Almost Sure Central Limit Theorem of Sample Quantiles

Copyright q 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.


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 F n x denote the sample distribution function, that is, Let us define the pth quantile of F by and the sample quantile ξ np by ξ np inf x : F n x ≥ p , p ∈ 0, 1 .

1.3
It is well known that ξ np is a natural estimator of ξ p .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 quantilehedging, 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 ξ p is the unique solution x of F x− ≤ p ≤ F x , then ξ np a.e.− −− → ξ p see 1 .In addition, if F x possesses a continuous density function f x in a neighborhood of ξ p and f ξ p > 0, then where N 0, 1 denotes the standard normal variable see for any x ∈ R, where S k denotes the partial sums S k X 1 • • • X k .Moreover, from the method to prove the ASCLT of sample quantiles, in Section 3, we obtain the ASCLT of order statistics.

Main Results
Theorem 2.1.Let X 1 , X 2 , . . ., X n be a sequence of independent identically distributed random variables from a cumulative distribution function F. Let p ∈ 0, 1 and suppose that f ξ p : F ξ p exists and is positive.Then one has for any x ∈ R, where σ 2 p 1 − p /f 2 ξ p .

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Proof.Firstly, it is not difficult to check where

2.3
From the Taylor's formula, it follows By the Lindeberg's central limit theorem, we can get Throughout the following proof, C denotes a positive constant, which may take different values whenever it appears in different expressions.
Put that Let g be a bounded Lipschitz function bounded by C, then from 2.6 , we have where N 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 n → ∞,

2.11
It is obvious that Since g is bounded, we have Furthermore, for 1 ≤ k < j ≤ n, we have where

2.16
From the above discussions, it follows that 2.17 Take n k e k τ , where τ > 1.Then by Borel-Cantelli lemma, we have Since g is bounded function, then for n k < n ≤ n k 1 , we obtain

2.19
where we used the fact So, the proof of the theorem is completed.

Further Results
Another method to estimate the quantile is to use the order statistics.Based on the sample {X 1 , . . ., X n } of observations on F x , the ordered sample values:

3.6
Hence, by the same proof of Theorem 2.1, we can obtain the desired result.
−3/4log n 3/4 , a.e, as n → ∞.Very recently, Xu and Miao 4 obtained the moderate deviation, large deviation and Bahadur asymptotic efficiency of the sample quantiles ξ np .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 ξ np .The theory of ASCLT has been first introduced independently by Brosamler 6 and Schatte 7 .The classical ASCLT states that when 1 are called the order statistics.For more details about order statistics, one can refer to Serfling 1 or David and Nagaraja 10 .Suppose that F is twice differentiable at ξ p with F ξ p f ξ p > 0, then the Bahadur representation for order statistics was first established byBahadur 3, as n → ∞ 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 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 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.Let X 1 , X 2 , . . ., X n be a sequence of independent identically distributed random variables from a cumulative distribution function F. Let p ∈ 0, 1 and suppose that f ξ p : F ξ p exists and is positive.Let k n np o Firstly, it is easy to see that the following two events are equivalent: E1 {X i ≤ξ p σx/ √ j} − 1 {X i ≤ξ p σx/ √ j X k j −ξ p ≤σx} Φ x , a.s.3.4foranyx ∈ R, where σ 2 p 1 − p /f 2 ξ p .Proof.