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On complete moment convergence for arrays of rowwise pairwise negatively quadrant dependent random variables
Journal of Inequalities and Applications volume 2019, Article number: 46 (2019)
Abstract
In this paper, we establish some results on the complete moment convergence for weighted sums of pairwise negatively quadrant dependent (PNQD) random variables. The obtained results improve the corresponding ones of Ko (Stoch. Int. J. Probab. Stoch. Process. 85:172–180, 2013).
1 Introduction
The following concept of PNQD random variables was introduced by Lehmann [2].
Definition 1.1
A sequence \(\{X_{n}, n\geq 1\}\) of random variables is said to be pairwise negatively quadrant dependent (PNQD) if for any \(r_{i}\), \(r_{j}\) and \(i\neq j\),
Negative quadrant dependence is shown to be a stronger notion of dependence than negative correlation but weaker than negative association. The convergence properties of NQD random sequences have been studied in many papers. We refer to Wu [3] for Kolmogorov-type three-series theorem, Matula [4] for the Kolmogorov-type strong law of large numbers, Jabbari [5] for the almost sure limit theorems for weighted sums of pairwise NQD random variables under some fragile conditions, Li and Yang [6], Wu [7], and Xu and Tang [8] for strong convergence, Gan and Chen [9] for complete convergence and complete moment convergence, Wu and Guan [10] for a mean convergence theorem and weak laws of large numbers for dependent random variables, and so on.
The concept of complete convergence of a sequence of random variables was first given by Hsu and Robbins [11].
Definition 1.2
A sequence of random variables \(\{U_{n}, n \in N\}\) is said to converge completely to a constant a if for any \(\varepsilon >0\),
By the Borel–Cantelli lemma this result implies that \(U_{n}\rightarrow a\) almost surely. Therefore, the complete convergence is a very important tool in establishing the almost sure convergence of sums of random variables and weighted sums of random variables.
Recently, Ko [1] proved the following complete convergence theorem for arrays of PNQD random variables.
Theorem A
Let \(\{X_{nj}, 1\leq j\leq b_{n}, n\geq 1\}\) be an array of rowwise and PNQD random variables with mean zero, and let \(\{a_{nj}, j\geq 1, n\geq 1\}\) be an array of positive numbers. Let \(\{b_{n}, n\geq 1\}\) be a nondecreasing sequence of positive numbers. Assume that, for some \(0< t<2\) and all \(\varepsilon >0\),
and
Then
Chow [12] was the first who showed the complete moment convergence for a sequence of independent and identically distributed random variables by generalizing the result of Baum and Katz [13]. The concept of complete moment convergence is as follows.
Definition 1.3
Let \(\{Z_{n}, n\geq 1\}\) be a sequence of random variables, and let \(a_{n}> 0\), \(b_{n}> 0\), and \(q> 0\). If for any \(\varepsilon >0\),
then this is called the complete moment convergence.
It is easily seen that complete moment convergence is stronger than complete convergence. There are many papers on complete moment convergence; see, for example, Sung [14] for independent random variables, Wang and Hu [15] for the maximal partial sums of a martingale difference sequence, Shen et al. [16] for arrays of rowwise negatively superadditive dependent (NSD) random variables. Wu et al. [17] for arrays of rowwise END random variables, Wu [7] for negatively associated random variables, Wu et al. [18] for weighted sums of weakly dependent random variables, Wang et al. [19] for double indexed randomly weighted sums and its applications, Wu and Wang [20] for a class of dependent random variables, and so forth.
In this work, we improve Theorem A from complete convergence to complete moment convergence for PNQD random variables under some stronger conditions. In addition, we obtain some much stronger conclusions under the same conditions of the corresponding theorems in Ko [1].
Throughout this paper, the symbol C always stands for a generic positive constant which may differ from one place to another. By \(I(A)\) we denote the indicator function of a set A. We also denote \(x_{+}=xI(x\geq 0)\).
2 Main results
Now we state the main results of this paper. The proofs are given in next section.
Theorem 2.1
Let \(\{X_{nj}, 1\leq j\leq b_{n}, n\geq 1\}\) be an array of rowwise and PNQD random variables with mean zero, and let \(\{a_{nj}, j\geq 1, n\geq 1\}\) be an array of positive numbers. Let \(\{b_{n}, n\geq 1\}\) be a nondecreasing sequence of positive numbers. Assume that, for some \(0< t<2\) and all \(\varepsilon >0\),
and
Then, for all \(\varepsilon >0\),
Remark 2.1
Let \(H_{nj}=\sum_{j=1}^{k} (a_{nj}X_{nj}-a_{nj}EX _{nj} I[|a_{nj} X_{nj}| < \varepsilon b_{n}^{\frac{1}{t}}] ) \). Note that
Thus (2.3) is much stronger than (1.3).
