ADS Advances in Decision Sciences 2090-3367 2090-3359 Hindawi Publishing Corporation 671942 10.1155/2012/671942 671942 Research Article Almost Sure Central Limit Theorem of Sample Quantiles Miao Yu 1 Xu Shoufang 2 Peng Ang 3 Nadarajah Saralees 1 College of Mathematics and Information Science Henan Normal University Henan Province Xinxiang 453007 China htu.cn 2 Department of Mathematics and Information Science, Xinxiang University, Henan Province, Xinxiang 453000 China xxmu.edu.cn 3 Heze College of Finance and Economics, Shandong Province, Heze 274000 China xxmu.edu.cn 2012 4 9 2012 2012 19 04 2012 12 08 2012 2012 Copyright © 2012 Yu Miao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We obtain the almost sure central limit theorem (ASCLT) of sample quantiles. Furthermore, based on the method, the ASCLT of order statistics is also proved.

1. Introduction

To describe the results of the paper, suppose that we have an independent and identically distributed sample of size n from a distribution function F(x) with a continuous probability density function f(x). Let Fn(x) denote the sample distribution function, that is, (1.1)Fn(x)=1ni=1n1{Xix},-<x<. Let us define the pth quantile of F by (1.2)ξp=inf{x:F(x)p},p(0,1), and the sample quantile ξ^np by (1.3)ξ^np=inf{x:Fn(x)p},p(0,1). It is well known that ξ^np is a natural estimator of ξp. Since the quantile can be used for describing some properties of random variables, and there are not the restrictions of moment conditions, it is being widely employed in diverse problems in finance, such as quantile-hedging, optimal portfolio allocation, and risk management.

In practice, the large sample theory which can give the asymptotic properties of sample estimator is an important method to analyze statistical problems. There are numerous literatures to study the sample quantiles. Let p(0,1), if ξp is the unique solution x of F(x-)pF(x), then ξ^npa.e.ξp (see ). In addition, if F(x) possesses a continuous density function f(x) in a neighborhood of ξp and f(ξp)>0, then (1.4)n1/2f(ξp)(ξ^np-ξp)[p(1-p)]1/2N(0,1),asn, where N(0,1) denotes the standard normal variable (see [1, 2]). Suppose that F(x) is twice differentiable at ξp, with F(ξp)=f(ξp)>0, then Bahadur  proved (1.5)ξ^np=ξp+p-Fn(ξp)f(ξp)+R~n,a.e., where R~n=O(n-3/4(logn)3/4), a.e, as n. Very recently, Xu and Miao  obtained the moderate deviation, large deviation and Bahadur asymptotic efficiency of the sample quantiles ξ^np. Xu et al.  studied the Bahadur representation of sample quantiles for negatively associated sequences under some mild conditions.

Based on the above works, in the paper, we are interested in the almost sure central limit theorem (ASCLT) of sample quantiles ξ^np. The theory of ASCLT has been first introduced independently by Brosamler  and Schatte . The classical ASCLT states that when 𝔼X=0,  Var(X)=σ2, (1.6)limn1lognk=1n1k1{Skkσx}=Φ(x),a.s. for any x, where Sk denotes the partial sums Sk=X1++Xk. Moreover, from the method to prove the ASCLT of sample quantiles, in Section 3, we obtain the ASCLT of order statistics.

2. Main Results Theorem 2.1.

Let X1,X2,,Xn be a sequence of independent identically distributed random variables from a cumulative distribution function F. Let p(0,1) and suppose that f(ξp):=F(ξp) exists and is positive. Then one has (2.1)limn1lognk=1n1k1{k(ξ^kp-ξp)σx}=Φ(x), a.s. for any x, where σ2=p(1-p)/f2(ξp).

Proof.

Firstly, it is not difficult to check (2.2){k(ξ^kp-ξp)σx}={Fk(ξp+σxk)p}=:{1ki=1kYi,kωk}, where (2.3)Yi,k=E1{Xiξp+σx/k}-1{Xiξp+σx/k},ωk=k(E1{Xiξp+σx/k}-p). From the Taylor's formula, it follows (2.4)E1{Xiξp+σx/k}=F(ξp+σxk)=F(ξp)+F(ξp)σxk+o(1k)=p+f(ξp)σxk+o(1k), which implies (2.5)ωk=f(ξp)σx+o(1). By the Lindeberg's central limit theorem, we can get (2.6)1f(ξp)σki=1kYi,kdN(0,1),ask. Hence, (2.1) is equivalent to (2.7)limn1lognk=1n1k1{(1/f(ξp)σk)i=1kYi,kx+o(1)}=Φ(x),a.s.

