Advances in Decision Sciences

Volume 2012, Article ID 671942, 7 pages

http://dx.doi.org/10.1155/2012/671942

## Almost Sure Central Limit Theorem of Sample Quantiles

^{1}College of Mathematics and Information Science, Henan Normal University, Henan Province, Xinxiang 453007, China^{2}Department of Mathematics and Information Science, Xinxiang University, Henan Province, Xinxiang 453000, China^{3}Heze College of Finance and Economics, Shandong Province, Heze 274000, China

Received 19 April 2012; Accepted 12 August 2012

Academic Editor: Saralees Nadarajah

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.

#### Abstract

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 from a distribution function with a continuous probability density function . Let denote the sample distribution function, that is, Let us define the th quantile of by and the sample quantile by It is well known that is a natural estimator of . 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 , if is the unique solution of , then (see [1]). In addition, if possesses a continuous density function in a neighborhood of and , then where denotes the standard normal variable (see [1, 2]). Suppose that is twice differentiable at , with , then Bahadur [3] proved where , a.e, as . Very recently, Xu and Miao [4] obtained the moderate deviation, large deviation and Bahadur asymptotic efficiency of the sample quantiles . Xu et al. [5] 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 . The theory of ASCLT has been first introduced independently by Brosamler [6] and Schatte [7]. The classical ASCLT states that when , , for any , where denotes the partial sums . 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 be a sequence of independent identically distributed random variables from a cumulative distribution function . Let and suppose that exists and is positive. Then one has
**
for any , where .*

*Proof. *Firstly, it is not difficult to check
where
From the Taylor's formula, it follows
which implies
By the Lindeberg's central limit theorem, we can get
Hence, (2.1) is equivalent to

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

Put that
Let be a bounded Lipschitz function bounded by , then from (2.6), we have
where denotes the standard normal random variable. Next, we should notice that (2.7) is equivalent to
from Section 2 of Peligrad and Shao [8] and Theorem 7.1 of Billingsley [9]. Hence, to prove (2.7), it suffices to show that as ,
It is obvious that
Since is bounded, we have
Furthermore, for , we have
where
Therefore, we have
From the above discussions, it follows that
Take , where . Then by Borel-Cantelli lemma, we have
Since is bounded function, then for , we obtain
where we used the fact
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 of observations on , the ordered sample values: are called the order statistics. For more details about order statistics, one can refer to Serfling [1] or David and Nagaraja [10]. Suppose that is twice differentiable at with , then the Bahadur representation for order statistics was first established by Bahadur [3], as where 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. [11] proved asymptotic properties of the deviation between order statistics and th 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 [12], Hörmann [13], Tong et al. [14], 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 be a sequence of independent identically distributed random variables from a cumulative distribution function . Let and suppose that exists and is positive. Let , then one has
**
for any , where .*

*Proof. *Firstly, it is easy to see that the following two events are equivalent:
where
Hence, by the same proof of Theorem 2.1, we can obtain the desired result.

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