, Theorem 2.3.2 and Lemma 2.5.4 B], we obtain the following result.
Lemma 3.3. Let and be a sequence of -mixing random variables with . Assume that the common marginal distribution function is differentiable at , with . Suppose that is bounded in a neighborhood of , say . Then for any , with probability 1
Proof. Let , , . Write
By Lemma 3.2 ,
where and . Likewise, by Lemma 3.2 ,
where and . It is easy to see that and are still -mixing random variables with mean zero and same mixing coefficients. Since , , , , it follows from Lemma 3.1 that
where , and . Consequently,
Since is continuous at with , is the unique solution of and . By the assumption on and Taylor’s expansion,
Since , it has as . So we can choose such that
for all sufficiently large. Hence,
for all sufficiently large.
Since as and , it follows that . Therefore,
which implies that with probability 1 (), the relations hold for only finitely many by Borel-Cantelli Lemma. Thus (3.1) holds.
Lemma 3.4. (Wang et al. [11, Lemma 3.4]) Let and . For any , we assume that . Then
4 Proofs of main results
Proof of Theorem 2.1. The proof is inspired by Serfling [9, Lemma 2.5.4 E]. Let be a sequence of positive integers such that as . For integers , put
where . Denote
Then for all ,
So it has
By the fact , , we have by the Mean Value Theorem that
Taking , we can check that are still -mixing random variables with , . Applying Lemma 3.1 to and , we obtain that
where . Since , let we have that , and
Let be some positive constant such that . Then there exists such that
for all . Thus
for all sufficiently large. Therefore,
Together with Borel–Cantelli Lemma, it follows that with probability 1 (), the relations hold for only finitely many . Hence , , for all sufficiently large, i.e., ,