- Research
- Open access
- Published:
Complete moment convergence for moving average process generated by \(\rho^{-}\)-mixing random variables
Journal of Inequalities and Applications volume 2015, Article number: 245 (2015)
Abstract
Let \(\{Y_{i},-\infty< i<\infty\}\) be a sequence of \(\rho^{-}\)-mixing random variables without the assumption of identical distributions, and \(\{a_{i},-\infty< i<\infty\}\) be an absolutely summable sequence of real numbers. In this paper, under some suitable conditions, we establish the complete moment convergence for the partial sum of moving average processes \(\{X_{n}=\sum_{i=-\infty}^{\infty}a_{i}Y_{i+n},n\geq 1\}\). These results promote and improve the corresponding results obtained by Li and Zhang (Stat. Probab. Lett. 70:191-197, 2004) from NA to the case of a \(\rho^{-}\)-mixing setting.
1 Introduction
Let \(\{Y_{i},-\infty< i<\infty\}\) be a sequence of random variables and \(\{a_{i},-\infty< i<\infty\}\) be an absolutely summable sequence of real numbers, and for \(n\geq1\) set \(X_{n}=\sum_{i=-\infty}^{\infty}a_{i}Y_{i+n}\). The limit behavior of the moving average process \(\{X_{n},n\geq1\}\) has been extensively investigated by many authors. For example, Baek et al. [1] have obtained the convergence of moving average processes, Burton and Dehling [2] have obtained a large deviation principle, Ibragimov [3] has established the central limit theorem, Račkauskas and Suquet [4] have proved the functional central limit theorems for self-normalized partial sums of linear processes, and Chen et al. [5], Guo [6], Kim et al. [7, 8], Ko et al. [9], Li et al. [10], Li and Zhang [11], Qiu et al. [12], Wang and Hu [13], Yang and Hu [14], Zhang [15], Zhen et al. [16], Zhou et al. [17], Zhou and Lin [18], Shen et al. [19] have obtained the complete (moment) convergence of moving average process based on a sequence of dependent (or mixing) random variables, respectively. But very few results for moving average process based on a \(\rho^{-}\)-mixing random variables are known. Firstly, we recall some definitions.
For two nonempty disjoint sets S, T of real numbers, we define \(\operatorname{dist}(S, T)=\min\{\vert j-k\vert ; j\in S, k\in T\}\). Let \(\sigma(S)\) be the σ-field generated by \(\{Y_{k}, k\in S\}\), and define \(\sigma(T)\) similarly.
Definition 1.1
A sequence \(\{Y_{i},-\infty< i<\infty\}\) is called \(\rho^{-}\)-mixing, if
where
where the supremum is taken over all coordinatewise increasing real functions f on \(R^{S}\) and g on \(R^{T}\).
Definition 1.2
A sequence \(\{Y_{i},-\infty< i<\infty\}\) is called \(\rho^{*}\)-mixing if
where
Definition 1.3
A sequence \(\{Y_{i},i\in Z\}\) is called negatively associated (NA) if for every pair of disjoint subsets S, T of Z and any real coordinatewise increasing functions f on \(R^{S}\) and g on \(R^{T}\)
Definition 1.4
A sequence \(\{Y_{i},-\infty< i<\infty\}\) of random variables is said to be stochastically dominated by a random variable Y if there exists a constant C such that
Definition 1.5
A real valued function \(l(x)\), positive and measurable on \([0,\infty)\), is said to be slowly varying at infinity if for each \(\lambda>0\), \(\lim_{x\to\infty}\frac{l(\lambda x)}{l(x)}=1\).
Li and Zhang [11] obtained the following complete moment convergence of moving average processes under NA assumptions.
Theorem A
Suppose that \(\{X_{n}=\sum_{i=-\infty}^{\infty} a_{i}\varepsilon_{i+n}, n\geq1\}\), where \(\{a_{i},-\infty< i<\infty\}\) is a sequence of real numbers with \(\sum_{i=-\infty}^{\infty} \vert a_{i}\vert <\infty\) and \(\{\varepsilon_{i},-\infty< i<\infty\}\) is a sequence of identically distributed NA random variables with \(E\varepsilon_{1}=0\), \(E\varepsilon_{1}^{2}<\infty\). Let h be a function slowly varying at infinity, \(1\leq q<2 \), \(r>1+q/2\). Then \(E\vert \varepsilon_{1}\vert ^{r}h(\vert \varepsilon_{1}\vert ^{q})<\infty\) implies
for all \(\varepsilon>0\).
Chen et al. [20] also established the following results for moving average processes under NA assumptions.
