© Wang Xuejun et al. 2010
Received: 4 February 2010
Accepted: 11 June 2010
Published: 6 July 2010
-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.
2. Main Results and Their Proofs
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).
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