 # The Bahadur representation for sample quantiles under dependent sequence

On the one hand, we investigate the Bahadur representation for sample quantiles under φ-mixing sequence with φ(n)=O(n^-3) and obtain a rate as O(n^-3/4 n), a.s.. On the other hand, by relaxing the condition of mixing coefficients to ∑_n=1^∞φ^1/2(n)<∞, a rate O(n^-1/2( n)^1/2), a.s., is also obtained.

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## 1 Introduction

Assume that

is a sequence of random variables defined on a fixed probability space

with a common marginal distribution function . Let be a distribution function (continuous from the right, as usual). For , the th quantile of is defined as

 ξp=inf{x:F(x)≥p}

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,

 ξp,n=ξp−Fn(ξp)−pf(ξp)+O(n−34(logn)3/4), n→∞.

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

 vp=inf{x:F(x)≥p},

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

 φ(B,R)=supA∈B,B∈R,P(A)>0|P(B|A)−P(B)|.

Define the -mixing coefficients by

 φ(n)=supk≥1φ(Fk1,F∞k+n), n≥0.

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

 Hp,n≐supx∈Dn|(Fn(x)−F(x))−(Fn(ξp)−p)|.

Then with probability 1,

 Hp,n≤(d+4C3)n−34logn,   for all n sufficiently large, (2.1)

where .

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,

 ξp,n=ξp−Fn(ξp)−pf(ξp)+O(n−34logn), n→∞. (2.2)

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

 supx∈En|(Fn(x)−F(x))−(Fn(ξp)−p)|≤(1+d)((16C3+θ)lognn)1/2 (2.3)

for all sufficiently large.

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

 supx∈En|Fn(x)−F(x)|≤(1+d)((16C3+θ)logn4n)1/2,  for all n sufficiently large. (2.4)

Theorem 2.5. Let the conditions of Theorem 2.3 be satisfied and be bounded in some neighborhood of . Then with probability 1

 ξp,n=ξp−Fn(ξp)−pf(ξp)+O((logn)1/2n1/2), n→∞. (2.5)

## 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

 P(∣∣ ∣∣n∑i=1Xi∣∣ ∣∣>ε)≤2eC1exp{−ε22C2(2Δ2+nβdε)},

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

(i)   ;

(ii)  ;

(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

 |ξp,n−ξp|≤(2√C3+δ)(logn)1/2f(ξp)n1/2,  for all n sufficiently large, (3.1)

where .

Proof.  Let , , . Write

 P(|ξp,n−ξp|>εn)=P(ξp,n>ξp+εn)+P(ξp,n<ξp−εn).

By Lemma 3.2 ,

 P(ξp,n>ξp+εn) = P(p>Fn(ξp+εn))=P(1−Fn(ξp+εn)>1−p) = P(n∑i=1I(Xi>ξp+εn)>n(1−p)) = P(n∑i=1(Vi−EVi)>nδn1),

where and . Likewise, by Lemma 3.2 ,

 P(ξp,n<ξp−εn)≤P(p≤Fn(ξp−εn))=P(n∑i=1(Wi−EWi)≥nδn2),

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

 P(ξp,n>ξp+εn)≤2eC1exp{−nδ2n12C3(2+nβδn1)},
 P(ξp,n<ξp−εn)≤2eC1exp{−nδ2n22C3(2+nβδn2)},

where , and . Consequently,

 P(|ξp,n−ξp|>εn)≤4eC1exp{−n[min(δn1,δn2)]22C3(2+nβ[max(δn1,δn2)])}.

Since is continuous at with , is the unique solution of and . By the assumption on and Taylor’s expansion,

 δn1=F(ξp+εn)−p=f(ξp)εn+o(εn)=(2√C3+δ)(logn)1/2n1/2+o(εn),

and

 δn2=p−F(ξp−εn)=f(ξp)εn+o(εn)=(2√C3+δ)(logn)1/2n1/2+o(εn).

So

 [min(δn1,δn2)]2≥(2√C3+δ/2)2⋅lognn, for all n sufficiently large.

Since , it has as . So we can choose such that

 nβ[max(δn1,δn2)]≤δ1, and δ2≐(2√C3+δ/2)22C3(2+δ1)>1

for all sufficiently large. Hence,

 P(|ξp,n−ξp|>εn)≤4eC1exp{−(2√C3+δ/2)22C3(2+δ1)logn}=4eC1nδ2

for all sufficiently large.

