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Complete moment convergence of extended negatively dependent random variables
Journal of Inequalities and Applications volume 2020, Article number: 150 (2020)
Abstract
In this paper, some results on the complete moment convergence of extended negatively dependent (END) random variables are established. The results in the paper improve and extend the corresponding ones of Qiu et al. (Acta Math. Appl. Sin. 40(3):436–448, 2017) under some weaker conditions. Our results also improve and extend the related known works in the literature.
1 Introduction
In many statistical models, it is not reasonable to assume that random variables are independent, and so it is very meaningful to extend the concept of independence to dependence cases. One important dependence sequence of these dependences is extended negatively dependent (END) random variables, we recall the concept of END random variables as follows.
Definition 1.1
The random variables \(\{X_{n},n\geq 1\}\) are said to be extended negatively dependent (END) random variables if there exists a positive constant \(M>0\) such that both
and
hold for each \(n\geq 1\) and all real \(x_{1},x_{2},\ldots,x_{n}\).
The concept of END random variables was introduced by Liu [2]. Obviously, END random variables (\(M=1\)) imply NOD (negatively orthant dependent) random variables (Joag-Dev and Proschan [3]). Liu [2] pointed out that the END random variables are more comprehensive, and they can reflect not only negative dependence random variables but also positive ones, to some extent. Joag-Dev and Proschan [3] once pointed out that NOD random variables imply NA (negatively associated) random variables, but NA random variables do not imply NOD random variables, so END random variables imply NA random variables. Thus, it is interesting to investigate convergence properties for END random variables.
After the appearance of Liu [2], many scholars have focused on the properties of END random variables, and a lot of results have been gained. For example, Liu [4] studied necessary and sufficient conditions for moderate deviations of dependent random variables with heavy tails; Chen et al. [5] established strong law of large numbers for END random variables; Wu and Guan [6] presented convergence properties of the partial sums of END random variables; Shen [7] presented probability inequalities for END sequence and their applications; Wang and Wang [8] investigated large deviations for random sums of END random variables; Wang et al. and Qiu et al. [9–13] studied complete convergence of END random variables, etc.
The complete convergence plays a very important role in the probability theory and mathematical statistics. The concept of complete convergence was introduced by Hsu and Robbins [14] as follows: A sequence \(\{U_{n},n\geq 1\}\) of random variables is said to converge completely to a constant θ if, for \(\forall \varepsilon >0\), \(\sum_{n=1}^{\infty }P(\vert U_{n}-\theta \vert > \varepsilon ) <\infty \). In view of the Borel–Cantelli lemma, the complete convergence implies that \(U_{n}\rightarrow \theta \) almost surely. Therefore, complete convergence is a very important tool in establishing almost sure convergence for partial of random variables as well as weighted sums of random variables.
Let \(\{X_{n},n\geq 1\}\) be a sequence of random variables, \(a_{n} > 0\), \(b_{n} > 0\), \(\gamma > 0\). If for \(\forall \varepsilon >0\), \(\sum_{n=1}^{\infty }a_{n}E\{b^{-1}_{n}\vert X_{n}\vert - \varepsilon \}_{+}^{\gamma } <\infty \), then \(\{X_{n},n\geq 1\}\) is called the complete moment convergence (Chow [15]). It is well known that complete moment convergence implies complete convergence, i.e., the complete moment convergence is more general than complete convergence. The following result is from Chow [15].
Theorem A
Let\(r>1\), \(1\leq p<2\), \(\{X,X_{n}, n \geq 1\}\)be a sequence of independent identically distributed random variables and\(EX_{1}=0\), if\(E\{\vert X_{1}\vert ^{rp}+\vert X_{1}\vert \log (1+\vert X_{1}\vert )\}<\infty \), then
It should be noted that Theorem A has been extended and improved by many scholars (see [16–19]).
Recently, Chen and Sung [20] obtained complete and complete moment convergence of ρ-mixing random variables, and Qiu et al. [1] obtained the following complete moment convergence for weighted sums of END random variables.
Theorem B
Let\(r>1\), \(1\leq p<2\), \(\lambda >0\), \(\alpha >1\), \(\beta >1\)with\(1/\alpha +1/\beta =1/p\). Let\(\{a_{ni}, 1\leq i\leq n, n\geq 1\}\)be an array of constants satisfying
whereDis a positive constant. \(\{X,X_{n}, 1\leq n\}\)is a sequence of identically distributed END random variables with\(EX=0 \). If
then
In this article, our goal is to further study complete moment convergence for weighted sums of END random variables with suitable conditions. By using the truncated method, we obtain a novel result, which extends that in Qiu et al. [1] under some weaker conditions. Our result also improves and extends those in Chen and Sung [20], Sung [21], and Qiu and Xiao [22].
