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On the strong convergence and some inequalities for negatively superadditive dependent sequences
Journal of Inequalities and Applicationsvolume 2013, Article number: 448 (2013)
In this paper, we study the Marcinkiewicz-type strong law of large numbers, Hajek-Renyi-type inequality and other inequalities for negatively superadditive dependent (NSD) sequences. As an application, the integrability of supremum for NSD random variables is obtained. Our results extend the corresponding ones of Christofides and Vaggelatou (J. Multivar. Anal. 88:138-151, 2004) and Liu et al. (Stat. Probab. Lett. 43:99-105, 1999).
Let be a sequence of random variables defined on a fixed probability space . , , . The concept of negatively associated (NA) random variables was introduced by Joag-Dev and Proschan . A finite family of random variables is said to be negatively associated if for every pair of disjoint subsets ,
whenever f and g are coordinatewise nondecreasing such that this covariance exists. An infinite family of random variables is NA if every finite subfamily is NA.
The concept of negatively superadditive dependent (NSD) random variables was introduced by Hu  based on the class of superadditive functions. Superadditive structure functions have important reliability interpretations, which describe whether a system is more series-like or more parallel-like .
Definition 1.1 (Kemperman )
A function is called superadditive if for all , where ∨ is for componentwise maximum and ∧ is for componentwise minimum.
Definition 1.2 (Hu )
A random vector is said to be negatively superadditive dependent (NSD) if
where are independent such that and have the same distribution for each i and ϕ is a superadditive function such that the expectations in (1.1) exist.
Hu  gave an example illustrating that NSD does not imply NA, and Hu posed an open problem whether NA implies NSD. Christofides and Vaggelatou  solved this open problem and indicated that NA implies NSD. Negatively superadditive dependent structure is an extension of negatively associated structure and sometimes more useful than negatively associated structure. Moreover, we can get many important probability inequalities for NSD random variables. For example, the structure function of a monotone coherent system can be superadditive , so inequalities derived from NSD can give one-side or two-side bounds of the system reliability. The notion of NSD random variables has wide applications in multivariate statistical analysis and reliability theory.
Eghbal et al.  derived two maximal inequalities and strong law of large numbers of quadratic forms of NSD random variables under the assumption that is a sequence of nonnegative NSD random variables with for all and some . Eghbal et al.  provided some Kolmogorov inequality for quadratic forms and weighted quadratic forms , where is a sequence of nonnegative NSD uniformly bounded random variables. Shen et al.  obtained the Khintchine-Kolmogorov convergence theorem and strong stability for NSD random variables. Strong convergence for NA sequences and other dependent sequences has been extensively investigated. For example, Sung [9, 10] obtained the complete convergence results for identically distributed NA random variables and -mixing random variables respectively, Zhou et al.  studied complete convergence for identically distributed -mixing random variables under a suitable moment condition, Zhou  discussed complete moment convergence of moving average process under φ-mixing assumptions.
This paper is organized as follows. In Section 2, some preliminary lemmas and inequalities for NSD random variables are provided. In Section 3, Marcinkiewicz-type strong law of large numbers, Hajek-Renyi-type inequalities and the integrability of supremum for NSD random variables are presented. These results extend the corresponding results of Christofides and Vaggelatou  and Liu et al. .
Throughout the paper, are independent such that and have the same distribution for each i. C denotes a positive constant not depending on n, which may be different in various places. represents that there exists a constant such that for all sufficiently large n.
Lemma 2.1 (Hu )
If is NSD, then is NSD for any , .
Lemma 2.2 (Hu )
Let be an NSD random vector, and let be an independent vector such that and have the same distribution for each i. Then, for any nondecreasing convex function f and ,
Lemma 2.3 (Toeplitz lemma)
Let be a sequence of real numbers. If and (finite), then .
Lemma 2.4 (Shen et al. )
Let be NSD random variables with mean zero and finite second moments. Then, for ,
Lemma 2.5 (Shen et al. )
Let be a sequence of NSD random variables. Assume that
then converges almost surely.