Throughout the following proof, C denotes a positive constant, which may take different values whenever it appears in different expressions.

Put that (2.8)Zi,k:=1f(ξp)σYi,k. Let g be a bounded Lipschitz function bounded by C, then from (2.6), we have (2.9)Eg(1ki=1kZi,k)Eg(N),ask, where N denotes the standard normal random variable. Next, we should notice that (2.7) is equivalent to (2.10)limn1lognk=1n1kg(1ki=1kZi,k)=Eg(N)a.s. from Section 2 of Peligrad and Shao  and Theorem  7.1 of Billingsley . Hence, to prove (2.7), it suffices to show that as n, (2.11)Rn=1lognk=1n1k[g(1ki=1kZi,k)-Eg(1ki=1kZi,k)]=:1lognk=1n1kTk0,a.s. It is obvious that (2.12)ERn2=1log2n[k=1n1k2ETk2+2k=1n-1j=k+1n1kjETkTj]. Since g is bounded, we have (2.13)1log2nk=1n1k2ETk2Clogn. Furthermore, for 1k<jn, we have (2.14)|ETkTj|=|Cov(g(1ki=1kZi,k),g(1ji=1jZi,j))|=|Cov(g(i=1kZi,kk),g(i=1jZi,jj)-g(i=k+1jZi,jj))|CjE|i=1kZi,j|Ckj(EZ1,j2)1/2, where (2.15)EZ1,j2=1+O(1j). Therefore, we have (2.16)1log2nk=1n-1j=k+1n1kj|ETkTj|Clog2nk=1n-1j=k+1n1k1/2j3/2(EZ1,j2)1/2=Clog2nj=2nk=1j-11k1/2j3/2(EZ1,j2)1/2Clogn. From the above discussions, it follows that (2.17)ERn2Clogn. Take nk=ekτ, where τ>1. Then by Borel-Cantelli lemma, we have (2.18)Rnk0,a.s.ask. Since g is bounded function, then for nk<nnk+1, we obtain (2.19)|Rn|1lognk|l=1nk1l[g(1li=1lZi,l)-Eg(1li=1lZi,l)]|+1lognkl=nk+1nk+11l|g(1li=1lZi,l)-Eg(1li=1lZi,l)||Rnk|+Clognkl=nk+1nk+11l0,a.s.,asn, where we used the fact (2.20)lognk+1lognk=(k+1)τkτ1,ask. So, the proof of the theorem is completed.

3. Further Results

Another method to estimate the quantile is to use the order statistics. Based on the sample {X1,,Xn} of observations on F(x), the ordered sample values: (3.1)X(1)X(2)X(n) are called the order statistics. For more details about order statistics, one can refer to Serfling  or David and Nagaraja . Suppose that F is twice differentiable at ξp with F(ξp)=f(ξp)>0, then the Bahadur representation for order statistics was first established by Bahadur , as n(3.2)X(kn)=ξp+(kn/n)-Fn(ξp)f(ξp)+O(n-3/4(logn)(1/2)(δ+1))a.e., where (3.3)kn=np+o(n(logn)δ),n,forsomeδ12. From the idea of the Bahadur representation for order statistics, many important properties of order statistics can be easily proved. For example, Miao et al.  proved asymptotic properties of the deviation between order statistics and pth quantile, which included large and moderate deviation, Bahadur asymptotic efficiency.

Though there are some papers to study the ASCLT for the order statistics (e.g., Peng and Qi , Hörmann , Tong et al. , etc.), based on the method to deal with the sample quantile, we can also obtain the ASCLT of the order statistics.

Theorem 3.1.

Let X1,X2,,Xn be a sequence of independent identically distributed random variables from a cumulative distribution function F. Let p(0,1) and suppose that f(ξp):=F(ξp) exists and is positive. Let kn=np+o(n), then one has (3.4)limn1lognj=1n1j1{j(X(kj)-ξp)σx}=Φ(x), a.s. for any x, where σ2=p(1-p)/f2(ξp).

Proof.

Firstly, it is easy to see that the following two events are equivalent: (3.5){j(X(kj)-ξp)σx}={i=1j1{Xiξp+σx/j}kj}=:{i=1jY-i,jjω-j}, where (3.6)Y-i,j=E1{Xiξp+σx/j}-1{Xiξp+σx/j},ω-j=1j(jF(ξp+σxj)-kj)=f(ξp)σx+o(1). Hence, by the same proof of Theorem 2.1, we can obtain the desired result.

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