- Research
- Open Access
A generalization of almost sure local limit theorem of uniform empirical process
- Chenglian Zhu^{1}Email author
https://doi.org/10.1186/s13660-015-0626-3
© Zhu; licensee Springer. 2015
- Received: 31 July 2014
- Accepted: 9 March 2015
- Published: 18 March 2015
Abstract
Let \(\{X_{n}; n\geq1\}\) be a sequence of independent and identically distributed \(U[0, 1]\)-distributed random variables. In this paper, we are concerned with the almost sure local central limit theorem of \(\|F_{n}\|\) and \(\sup_{0\leq t\leq1}F_{n}(t)\), and some corresponding results are derived.
Keywords
- almost sure central limit theorem
- almost sure local central limit theorem
- uniform empirical process
MSC
- 60E15
- 60F15
1 Introduction
For Gaussian sequences, Csáki and Gonchigdanzan [5] presented the validity of (1.2) for maxima of stationary Gaussian sequences under some mild conditions. Furthermore, Chen and Lin [6] extended it to non-stationary Gaussian sequences. As for some other dependent random variables, Peligrad and Shao [7] and Dudziński [8] derived some corresponding results about an almost sure central limit theorem. The almost sure central limit theorem in a joint version for log average in the case of independent and identically distributed random variables was obtained by Peng et al. [9], a joint version of almost sure limit theorem for log average of maxima and partial sums in the case of stationary Gaussian random variables was derived by Dudziński [10]. In this direction, an extension of almost sure central limit theory was studied by Hörmann [11].
Moreover, Wu [12–14] explored the almost sure limit theorem for product of partial sums, stable distribution and product of sums of partial sums, respectively. Zang [15] derived the almost sure limit theorem of random fields for more general weights than the usual logarithmic average. Recently, Zhang [16] established the almost sure central limit theorem for uniform empirical processes with logarithmic average. And then, under some regular conditions, a general result of almost sure central limit theorem for uniform empirical processes with general weights was derived by Zang [17] with the methodology of Hörmann [11].
On the other hand, Chung and Erdős [18] proved the following result.
Theorem A
This result may be called almost sure local central limit theorem (ASLCLT), while (1.1) may be called almost sure global central limit theorem.
In this paper, under some mild conditions, we are concerned with the almost sure local central limit theorems of \(\|F_{n}\|\) and \(\sup_{0\leq t\leq1}F_{n}(t)\), which is inspired by the above result, especially Csáki et al. [19] and the references therein concerning the almost sure local central limit theorem.
The rest of this paper is organized as follows. In Section 2, a generalization of almost sure local central limit theorem of uniform empirical process is formulated. In Section 3, proofs of our main results are established. In Section 4, the paper is concluded and some statistical applications for future research are outlined.
2 Main results
Theorem 2.1
Theorem 2.2
Remark 2.3
We believe that condition (2.1) can be weakened through more complicated calculating procedures, so we will study it in the future work.
3 The proofs of the main results
In this section, we shall give some auxiliary lemmas which will be used to prove our main result. The first lemma comes from Gonchigdanzan [20].
Lemma 3.1
Lemma 3.2
Proof
This lemma is due to Kiefer and Wolfowitz [21]. □
Now, we give the proofs of our main results.
Proof of Theorem 2.1
Proof of Theorem 2.2
The proof is similar to the above procedures, so we omit it here. □
4 Concluding remarks
In this paper, we are concerned with the limit theory of uniform empirical process. A generalization of almost sure local central limit theorem of uniform empirical process has been established.
Some statistical applications related to our main result deserve further investigation. By virtue of being a new approach of testing based on ASCLT, Thangavelu [22] investigated hypothesis testing via the estimation of quantiles of the distribution of the concerned statistics. Based on the theorem on ASCLT for rank statistics, he also proposed a small-sample approximation for the two-sample nonparametric Behrens-Fisher problem. These statistical applications concerning our work will be discussed in the future work.
Declarations
Acknowledgements
This work is supported by the National Natural Science Foundation of China (Grant Nos. 11326174 and 11401245), the Natural Science Foundation of Jiangsu Province (Grant No. BK20130412), the Natural Science Research Project of Ordinary Universities in Jiangsu Province (Grant No. 12KJB110003).