The layout of this paper is as follows. Main results and some lemmas are provided in Sect. 2. Proofs of the main results are given in Sect. 3. Throughout the paper, the symbol C denotes a positive constant, which may take different values in different places. \(I(A)\) is the indicator function of an event A.
2 Main results and some lemmas
Theoremm 2.1
Let\(r>1\), \(1\leq p<2\), \(\lambda >0\), \(\alpha >0\), \(\beta >0\)with\(1/\alpha +1/\beta =1/p\). Let\(\{a_{ni}, 1\leq i\leq n, n\geq 1\}\)be an array of constants satisfying (1.1). \(\{X,X_{n}, {n\geq 1}\}\)is a sequence of identically distributed END random variables with\(EX=0 \). Assume that one of the following conditions holds:
- (1)
If\(\alpha < rp\), then
$$ \textstyle\begin{cases} E \vert X \vert ^{(r-1)\beta }< \infty &\textit{if } \lambda < (r-1)\beta , \\ E \vert X \vert ^{(r-1)\beta }\log (1+ \vert X \vert )< \infty &\textit{if } \lambda = (r-1)\beta , \\ E \vert X \vert ^{\lambda }< \infty &\textit{if } \lambda >(r-1)\beta . \end{cases} $$(2.1) - (2)
If\(\alpha =rp\), then
$$ \textstyle\begin{cases} E \vert X \vert ^{(r-1)\beta }\log (1+ \vert X \vert )< \infty &\textit{if } \lambda \leq (r-1)\beta =rp, \\ E \vert X \vert ^{\lambda }< \infty &\textit{if } \lambda >(r-1)\beta =rp. \end{cases} $$(2.2) - (3)
If\(\alpha >rp\), then
$$ \textstyle\begin{cases} E \vert X \vert ^{rp}< \infty &\textit{if } \lambda < rp, \\ E \vert X \vert ^{rp}\log (1+ \vert X \vert )< \infty &\textit{if } \lambda =rp, \\ E \vert X \vert ^{\lambda } < \infty &\textit{if } \lambda >rp. \end{cases} $$(2.3)
Then
Conversely, if (2.4) holds for any array\(\{a_{ni}, 1\leq i\leq n, n\geq 1\}\)satisfying (1.1), then\(EX=0\), \(E\vert X\vert ^{(r-1)\beta }<\infty \), \(E\vert X\vert ^{rp}<\infty \).
Remark 2.1
The Rademacher–Menshov inequality is only used in the proof process of Theorem 2.1. The results in this paper still hold for random variable satisfying Rosenthal’s inequality. Therefore, our results improve and extend the result of Chen and Sung [20].
Remark 2.2
In this paper, the conditions of Theorem 2.1 are weaker than those in Theorem 1.1 of Qiu et al. [1], and the condition of “if \(\alpha >rp\), assume \(\lambda <\alpha \) (Qiu et al. [1])” is not necessary for (2.4) in our paper. Therefore our results improve and extend the result of Qiu et al. [1]. It is worth pointing out that the method applied in this article is different from that in Qiu et al. [1].
To prove Theorem 2.1 of the paper, we need the following important lemmas.
Lemma 2.1
(Qiu [22]; Rademacher–Menshov inequality)
Let\(p>1\), \(\{X_{n},n\geq 1\}\)be a sequence of END random variables with\(EX_{n}=0\)and\(E\vert X_{n}\vert ^{p}<\infty \). Then there exists a positive constant\(C_{p}\)only depending onpsuch that
Lemma 2.2
(Qiu [22])
Let\(p\geq 1\), \(\{X_{n},n\geq 1\}\)be a sequence of END random variables with\(EX_{n}=0\)and\(E\vert X_{n}\vert ^{p}<\infty \). Then there exists a positive constant\(C_{p}\)only depending onpsuch that
Lemma 2.3
(Liu [2])
Let\(\{X_{n},n\geq 1\}\)be a sequence of END random variables. If\(f_{1},f_{2},\ldots,f_{n}\)are all nondecreasing (or nonincreasing) functions, then random variables\(f_{1}(X_{1}),f_{2}(X_{2}),\ldots, f_{n}(X_{n})\)are still END random variables.
Lemma 2.4
(Wu [23])
Let\(\{X_{n},n\geq 1\}\)and\(\{Y_{n},n\geq 1\}\)be sequences of random variables, for any\(q>r>0\), \(\varepsilon >0\), \(a>0\), then
where\(C_{r}=1\)if\(0< r\leq 1\)or\(C_{r}=2^{r-1}\)if\(r>1\).
Chen and Sung [20] obtained the following theorems (see Lemmas 2.5–2.7).
