Research Article | Open Access
Yu Miao, Shoufang Xu, Ang Peng, "Almost Sure Central Limit Theorem of Sample Quantiles", Advances in Decision Sciences, vol. 2012, Article ID 671942, 7 pages, 2012. https://doi.org/10.1155/2012/671942
Almost Sure Central Limit Theorem of Sample Quantiles
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.
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 ). 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  proved where , a.e, as . Very recently, Xu and Miao  obtained the moderate deviation, large deviation and Bahadur asymptotic efficiency of the sample quantiles . 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 . The theory of ASCLT has been first introduced independently by Brosamler  and Schatte . 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
From the Taylor's formula, it follows
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  and Theorem 7.1 of Billingsley . 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  or David and Nagaraja . Suppose that is twice differentiable at with , then the Bahadur representation for order statistics was first established by Bahadur , 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.  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 , 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 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.
- R. J. Serfling, Approximation Theorems of Mathematical Statistics, John Wiley & Sons, New York, NY, USA, 1980.
- S. S. Wilks, Mathematical Statistics, John Wiley & Sons, New York, NY, USA, 1962.
- R. R. Bahadur, “A note on quantiles in large samples,” Annals of Mathematical Statistics, vol. 37, pp. 577–580, 1966.
- S. F. Xu and Y. Miao, “Limit behaviors of the deviation between the sample quantiles and the quantile,” Filomat, vol. 25, no. 2, pp. 197–206, 2011.
- S. F. Xu, L. Ge, and Y. Miao, “On the Bahadur representation of sample quantiles and order statistics for NA sequences,” Journal of the Korean Statistical Society. In press.
- G. A. Brosamler, “An almost everywhere central limit theorem,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 104, no. 3, pp. 561–574, 1988.
- P. Schatte, “On strong versions of the central limit theorem,” Mathematische Nachrichten, vol. 137, pp. 249–256, 1988.
- M. Peligrad and Q. M. Shao, “A note on the almost sure central limit theorem for weakly dependent random variables,” Statistics & Probability Letters, vol. 22, no. 2, pp. 131–136, 1995.
- P. Billingsley, Convergence of Probability Measures, John Wiley & Sons, New York, NY, USA, 1968.
- H. A. David and H. N. Nagaraja, Order Statistics, Wiley Series in Probability and Statistics, Wiley-Interscience, John Wiley & Sons, Hoboken, NJ, USA, 3rd edition, 2003.
- Y. Miao, Y.-X. Chen, and S.-F. Xu, “Asymptotic properties of the deviation between order statistics and -quantile,” Communications in Statistics, vol. 40, no. 1, pp. 8–14, 2011.
- L. Peng and Y. C. Qi, “Almost sure convergence of the distributional limit theorem for order statistics,” Probability and Mathematical Statistics, vol. 23, no. 2, pp. 217–228, 2003.
- S. Hörmann, “A note on the almost sure convergence of central order statistics,” Probability and Mathematical Statistics, vol. 25, no. 2, pp. 317–329, 2005.
- B. Tong, Z. X. Peng, and S. Nadarajah, “An extension of almost sure central limit theorem for order statistics,” Extremes, vol. 12, no. 3, pp. 201–209, 2009.
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