Complete convergence for Sung’s type weighted sums of END random variables
© Zhang; licensee Springer 2014
Received: 14 May 2014
Accepted: 11 September 2014
Published: 24 September 2014
In this paper, the author studies the complete convergence results for Sung’s type weighted sums of sequences of END random variables and obtains some new results. These results extend and improve the corresponding theorems of Sung (Discrete Dyn. Nat. Soc. 2010:630608, 2010, doi:10.1155/2010/630608).
1 Introduction and main results
Recently, Sung  obtained a complete convergence result for weighted sums of identically distributed -mixing random variables (we call these Sung’s type weighted sums).
Conversely, (1.2) implies and if (1.2) holds for any array with (1.1) for some .
In this paper, we will extend Theorem A under the END setup. We firstly introduce the concept of END random variables.
hold for every sequence of real numbers.
The concept was introduced by Liu . When , the notion of END random variables reduces to the well-known notion of so-called negatively dependent (ND) random variables, which was firstly introduced by Embrahimi and Ghosh ; some properties and limit results can be found in Alam and Saxena , Block et al. , Joag-Dev and Proschan , and Wu and Zhu . As is mentioned in Liu , the END structure is substantially more comprehensive than the ND structure in that it can reflect not only a negative dependence structure but also a positive one, to some extent. Liu  pointed out that the END random variables can be taken as negatively or positively dependent and provided some interesting examples to support this idea. Joag-Dev and Proschan  also pointed out that negatively associated (NA) random variables must be ND and ND is not necessarily NA, thus NA random variables are END. A great number of articles for NA random variables have appeared in the literature. But very few papers are written for END random variables. For example, for END random variables with heavy tails Liu  obtained the precise large deviations and Liu  studied sufficient and necessary conditions for moderate deviations, and Qiu et al.  and Wu and Guan  studied complete convergence for weighted sums and arrays of rowwise END, and so on.
Now we state the main results; some lemmas and the proofs will be detailed in the next section.
Theorem 1.1 Let and . Let be a sequence of identically distributed END random variables with and . Assume that is an array of real numbers satisfying (1.1) for some . Then (1.2) holds. Conversely, (1.2) implies and if (1.2) holds for any array with (1.1) for some .
Remark 1.1 The tool is the maximal Rosenthal’s moment inequality in the proof of Theorem A. But we do not know whether the maximal Rosenthal’s moment inequality holds or not for an END sequence. So the proof of Theorem 1.1 is different from that of Theorem A.
Remark 1.2 Theorem 1.1 does not discuss the very interesting case: . In fact, it is still an open problem whether (1.2) holds or not even in the partial sums of an END sequence when . But we have the following partial result.
Conversely, (1.3) implies if (1.3) holds for any array with (1.1) for some .
Throughout this paper, C always stands for a positive constant which may differ from one place to another.
2 Lemmas and proofs of main results
To prove the main result, we need the following lemmas.
Lemma 2.1 ()
Let be END random variables. Assume that are Borel functions all of which are monotone increasing (or all are monotone decreasing). Then are END random variables.
holds when .
By Lemma 2.2 and the same argument as Theorem 2.3.1 in Stout , the following lemma holds.
holds when .
Lemma 2.4 Let and . Let be a sequence of identically distributed END random variables with and . Assume that is an array of real numbers satisfying for and . Then (1.2) holds.
Since and , we have . We take t as given such that .
By the same argument as (2.3), we also have .
If , then we can take , in this case in (2.5). Since and , (2.6) still holds. Therefore .
If , then we take , in this case . Since , (2.9) still holds. Therefore, . Similar to the proof of , we also have . Thus, (1.2) holds. □
Lemma 2.5 Let and . Let be a sequence of identically distributed END random variables with . Assume that is an array of real numbers satisfying (1.1) for some and or . Then (1.2) holds.
Similarly, we have .
If , then we take , in this case in (2.13). Since and , (2.14) still holds. Therefore .
If , then we take , in this case in (2.16). Similar to the proof of Lemma 2.4 of Sung , (2.17) still holds. Therefore, . Similar to the proof of , we have . Thus, (1.2) holds. □
Proof of Theorem 1.1 By Lemmas 2.4 and 2.5, the proof is similar to that in Sung , so we omit the details. □
where and .
By Lemma 2.1, is a sequence of identically distributed END with zero mean. Then by taking instead of in (2.18). Hence (2.19) holds.
The proof of (2.20) is the same as that of (2.19).
Necessity. It is similar to the proof of Theorem 2.2 in Sung . Here we omit the details. So we complete the proof. □
The author would like to thank the referees and the editors for the helpful comments and suggestions. The research is supported by the General Project of Science and Technology Department of Hunan Province (2013NK3017) and by the Agriculture Science and Ttechnology Supporting Programme of Hengyang Municipal Bureau of Science and Technology (2013KN36).
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