Lemma 2.5
(Chen [20])
Let\(r>1\), \(1\leq p<2\), \(\alpha >0\), \(\beta >0\)with\(1/\alpha +1/\beta =1/p\). Let\(\{a_{ni}, 1\leq i\leq n, n\geq 1\}\)be an array of constants satisfying (1.1). Xis a random variable, then
Lemma 2.6
(Chen [20])
Let\(r>1\), \(1\leq p<2\), \(\alpha >0\), \(\beta >0\)with\(1/\alpha +1/\beta =1/p\). Let\(\{a_{ni}, 1\leq i\leq n, n\geq 1\}\)be an array of constants satisfying (1.1). IfXis a random variable, then for any\(v>\max \{\alpha ,(r-1)\beta \}\)
Lemma 2.7
(Chen [20])
Let\(\lambda >0\), \(r>1\), \(1\leq p<2\), \(\alpha >0\), \(\beta >0\)with\(1/\alpha +1/\beta =1/p\). Let\(\{a_{ni}, 1\leq i\leq n, n\geq 1\}\)be an array of constants satisfying (1.1) andXbe a random variable. Then the following statements hold:
- (1)
If\(\alpha < rp\), then
$$\begin{aligned}& \sum_{n=1}^{\infty }n^{r-2-\lambda /p}\sum _{i=1}^{n}E \vert a_{ni}X \vert ^{\lambda }I\bigl( \vert a_{ni}X \vert > n^{1/p}\bigr) \\& \quad \leq \textstyle\begin{cases} CE \vert X \vert ^{(r-1)\beta } &\textit{if } \lambda < (r-1)\beta , \\ CE \vert X \vert ^{(r-1)\beta }\log (1+ \vert X \vert ) &\textit{if } \lambda = (r-1)\beta , \\ CE \vert X \vert ^{\lambda } &\textit{if } \lambda >(r-1)\beta . \end{cases}\displaystyle \end{aligned}$$ - (2)
If\(\alpha =rp\), then
$$\begin{aligned}& \sum_{n=1}^{\infty }n^{r-2-\lambda /p}\sum _{i=1}^{n}E \vert a_{ni}X \vert ^{\lambda }I\bigl( \vert a_{ni}X \vert > n^{1/p}\bigr) \\& \quad \leq \textstyle\begin{cases} CE \vert X \vert ^{(r-1)\beta }\log (1+ \vert X \vert ) &\textit{if } \lambda \leq (r-1)\beta =rp, \\ CE \vert X \vert ^{\lambda } &\textit{if } \lambda >(r-1)\beta =rp. \end{cases}\displaystyle \end{aligned}$$ - (3)
If\(\alpha >rp\), then
$$\begin{aligned}& \sum_{n=1}^{\infty }n^{r-2-\lambda /p}\sum _{i=1}^{n}E \vert a_{ni}X \vert ^{\lambda }I\bigl( \vert a_{ni}X \vert > n^{1/p}\bigr) \\& \quad\leq \textstyle\begin{cases} CE \vert X \vert ^{rp} &\textit{if } \lambda < rp, \\ CE \vert X \vert ^{rp}\log (1+ \vert X \vert ) &\textit{if } \lambda =rp, \\ CE \vert X \vert ^{\lambda } &\textit{if } \lambda >rp. \end{cases}\displaystyle \end{aligned}$$
3 Proofs of theorems
Proof of Theorem 2.1
Noting \(\alpha >0\), \(\beta >0\), \(1/\alpha +1/\beta =1/p\), we have
For \(\forall t:0< t\leq \alpha \), by the Hölder inequality and (1.1), we have
For \(\forall t: t>\alpha \), it follows from the \(C_{r}\) inequality and (1.1) that
Noting that \(a_{ni}=a_{ni}^{+}-a_{ni}^{-}\), without loss of generality, we can assume \(a_{ni}>0\).
Sufficiency. Set \(\theta \in (\frac{p}{\alpha \wedge rp},1)\) for \(1\leq i\leq n\), \(n\geq 1\), and let
Then \(a_{ni}X_{i}=\sum_{l=1}^{5}X^{(l)}_{ni}\). It follows from the definition of \(X^{(2)}_{ni},\theta \in (\frac{p}{\alpha \wedge rp},1)\), (3.1), and (2.1)–(2.3) that
By the definition \(X^{(4)}_{ni}\) and (3.1), from the above proof process, we have
Similarly, we can obtain
and
Noting that \(EX_{i}=0\), it follows from Lemma 2.4 and the \(C_{r}\) inequality that, for \(v>\lambda \geq 1\),
Similarly, for \(v>\lambda \), \(0<\lambda <1\), we have
In order to prove Theorem 2.1, we need to prove \(I_{i}<\infty \), \(i=1,2,\ldots,5\).
