- Research Article
- Open Access
© Wang Xuejun et al. 2010
- Received: 4 February 2010
- Accepted: 11 June 2010
- Published: 6 July 2010
- Positive Integer
- Real Number
- Limit Theorem
- Probability Space
- Central Limit
-mixing random variables were introduced by Dobrushin  and many applications have been found. See, for example, Dobrushin , Utev , and Chen  for central limit theorem, Herrndorf  and Peligrad  for weak invariance principle, Sen [6, 7] for weak convergence of empirical processes, Shao  for almost sure invariance principles, Hu and Wang  for large deviations, and so forth. When these are compared with the corresponding results of independent random variable sequences, there still remains much to be desired.
Throughout the paper, let be the indicator function of the set . We assume that is a positive increasing function on satisfying as and is the inverse function of . Since , it follows that . For easy notation, we let and . denotes that there exists a positive constant such that . denotes a positive constant which may be different in various places.
Let , be a sequence of random variables and let , be an array of constants. The almost sure limiting behavior of weighted sums was studied by many authors; see, for example, Choi and Sung , Cuzick , Wu , and Sung [13, 14], and so forth.
Lemma 1.5 (cf. [15, Lemma ]).
Lemma 1.6 (cf. [8, Lemma 2.2]).
which implies (1.7). By Lemma 1.6, we can get the desired result (1.9) immediately. The proof is complete.
The proof is similar to that of Lemma by Sung . So we omit it.
We complete the proof of the theorem.
Therefore, we can conclude that (2.15) holds by (2.12), (2.13), (2.14), and (2.19).
In order to prove that (2.14) holds, we should consider the following two cases.
Thus we get the desired result.
The authors are most grateful to the Editor Andrei I. Volodin and anonymous referee for the careful reading of the manuscript and valuable suggestions which helped in significantly improving an earlier version of this paper. This work was supported by the National Natural Science Foundation of China (10871001, 60803059), Provincial Natural Science Research Project of Anhui Colleges (KJ2010A005), Talents Youth Fund of Anhui Province Universities (2010SQRL016ZD), and Youth Science Research Fund of Anhui University (2009QN011A).
- Dobrushin RL: The central limit theorem for non-stationary Markov chain. Theory of Probability and Its Applications 1956, 1: 72–88.MathSciNetMATHGoogle Scholar
- Utev SA: The central limit theorem for -mixing arrays of random variables. Theory of Probability and Its Applications 1990, 35(1):131–139. 10.1137/1135013MathSciNetView ArticleMATHGoogle Scholar
- Chen DC: A uniform central limit theorem for nonuniform -mixing random fields. The Annals of Probability 1991, 19(2):636–649. 10.1214/aop/1176990445MathSciNetView ArticleMATHGoogle Scholar
- Herrndorf N: The invariance principle for -mixing sequences. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 1983, 63(1):97–108. 10.1007/BF00534180MathSciNetView ArticleMATHGoogle Scholar
- Peligrad M: An invariance principle for -mixing sequences. The Annals of Probability 1985, 13(4):1304–1313. 10.1214/aop/1176992814MathSciNetView ArticleMATHGoogle Scholar
- Sen PK: A note on weak convergence of empirical processes for sequences of -mixing random variables. Annals of Mathematical Statistics 1971, 42: 2131–2133. 10.1214/aoms/1177693079MathSciNetView ArticleMATHGoogle Scholar
- Sen PK: Weak convergence of multidimensional empirical processes for stationary -mixing processes. The Annals of Probability 1974, 2(1):147–154. 10.1214/aop/1176996760View ArticleMathSciNetMATHGoogle Scholar
- Shao QM: Almost sure invariance principles for mixing sequences of random variables. Stochastic Processes and Their Applications 1993, 48(2):319–334. 10.1016/0304-4149(93)90051-5MathSciNetView ArticleMATHGoogle Scholar
- Hu S, Wang X: Large deviations for some dependent sequences. Acta Mathematica Scientia. Series B 2008, 28(2):295–300. 10.1016/S0252-9602(08)60030-2MathSciNetView ArticleMATHGoogle Scholar
- Choi BD, Sung SH: Almost sure convergence theorems of weighted sums of random variables. Stochastic Analysis and Applications 1987, 5(4):365–377. 10.1080/07362998708809124MathSciNetView ArticleMATHGoogle Scholar
- Cuzick J: A strong law for weighted sums of i.i.d. random variables. Journal of Theoretical Probability 1995, 8(3):625–641. 10.1007/BF02218047MathSciNetView ArticleMATHGoogle Scholar
- Wu WB: On the strong convergence of a weighted sum. Statistics & Probability Letters 1999, 44(1):19–22. 10.1016/S0167-7152(98)00287-9MathSciNetView ArticleMATHGoogle Scholar
- Sung SH: Strong laws for weighted sums of i.i.d. random variables. Statistics & Probability Letters 2001, 52(4):413–419. 10.1016/S0167-7152(01)00020-7MathSciNetView ArticleMATHGoogle Scholar
- Sung SH: Strong laws for weighted sums of i.i.d. random variables. II. Bulletin of the Korean Mathematical Society 2002, 39(4):607–615.MathSciNetView ArticleMATHGoogle Scholar
- Lu CR, Lin ZY: Limit Theory for Mixing Dependent Sequences. Science Press, Beijing, China; 1997.Google Scholar
This article is published under license to BioMed Central Ltd. 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.