On the strong convergence for weighted sums of negatively superadditive dependent random variables

In this article, some strong convergence results for weighted sums of negatively superadditive dependent random variables are studied without assumption of identical distribution. The results not only generalize the corresponding ones of Cai (Metrika 68:323-331, 2008) and Sung (Stat. Pap. 52:447-454, 2011), but also extend and improve the corresponding one of Chen and Sung (Stat. Probab. Lett. 92:45-52, 2014).

where X *  , X *  , . . . , X * n are independent such that X * i and X i have the same distribution for each i and φ is a superadditive function such that the expectations in (.) exist. A sequence of random variables {X n ; n ≥ } is said to be NSD if for every n ≥ , (X  , X  , . . . , X n ) is NSD.
The concept of NA was given by Joag-Dev and Proschan [], and the concept of NSD was introduced by Hu [], which was based on the class of superadditive functions. Hu [] gave an example illustrating that NSD random variables are not necessarily NA, and left an open problem whether NA random variables implies NSD. Christofides and Vaggelatou [] solved this open problem and showed that NA implies NSD. Thus, it is shown that NSD is much weaker than NA. Because of the wide application of NSD random variables, many authors have studied this concept and obtained some interesting results and applications. For example, we refer to [-]. Hence, it is of important significance to extend the limit properties of NA to the case of NSD random variables.
The concept of complete convergence was introduced by Hsu and Robbins [] as follows. A sequence of random variables {X n ; n ≥ } is said to converge completely to a constant λ if ∞ n= P |X n -λ| > ε < ∞ for all ε > . (.) In view of the Borel-Cantelli lemma, the sequence of random variables {X n ; n ≥ } converging completely to a constant λ implies X n → λ almost surely (a.s.). Therefore, the complete convergence of random variables is a very important tool in establishing almost sure convergence. The first results concerning complete convergence for normed sums of random variables were due to Hsu   for some  < α ≤ . Let b n = n /α (log n) /γ for some γ > . Furthermore, suppose that EX =  when  < α ≤ . Then: In the case α > γ , Chen and Sung [] studied the complete convergence for weighted sums of NA random variables under the moment condition E|X| α /(log( + |X|)) α/γ - < ∞, which is weaker than Theorem .. Li [], in this paper, we will further study the complete convergence for weighted sums of NSD random variables. Some complete convergence results for the maximum weighted sums of NSD random variables are obtained without the assumption of an identical distribution. As an application, the Marcinkiewicz-Zygmund type strong law of large numbers for weighted sums of NSD random variables is obtained. Our results not only generalize the corresponding ones of Cai [] and Sung [], but they also extend and improve the corresponding one of Chen and Sung [].

Preliminaries
Throughout this paper, C represents a generic positive constant whose value may change from one appearance to the next, and a n = O(b n ) means a n ≤ Cb n . Let I(A) be the indicator function of the set A.
Definition . A sequence of random variables {X n ; n ≥ } is said to be stochastically dominated by a random variable X if there exists a positive constant C such that for all x ≥  and n ≥ .
In order to prove our main results, we introduce the following lemmas.
and, for p > ,

Lemma . (Sung []) Let X be a random variable and {a
) Let X be a random variable and {a ni ;  ≤ i ≤ n, n ≥ } be an array of constants satisfying a ni =  or |a ni | > , and n i= |a n ≥ } be a sequence of random variables which is stochastically dominated by a random variable X. For any u > , t >  and n ≥ , the following two statements hold:

Main results and proofs
Now we state and prove our main results.
Theorem . Let {X n ; n ≥ } be a sequence of NSD random variables which is stochastically dominated by a random variable X, and b n = n /α (log n) /γ for some  < α ≤  and γ > . Let {a ni ;  ≤ i ≤ n, n ≥ } be an array of constants satisfying n i= |a ni | γ = O(n). Assume further that EX n =  when  < α ≤ . Then: Theorem . Let {X n ; n ≥ } be a sequence of NSD random variables which is stochastically dominated by a random variable X, and b n = n /α (log n) /γ for some  < α ≤  and γ > . Let {a ni ;  ≤ i ≤ n, n ≥ } be an array of constants satisfying n i= |a ni | α = O(n). Assume further that EX n =  when Remark . In Theorem . and Theorem ., we use different methods from those of Sung [] and Chen and Sung [] to prove the results, and obtain some strong convergence results for weighted sums of NSD random variables without assumptions of identical distribution. The obtained theorems not only extend the corresponding results of Cai [] and Sung [] and Chen and Sung [] to the case of NSD random variables, but they also improve them.
Proof of Theorem . Without loss of generality, we suppose that a ni > . For ∀i ≥ , define It is easy to check that, for all ε > , If  < α ≤ , then by EX n = , Lemma ., Definition ., the C r inequality, the Markov inequality and the Hölder inequality, we get If  < α ≤ , then by Lemma ., Definition ., the C r inequality and the Markov inequality, we get again It immediately follows from (.) and (.), that (.) holds. Hence, for n large enough, Then, to prove (.), it suffices to prove that By Lemma ., we can easily obtain For fixed n ≥ , it is easily seen that {Y i ;  ≤ i ≤ n} is still a sequence of NSD random variables by Lemma .. Hence, for p > , it follows from Lemma ., the Markov inequality and the Jensen inequality that Firstly, we prove J  < ∞. By Lemma ., we obtain Actually, by Lemma ., we can directly obtain J  < ∞. Hence, we only need to prove J  < ∞ in the following two cases. (i) If α < γ , take p > max{, γ }, then by n i= |a ni | γ ≤ Cn and E|X| γ < ∞ it follows that (ii) If α = γ , we need to divide {a ni ;  ≤ i ≤ n, n ≥ } into three subsets: {a ni : |a ni | ≤ /(log n) t }, {a ni : /(log n) t < |a ni | ≤ } and {a ni : |a ni | > }, where t = /(pα). Then we obtain Obviously, by Lemma ., we directly obtain J  ≤ E|X| α log ( + |X|) < ∞. It follows from i:|a ni |≤/(log n) t |a ni | α ≤ Cn(log n) -tα and E|X| α < ∞ that It follows from i:/(log n) t <|a ni |≤ |a ni | p ≤ Cn, E|X| α < ∞ and t = /(pα) for p > ,  < α ≤  that Therefore, by (.)-(.), we can see that J  < ∞. Finally, we prove J  < ∞. Actually, take p > max{, γ α }, then by Lemma ., the Markov inequality and E|X| γ < ∞, we get Thus, the proof of Theorem . is completed.
Proof of Theorem . Without loss of generality, we suppose that a ni > . For ∀i ≥ , define It is easy to check that, for all ε > , To prove (.), it suffices to show that We first prove (.). Note that By the Markov inequality, we get for any  < θ < α and It is easy to show that Then, (.) holds by (.)-(.). Now we prove (.) in the following two cases.
(i) If  < α ≤ , similar to the proof of (.), we have Note that for any  < θ < α and By the Markov inequality, the C r inequality and (.)-(.), we obtain Ea ni X i I |a ni X i | ≤ b n → , as n → ∞.
(  .   ) By EX n = , we have and E|a ni X i |I |a ni X i | > b n , |X i | > b n ≤ Cb - n n -+/α (log n) /γ - Thus, the proof of Theorem . is completed.

Conclusions
In this paper, we use different methods from those of Sung [] and Chen and Sung [] to prove the results, and we obtain some strong convergence results for weighted sums of NSD random variables without the assumption of an identical distribution. Our results extend and improve the corresponding ones of Cai [] and Sung [] and Chen and Sung [] to the case of NSD random variables.