Theorem 2.2
Let \(\{X_{nj}, 1\leq j\leq b_{n}, n\geq 1\}\) be an array of rowwise and PNQD random variables with mean zero, and let \(\{a_{nj}, j\geq 1, n\geq 1\}\) be an array of positive numbers. Let \(\{b_{n}, n\geq 1\}\) be a nondecreasing sequence of positive numbers. Assume that, for some sequence \(\{\lambda _{n}, n\geq 1\}\) with \(0<\lambda _{n}\leq 1\), we have \(E|X_{nj}|^{1+\lambda _{n}}<\infty \) for \(1\leq j\leq b_{n}\), \(n\geq 1\). If for some sequence \(\{c_{n}, n \geq 1\}\) of positive real numbers and \(0< t<2\),
then for any \(\varepsilon >0\),
Remark 2.2
Noting that the conditions of Theorem 2.2 are the same as in Theorem 3.2 in Ko [1], we have
Therefore (2.5) is much stronger than (3.8) of Theorem 3.2 in Ko [1]. To sum up, Theorem 2.2 improves Theorem 3.2 in Ko [1].
Corollary 2.3
Let \(\{X_{nj}, 1\leq j\leq b_{n}, n\geq 1\}\) be an array of rowwise PNQD random variables, and let \(\{a_{nj}, j \geq 1, n\geq 1\}\) be an array of positive numbers. Let \(h(x)>0\) be a slowly varying function as \(x\rightarrow \infty \), and let \(\alpha > \frac{1}{2}\) and \(\alpha r\geq 1\). Suppose that, for \(0< t<2\), the following conditions hold for any \(\varepsilon >0\):
and
Then, for all \(\varepsilon >0\),
Theorem 2.4
Let \(\{X_{nj}, j\geq 1, n\geq 1\}\) be an array of rowwise identically distributed PNQD random variables with \(E X_{11}=0\), and let \(h(x)>0\) be a slowly varying function as \(x\rightarrow \infty \). If \(E|X_{11}|^{(\alpha r+2)t} h(|X_{11}|^{t})< \infty \) for \(\alpha >\frac{1}{2}\), \(\alpha r\geq 1\), and \(0< t<2\), then
Remark 2.3
Noting that the conditions of Theorem 2.2 are the same as in Theorem 3.4 in Ko [1], we have
Therefore (2.9) is much stronger than (3.13) of Theorem 3.4 in Ko [1]. Theorem 2.4 improves Theorem 3.4 in Ko [1].
Corollary 2.5
Let \(\{X_{nj}, 1\leq j\leq b_{n}, n\geq 1\}\) be an array of rowwise and PNQD random variables with mean zero, and let \(\{a_{nj}, j\geq 1, n\geq 1\}\) be an array of positive numbers. Let \(\{b_{n}, n\geq 1\}\) be a nondecreasing sequence of positive numbers, and let \(\{c_{n}, n\geq 1\}\) be a sequence of positive numbers. Assume that, for all \(\varepsilon >0\),
and
Then, for all \(\varepsilon >0\),
Corollary 2.6
Let \(\{X_{nj}, 1\leq j\leq b_{n}, n\geq 1\}\) be an array of rowwise and PNQD random variables with mean zero and finite variances. Let \(\{a_{nj}, j\geq 1, n\geq 1\}\) be an array of positive numbers satisfying
for some \(0<\delta <1\). Then, for all \(\varepsilon >0\) and \(\alpha >0\),
Remark 2.4
Note that
Therefore (2.14) is much stronger than (3.18) of Corollary 3.6 in Ko [1].
3 The proofs
To prove our results, we need some lemmas. The first one is the basic property for PNQD random variables, which can be referred to Lehmann [2].
Lemma 3.1
Let \(\{X_{n}, n\geq 1\}\) be a sequence of PNQD random variables, and let \(\{f_{n}, n\geq 1\}\) be a sequence of nondecreasing functions. Then \(\{f_{n}(X_{n}), n\geq 1\}\) is still a sequence of PNQD random variables.
The next lemma comes from Wu [3] and plays an essential role to prove the result of the paper.
Lemma 3.2
Let \(\{X_{n}, n\geq 1\}\) be a sequence of PNQD random variables with mean zero and finite second moments. Then
A positive measurable function \(h(x)\) on \([a, \infty )\) for some \(a>0\) is said to be slowly varying as \(x\rightarrow \infty \) if
The last lemma can be found in Wu [21].