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
References
- Brosamler, GA: An almost everywhere central limit theorem. Math. Proc. Camb. Philos. Soc. 104, 561-574 (1988) View ArticleMATHMathSciNetGoogle Scholar
- Schatte, P: On strong versions of the central limit theorem. Math. Nachr. 137, 249-256 (1988) View ArticleMATHMathSciNetGoogle Scholar
- Fahrner, I, Stadtmüller, U: On almost sure max-limit theorems. Stat. Probab. Lett. 37, 229-236 (1998) View ArticleMATHGoogle Scholar
- Cheng, SH, Peng, L, Qi, YC: Almost sure convergence in extreme value theory. Math. Nachr. 190, 43-50 (1998) View ArticleMATHMathSciNetGoogle Scholar
- Csáki, E, Gonchigdanzan, K: Almost sure limit theorems for the maximum of stationary Gaussian sequences. Stat. Probab. Lett. 58, 195-203 (2002) View ArticleMATHGoogle Scholar
- Chen, SQ, Lin, ZY: Almost sure max-limits for nonstationary Gaussian sequence. Stat. Probab. Lett. 76, 1175-1184 (2006) View ArticleMATHGoogle Scholar
- Peligrad, M, Shao, QM: A note on the almost sure central limit theorem for weakly dependent random variables. Stat. Probab. Lett. 22, 131-136 (1995) View ArticleMATHMathSciNetGoogle Scholar
- Dudziński, M: A note on the almost sure central limit theorem for some dependent random variables. Stat. Probab. Lett. 61, 31-40 (2003) View ArticleMATHGoogle Scholar
- Peng, ZX, Wang, LL, Nadarajah, S: Almost sure central limit theorem for partial sums and maxima. Math. Nachr. 282, 632-636 (2009) View ArticleMATHMathSciNetGoogle Scholar
- Dudziński, M: The almost sure central limit theorems in the joint version for the maxima and sums of certain stationary Gaussian sequences. Stat. Probab. Lett. 78, 347-357 (2008) View ArticleMATHGoogle Scholar
- Hörmann, S: An extension of almost sure central limit theory. Stat. Probab. Lett. 76, 191-202 (2006) View ArticleMATHGoogle Scholar
- Wu, QY: An improved result in almost sure central limit theory for products of partial sums with stable distribution. Chin. Ann. Math., Ser. B 33(6), 919-930 (2012) View ArticleMATHMathSciNetGoogle Scholar
- Wu, QY: Almost sure limit theorems for stable distributions. Stat. Probab. Lett. 81, 662-672 (2011) View ArticleMATHGoogle Scholar
- Wu, QY: Almost sure central limit theory for products of sums of partial sums. Stat. Probab. Lett. 81, 662-672 (2011) View ArticleMATHGoogle Scholar
- Zang, QP: A general result in almost sure central limit theory for random fields. Statistics 48, 965-970 (2014) View ArticleMATHMathSciNetGoogle Scholar
- Zhang, Y: A note on almost sure central limit theorem for uniform empirical processes. J. Jilin Univ. Sci. Ed. 49, 687-689 (2011) Google Scholar
- Zang, QP: A general result of the almost sure central limit theorem of uniform empirical process. Statistics 49, 98-103 (2015) View ArticleMathSciNetGoogle Scholar
- Chung, KL, Erdős, P: Probability limit theorems assuming only the first moment. Mem. Am. Math. Soc. 6, 1-19 (1951) Google Scholar
- Csáki, E, Földes, A, Révész, P: On almost sure local and global central limit theorems. Probab. Theory Relat. Fields 97, 321-337 (1993) View ArticleMATHGoogle Scholar
- Gonchigdanzan, K: Almost sure central limit theorems. Doctorate dissertation, University of Cincinnati, USA (2001) Google Scholar
- Kiefer, J, Wolfowitz, J: On the deviations of the empiric distribution function of vector chance variables. Trans. Am. Math. Soc. 87, 173-186 (1958) View ArticleMATHMathSciNetGoogle Scholar
- Thangavelu, K: Quantile estimation based on the almost sure central limit theorem. Dissertation, University of Göttingen, Germany (2005) Google Scholar