Taking \({v>\max \{2,2rp/[(2-p)(1-\theta )],2pr/(a-p),2pr/(2-p),a,(r-1)\beta , \lambda \}}\), it follows from Lemmas 2.1 and 2.3 that
By the definition of \(X^{(1)}_{ni}\) and \({v>2rp/[(2-p)(1-\theta )]>rp/(1-\theta )}\), we have
Since \(r>1\), \(1\leq p<2\), \(\alpha >0\), \(\beta >0\) with \(1/\alpha +1/\beta =1/p\), then \(p<\alpha \wedge rp\). By (3.1) and (2.1)–(2.3), we obtain
Then it follows from (3.5) and (3.6) that \(I_{1}<\infty \) holds.
By the definition of \(X^{(2)}_{ni}\), Lemmas 2.2 and 2.3, we get
Combining Lemmas 2.5 and 2.6, we obtain \(I_{21}<\infty \).
The proof of \(I_{22}<\infty \) will mainly be conducted under the following four cases.
Case 1:\(1<\alpha <2\), \(\alpha \leq rp\). Noting that \(p<\alpha \), by (2.1)–(2.2), we have \(E\vert X\vert ^{\alpha }<\infty \), then
Case 2:\(1<\alpha <2\), \(\alpha > rp\). Noting that \(rp<2\), by (2.3), we obtain \(E\vert X\vert ^{rp}<\infty \), then
Case 3:\(\alpha \geq 2\), \(\alpha \leq rp\). Noting that \(rp\geq 2\), by (2.1)–(2.2), we get \(E\vert X\vert ^{2}<\infty \), and then
Case 4:\(\alpha \geq 2\), \(\alpha > rp\), then \(E\vert X\vert ^{rp}<\infty \). If \(rp<2\), the proof is the same as that of Case 2. If \(rp\geq 2\), the proof is the same as that of Case 3.
Then it follows from (3.7)–(3.9) that \(I_{2}<\infty \) holds.
The proof of \(I_{4}<\infty \) will mainly be conducted under the following three cases.
Case 1:\(0<\lambda <1\). By (3.4), the \(C_{r}\) inequality, Lemma 2.7, and (2.1)–(2.3), we have
Case 2:\(1\leq \lambda \leq 2\). It follows from (3.3), the \(C_{r}\) inequality, Jensen’s inequality, Lemmas 2.2–2.3, 2.7, and (2.1)–(2.3) that
Case 3.\(\lambda >2\). By (3.3), the \(C_{r}\) inequality, Jensen’s inequality, Lemmas 2.2, 2.7, and (2.1)–(2.3), we have
From Lemma 2.7 and (2.1)–(2.3), we obtain \(I_{41}<\infty \).
The proof of \(I_{42}<\infty \) will mainly be conducted under the following two cases.
Case a:\(\alpha \leq rp\). Taking \(q=\max \{(r-1)\beta ,\lambda \}>2\), by (2.1)–(2.2), (3.2), we have \(E\vert X\vert ^{q}<\infty \) and
Case b:\(\alpha > rp\). Letting \(q=\max \{rp,\lambda \}>2\), it follows from (2.3) that \(E\vert X\vert ^{q}<\infty \). If \(\alpha \geq q \), by (3.1), we have
If \(\alpha < q \), then \((r-1)\beta < rp<\alpha <q \), by (3.2), we have
Then it follows from (3.10)–(3.14) that \(I_{4}<\infty \).
Similar to the proof of \(I_{2}<\infty \) and \(I_{4}<\infty \), we can get \(I_{3}<\infty \) and \(I_{5}<\infty \), too.
Necessity. By (2.4), we have
Set \(a_{ni}=1\) for \(\{1\leq i\leq n\), \(n\geq 1\}\), then (3.15) can be rewritten as follows:
which implies that \(EX=0\), \(E\vert X\vert ^{rp}<\infty \) (see Theorem 2 in Peligard and Gut [24]). Take \(a_{ni}=0\) for \(1\leq i\leq n-1\), \(n\geq 1\), and \(a_{nn}= n^{1/\alpha }\), then (3.15) can be rewritten as follows:
which is equivalent to \(E\vert X\vert ^{(r-1)\beta }<\infty \). The proof is completed. □
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The authors would like to thank the referees for their valuable comments and suggestions, which have improved the quality of the manuscript greatly.
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This work was jointly supported by the National Natural Science Foundation of China (61773217), the Key University Science Research Project of Anhui Province (KJ2019A0700), the Key Program in the Youth Talent Support Plan in Universities of Anhui Province (gxyqZD2016317), Hunan Provincial Science and Technology Project Foundation (2019RS1033), the Scientific Research Fund of Hunan Provincial Education Department (18A013), Hunan Normal University National Outstanding Youth Cultivation Project (XP1180101), and the Construct Program of the Key Discipline in Hunan Province.
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Song, M., Zhu, Q. Complete moment convergence of extended negatively dependent random variables. J Inequal Appl 2020, 150 (2020). https://doi.org/10.1186/s13660-020-02416-7
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DOI: https://doi.org/10.1186/s13660-020-02416-7