Lemma 3.3
If \(h(x)>0\) is a slowly varying function as \(x\rightarrow \infty \), then
-
(i)
\(\lim_{x\rightarrow \infty } \sup_{2^{k}\leq x<2^{k+1}} h(x)/ h \bigl(2^{k}\bigr) =1\), and
-
(ii)
\(c_{1} 2^{k} h( \varepsilon 2^{k} ) \leq \sum_{j=1}^{k} 2^{j} h ( \varepsilon 2^{j})\leq c_{2} 2^{k} h ( \varepsilon 2^{k})\)
for all \(r>0, \varepsilon >0\), and positive integers k and some positive constants \(c_{1}\) and \(c_{2}\).
Proof of Theorem 2.1
Let \(S_{j}= \sum_{j=1}^{k} (a_{nj}X _{nj}-a_{nj}EX_{nj} I[|a_{nj} X_{nj}| < \varepsilon b_{n}^{ \frac{1}{t}}]) \). For any fixed \(\varepsilon >0\),
Obviously, we have \(I_{1}<\infty \) by Theorem A. Hence we need only to prove \(I_{2}<\infty \). Clearly,
Then we can get
Firstly, we will prove that \(I_{3}<\infty \). Noting that
by (2.1) we have
To prove that \(I_{4}<\infty \), let
We have
Then we have
For \(I_{5}\), by the Markov inequality and (2.1) we have
Now consider \(I_{6}\). By the Markov inequality and Lemma 3.2 we have
Firstly, we will prove that \(I_{8}<\infty \). By (2.1) we have
Next, consider \(I_{7}<\infty \). We have
By (2.2) it is easy to see that
By the Markov inequality and (2.1) we have
This completes the proof of the theorem. □
Proof of Theorem 2.2
We estimate
and
Hence conditions (2.1) and (2.2) of Theorem 2.1 are satisfied. Since \(EX_{nj}=0\), we get
and thus (2.5) is completed. □
Proof of Corollary 2.3
Let \(c_{n}= n^{\alpha r-2} h(n) \) and \(b_{n}=n\). Then, by Theorem 2.1, (2.8) is completed. □
Proof of Theorem 2.4
For \(a_{nj}=1\), \(j\geq 1\), \(n\geq 1\), by Lemma 3.3 we have
and
and thus (2.6) and (2.7) are satisfied. Then, to complete the proof, it remains to show that, for \(1\leq j\leq n\), \(n^{-\frac{1}{t}} j |EX _{11} I[|X_{11}| < \varepsilon n^{\frac{1}{t}}] | \rightarrow 0\) as \(n \rightarrow \infty \).
If \((\alpha r+2)t <1\), then we have, as \(n \rightarrow \infty \),
and if \((\alpha r+2)t \geq 1\), then since \(|EX_{11}|=0\), we have, as \(n \rightarrow \infty \),
Hence the proof of Theorem 2.4 is completed. □
Proof of Corollary 2.5
Taking \(\log _{2} b_{n}\) instead of \(b_{n}^{\frac{1}{t}}\) in Theorem 2.1, we get (2.12). □
Proof of Corollary 2.6
In Theorem 2.1, let \(c_{n}= n^{2( \alpha -1)}\) and \(b_{n}^{-\frac{1}{t}}=n^{-\alpha } (\log _{2} n)^{-1}\). By (2.13) we have
and
Hence conditions of (2.1) and (2.2) of Theorem 2.1 are satisfied. Since \(EX _{nj}=0\), by (2.13) we get
and thus (2.14) is completed. □
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Acknowledgements
The authors are grateful to the referee for carefully reading the manuscript and for providing some comments and suggestions, which led to improvements in this paper.
Funding
The research of M. Ge is partially supported by the NSF of Anhui Educational Committe (KJ2017B11, KJ2018A0428). The research of Z. Dai is partially supported by the NSF of Anhui Educational Committe (KJ2017B09). The research of Y. Wu is partially supported by the Natural Science Foundation of Anhui Province (1708085MA04), the Key Program in the Young Talent Support Plan in Universities of Anhui Province (gxyqZD2016316), and Chuzhou University scientific research fund (2017qd17).
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Ge, M., Dai, Z. & Wu, Y. On complete moment convergence for arrays of rowwise pairwise negatively quadrant dependent random variables. J Inequal Appl 2019, 46 (2019). https://doi.org/10.1186/s13660-019-1995-9
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DOI: https://doi.org/10.1186/s13660-019-1995-9