3 Main results
Theorem 3.1 (Marcinkiewicz-type strong law of large numbers for NSD)
Let be a sequence of NSD identically distributed random variables. There exists some such that
If , we further assume that . Then
Proof Let . By (3.1),
Therefore follows from the Borel-Cantelli lemma. Thus (3.2) is equivalent to the following:
So, in order to prove (3.2), we need only to prove
Firstly, we prove (3.3). According to Lemma 2.5, it suffices to prove
By and (3.1),
Secondly, we prove (3.4).
If , then
It is easy to see that
Therefore, it has . By Lemma 2.3, we can get (3.4) immediately.
If , by (3.1)
we can get (3.4) from Kronecker’s lemma.
If , by and (3.1),
Thus we can also get (3.4) from Kronecker’s lemma. □
Theorem 3.2 Let be a sequence of NSD random variables with mean zero and finite second moments. Let be a sequence of positive nondecreasing real numbers. Then, for any and ,
Proof Without loss of generality, we may assume that for all . Let . For , define . Then may be an empty set. For , we let and be the index of the last nonempty set . Obviously, if and . It is easy to see that if . By Markov’s inequality and Lemma 2.4, we have
Now we estimate . Let . Then follows from the definition of . Therefore
Combining (3.6) with (3.7), we obtain (3.5) immediately. □
Theorem 3.3 Let be a sequence of NSD random variables with mean zero and finite second moments. Let be a sequence of positive nondecreasing real numbers. Then, for any and for any positive integer ,
Proof Observe that
For I, by Markov’s inequality, we have
For II, we will apply Theorem 3.2 to and . According to Lemma 2.1, is NSD. Noting that
thus, by Theorem 3.2, we obtain
Therefore the desired result (3.8) follows from (3.9)-(3.11) immediately. □
Theorem 3.4 Let be a sequence of positive nondecreasing real numbers. Let be a sequence of NSD random variables with mean zero and . If , then
Proof For , , by Theorem 3.2, it follows that
Example 3.5 Similar to the proof of Theorem 2.1 of Shen et al. , we can get the following inequalities for NSD random variables.
Let . Suppose that is a sequence of NSD random variables with mean zero and . Then, for ,
where is a positive constant depending only on p.
In fact, taking in (2.1), we can get (3.13) similar to the proof of Theorem 2.1 in . Following the same line of arguments as in the proof of Theorem 2 of Shao , we can obtain (3.14) and (3.15) immediately.
Equation (3.15) provides the Rosenthal-type inequality for NSD random variables, which is one of the most interesting inequalities in probability theory. Hu  pointed out that the Rosenthal-type inequality remains true for NSD random variables. Here we give the proof of this inequality and provide other two inequalities.
Remark 3.6 Theorem 3.1 provides Marcinkiewicz-type strong law of large numbers for NSD random variables. Marcinkiewicz strong law of large numbers for independent sequences and other dependent sequences was studied by many authors. See, for example, Lin et al.  for independent sequences, Wu and Jiang  for -mixing sequences, Wu  for PNQD sequences, Wang et al.  for Martingale difference sequences, and so forth.
Remark 3.7 If is a sequence of NA random variables with finite second moments, Chrisofides and Vaggelatou  obtained the following result: for any ,
So, if we further assume that , , then
If is a sequence of NA, then is a sequence of NSD, by and (3.8)
Hence Theorem 3.2 and Theorem 3.3 extend the Hajek-Renyi-type inequalities for NA random variables (Chrisofides and Vaggelatou ) and the factors are improved. As an application of Theorem 3.2, Theorem 3.4 extends the result of NA random variables (Liu et al. ) to NSD random variables.
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Article is supported by the National Natural Science Foundation of China (11171001, 11201001), Doctoral Research Start-up Funds Project of Anhui University, Key Program of Research and Development Foundation of Hefei University (13KY05ZD) and Natural Science Foundation of Anhui Province (1208085QA03).
The authors declare that they have no competing interests.
All authors read and approved the final manuscript.