Since as and , it follows that . Therefore,

 C1=exp{2en1−βφ(m)}≤Cexp{2en1−βn−2β}≤Cexp{2e},

and

 ∞∑n=1P(|ξp,n−ξp|>εn) = n0−1∑n=1P(|ξp,n−ξp|>εn)+∞∑n=n0P(|ξp,n−ξp|>εn) ≤ C+4eC1∞∑n=n01nδ2<∞,

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

 p≤Fn(ξp,n)

## 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

 ηr,n=ξp+anb−1nr,  αr,n=F(ηr+1,n)−F(ηr,n),

and

 Gr,n=Fn(ηr,n)−Fn(ξp)−F(ηr,n)+p≐1nn∑i=1(Yi−EYi),

where . Denote

 g(x)=Fn(x)−F(x)−Fn(ξp)+p,   x∈Dn.

Then for all ,

 g(x)≤Fn(ηr+1,n)−F(ηr,n)−Fn(ξp)+p=Gr+1,n+αr,n,

and

 g(x)≥Fn(ηr,n)−F(ηr+1,n)−Fn(ξp)+p=Gr,n−αr,n.

So it has

 Hp,n≤Kn+βn, (4.1)

where

 Kn=max{|Gr,n|:−bn≤r≤bn},    βn=max{αr,n:−bn≤r≤bn−1}.

By the fact , , we have by the Mean Value Theorem that

 αr,n≤d(ηr+1,n−ηr,n)=dn−34(logn)−14, −bn≤r≤bn−1,

thus

 0≤βn≤dn−34(logn)−14. (4.2)

Taking , we can check that are still -mixing random variables with , . Applying Lemma 3.1 to and , we obtain that

 P(|Gr,n|>γn) = P(∣∣ ∣∣n∑i=1(Yi−EYi)∣∣ ∣∣>nγn) ≤ 2eC1exp{−n2γ2n2C3(2∑ni=1EY2i+nβ⋅nγn)} ≤ 2eC1exp{−n2γ2n2C3(2nzr,n+nβ⋅nγn)} = 2eC1exp{−nγ2n2C3(2zr,n+nβγn)},

where . Since , let we have that , and

 P(|Gr,n|>γn)≤2eC1exp⎧⎪⎨⎪⎩−nγ2n2C3(2zr,n+n14γn)⎫⎪⎬⎪⎭. (4.3)

Let be some positive constant such that . Then there exists such that

 F(ξp+an)−F(ξp)=f(ξp)an+o(an)

and

 F(ξp)−F(ξp−an)=f(ξp)an+o(an)

for all . Thus

 zr,n≤C4an, for |r|≤bn and n>N. (4.4)

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

 P(|Gr,n|>γn) ≤ 2eC1exp⎧⎪⎨⎪⎩−n⋅16C23n−32(logn)22C3(2C4C0n−12(logn)34+n14⋅4C3n−34logn)⎫⎪⎬⎪⎭ ≤ 2eC1exp⎧⎪⎨⎪⎩−16C23n−12(logn)29C23n−12logn⎫⎪⎬⎪⎭ = 2eC1exp{−16logn9} = 2eC1n16/9

for all sufficiently large. Therefore,

 ∞∑n=1P(Kn>γn)≤C+C∞∑n=n0bn∑r=−bnP(|Gr,n|>γn)≤C+C∞∑n=n0n14lognn16/9<∞.

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., ,

 Kn≤4C3n−34logn,    for all n sufficiently large. (4.5)

Finally, (4.1), (4.2) and (4.5) yield (2.1).

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

 |ξp,n−ξp|≤(2√C3+δ)(logn)1/2f(ξp)n1/2,  for all n sufficiently large, (4.6)

which implies that , , for all sufficiently large. It follows from Theorem 2.1 that ,

 |(Fn(ξp,n)−F(ξp,n))−(Fn(ξp)−p)| ≤ supx∈Dn|(Fn(x)−F(x))−(Fn(ξp)−p)| = O(n−34logn), n→∞,

which implies that ,

 Fn(ξp)−p=Fn(ξp,n)−F(ξp,n)+O(n−34logn), n→∞. (4.7)

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

 Fn(ξp)−p = p+O(1n)−F(ξp,n)+O(n−34logn) = −(F(ξp,n)−F(ξp))+O(n−34logn) = −f(ξp)(ξp,n−ξp)−12f′(wn)(ξp,n−ξp)2+O(n−34logn) = −f(ξp)(ξp,n−ξ