On Complete Convergence for Weighted Sums of ρ ∗-Mixing Random Variables

and Applied Analysis 3 random variables to the case of ρ-mixing random variables. In addition, we will present some results on complete convergence for weighted sums of ρ-mixing random variables. Our main results are as follows. Theorem 7. Let {X n , n ≥ 1} be a sequence of ρ-mixing random variables, which is stochastically dominated by a random variableX with EX n = 0, and EX2 < ∞. Let {a ni , 1 ≤ i ≤ n, n ≥ 1} be a triangular array of constants such that

Definition 1.A sequence {  ,  ≥ 1} of random variables is said to be  * -mixing if there exists  ∈ N such that  * () < 1.
It is easily seen that  * -mixing (i.e., ρ-mixing) sequence contains independent sequence as a special case. * -mixing random variables were introduced by Bradley [13] and many applications have been found.ρ-mixing is similar to -mixing, but both are quite different.Many authors have studied this concept providing interesting results and applications.See, for example, Bradley [13] for the central limit theorem, Bryc and Smole ń ski [14], Peligrad and Gut [15], and Utev and Peligrad [16] for moment inequalities, Gan [17], Kuczmaszewska [18], Wu and Jiang [19] and Wang et al. [20,21] for almost sure convergence, Peligrad and Gut [15], Cai [22], Kuczmaszewska [23], Zhu [24], An and Yuan [25], Wang et al. [26], and Sung [27] for complete convergence, Peligrad [28] for invariance principle, Wu and Jiang [29] for strong limit theorems for weighted product sums of  * -mixing sequences of random variables, Wu and Jiang [30] for Chover-type laws of the -iterated logarithm, Wu [31] for strong consistency of estimator in linear model, Wang et al. [32] for complete consistency of the estimator of nonparametric regression models, Wu et al. [33] and Guo and Zhu [34] for complete moment convergence, and so forth.When these are compared with the corresponding results of independent random variable sequences, there still remains much to be desired.So studying the limit behavior of  *mixing random variables is of interest.
The following concepts of slowly varying function and stochastic domination will be used in this work.Definition 2. A real-valued function (), positive and measurable on (0, ∞), is said to be slowly varying if for each  > 0.
This work is organized as follows.Some important lemmas are presented in Section 2. Main results and their proofs are provided in Section 3.

Preliminaries
In this section, we will present some important lemmas which will be used to prove the main results of the paper.The first one is the Rosenthal type maximal inequality for  * -mixing random variables, which was obtained by Utev and Peligrad [16].
Lemma 4 (cf.Utev and Peligrad [16]).For a positive integer  ≥ 1 and positive real numbers  ≥ 2 and 0 ≤  < The next one is a basic property for stochastic domination.For the details of the proof, one can refer to Wu [35] or Tang [36].
The last one is the basic properties for slowly varying function, which was obtained by Bai and Su [37].

Main Results and Their Proofs
In this section, we will generalize and improve the result of Theorem A for independent and identically distributed random variables to the case of  * -mixing random variables.In addition, we will present some results on complete convergence for weighted sums of  * -mixing random variables.
Our main results are as follows.
Remark 8.The key to the proof of Theorem 7 is the Rosenthal type maximal inequality for  * -mixing sequences (i.e., Lemma 4).Similar to the proof of Theorem 7, we have the following result.
Remark 11.There are many sequences of random variables satisfying (12), such as negatively associated (NA, in short) sequence (see Shao [38]), negatively superadditivedependent (NSD, in short) sequence (see Wang et al. [39]), asymptotically almost negatively associated (AANA, in short) sequence (see Yuan and An [40]), -mixing sequence (see Wang et al. [12]), and  * -mixing sequence (see Utev and Peligrad [16]).Comparing Theorems 7 and 9 with Theorem A, conditions ( 1) and () in Theorem A can be removed.In addition, the condition "identical distribution" in Theorem A can be weakened by "stochastic domination." Hence, the results of Theorem 7 and Theorem 9 generalize and improve the corresponding one of Theorem A.
In the following, we will present some results on complete convergence for weighted sums of  * -mixing random variables.The main ideas are inspired by Kuczmaszewska [41].The first one is a very general result of complete convergence for weighted sums of  * -mixing random variables, which can be applied to obtain other result's of complete convergence, such as Baum-Katz type complete convergence and Hsu-Robbins type complete convergence.
In what follows, we will give some applications for Theorem 12.
Let () be the distribution function of .It follows, by Lemmas 5 and 6 and the inequality above, that where the fourth inequality above is followed by the proof of Corollary 2.8 in Kuczmaszewska [41].This shows that (23) holds.By Lemma 5,   -inequality, and Markov's inequality, we can see that where the last inequality is followed by the proof of Corollary 2.8 in Kuczmaszewska [41].Hence, condition ( 24) is satisfied.The proof will be completed if we show that This completes the proof of the theorem.
Remark 15.Noting that, for typical slowly varying functions () = 1 and () = log , we can get the simpler formulas in the above theorems.
The desired result (35) follows from the statements above and Theorem 12 immediately.The proof is completed.
1, there exists a positive constant  = (, , ) such that if {  ,  ≥ 1} is a sequence of random variables with  * () < ,   = 0, and |  |  < ∞ for every  ≥ 1, then for all  ≥ 1, 5.Let {  ,  ≥ 1} be a sequence of random variables which is stochastically dominated by a random variable .For any  > 0 and  > 0, the following two statements hold: Let {  ,  ≥ 1} be a sequence of random variables, which is stochastically dominated by a random variable  with   = 0 and  2 < ∞.Let {  ,  ≥ 1} be a sequence of constants such that ∑  =1  2  = (log −1− ) for some  > 0. Suppose that there exists a positive constant  such that (10)rem 9. Let {  ,  ≥ 1} be a sequence of random variables, which is stochastically dominated by a random variable  with   = 0 and  2 < ∞.Let {  , 1 ≤  ≤ ,  ≥ 1} be a triangular array of constants such that Then(10)holds for any  > 0.If the array of constants {  , 1 ≤  ≤ ,  ≥ 1} is replaced by the sequence of constants {  ,  ≥ 1}, then we can get the following strong law of large numbers for weighted sums ∑  =1     .The proof is standard, so we omit the details.Theorem 10.