Theorem B
Let \(q>0\), \(1\leq p<2\), \(r\geq1\), \(rp\neq1\). Suppose that \(\{X_{n}=\sum_{i=-\infty}^{\infty} a_{i}\varepsilon_{i+n}, n\geq1\}\), where \(\{a_{i},-\infty< i<\infty\}\) is a sequence of real numbers with \(\sum_{i=-\infty}^{\infty} \vert a_{i}\vert <\infty\) and \(\{\varepsilon_{i},-\infty< i<\infty\}\) is a sequence of identically distributed NA random variables. If \(E\varepsilon_{1}=0\) and \(E\vert \varepsilon_{1}\vert ^{rp}<\infty\), then
for all \(\varepsilon>0\). Furthermore if \(E\varepsilon_{1}=0\) and \(E\vert \varepsilon_{1}\vert ^{rp}<\infty\) for \(q< rp\), \(E\vert \varepsilon_{1}\vert ^{rp}\log(1+\vert \varepsilon_{1}\vert )<\infty\) for \(q=rp\), \(E\vert \varepsilon_{1}\vert ^{q}<\infty\) for \(q>rp\), then
for all \(\varepsilon>0\).
Recently, Zhou and Lin [18] obtained the following complete moment convergence of moving average processes under ρ-mixing assumptions.
Theorem C
Let h be a function slowly varying at infinity, \(p\geq1\), \(p\alpha>1\) and \(\alpha>1/2\). Suppose that \(\{X_{n},n\geq1\}\) is a moving average process based on a sequence \(\{ Y_{i},-\infty< i<\infty\}\) of identically distributed ρ-mixing random variables. If \(EY_{1}=0\) and \(E\vert Y_{1}\vert ^{p+\delta }h(\vert Y_{1}\vert ^{1/{\alpha}})<\infty\) for some \(\delta>0\), then for all \(\varepsilon>0\),
and
Obviously, \(\rho^{-}\)-mixing random variables include NA and \(\rho^{*}\)-mixing random variables, which have a lot of applications, their limit properties have aroused wide interest recently, and a lot of results have been obtained; we refer to Wang and Lu [21] for a Rosenthal-type moment inequality and weak convergence, Budsaba et al. [22, 23] for complete convergence for moving average process based on a \(\rho^{-}\)-mixing sequence, Tan et al. [24] for the almost sure central limit theorem. But there are few results on the complete moment convergence of moving average process based on a \(\rho^{-}\)-mixing sequence. Therefore, in this paper, we establish some results on the complete moment convergence for maximum partial sums with less restrictions. Throughout the sequel, C represents a positive constant although its value may change from one appearance to the next, \(I\{A\}\) denotes the indicator function of the set A.
2 Preliminary lemmas
In this section, we list some lemmas which will be useful to prove our main results.
Lemma 2.1
(Zhou [17])
If l is slowly varying at infinity, then
-
(1)
\(\sum_{n=1}^{m}n^{s}l(n)\leq C m^{s+1}l(m)\) for \(s>-1\) and positive integer m,
-
(2)
\(\sum_{n=m}^{\infty}n^{s}l(n)\leq C m^{s+1}l(m)\) for \(s<-1\) and positive integer m.
Lemma 2.2
(Wang and Lu [21])
For a positive real number \(q\geq2\), if \(\{X_{n},n\geq1\}\) is a sequence of \(\rho^{-}\)-mixing random variables, with \({E}X_{i}=0\), \({E}\vert X_{i}\vert ^{q}<\infty\) for every \(i \geq1\), then for all \(n\geq1\), there is a positive constant \(C=C(q,\rho^{-}(\cdot))\) such that
Lemma 2.3
(Wang et al. [25])
Let \(\{X_{n}, n\geq1\}\) be a sequence of random variables which is stochastically dominated by a random variable X. Then for any \(a>0\) and \(b>0\),
3 Main results and proofs
Theorem 3.1
Let l be a function slowly varying at infinity, \(p\geq1\), \(\alpha>1/2\), \(\alpha p> 1\). Assume that \(\{a_{i},-\infty< i<\infty\}\) is an absolutely summable sequence of real numbers. Suppose that \(\{X_{n}=\sum_{i=-\infty}^{\infty} a_{i}Y_{i+n}, n\geq1\}\) is a moving average process generated by a sequence \(\{Y_{i},-\infty< i<\infty\}\) of \(\rho^{-}\)-mixing random variables which is stochastically dominated by a random variable Y. If \(EY_{i}=0\) for \(1/2<\alpha\leq1\), \(E\vert Y\vert ^{p}l(\vert Y\vert ^{1/{\alpha}})<\infty\) for \(p>1\) and \(E\vert Y\vert ^{1+\delta}<\infty\) for \(p=1\) and some \(\delta>0\), then for any \(\varepsilon>0\)
and
Proof
Firstly to prove (3.1). Let \(f(n)=n^{\alpha p-2-\alpha}l(n)\) and \(Y^{(1)}_{xj}=-xI\{Y_{j}< -x\}+Y_{j}I\{\vert Y_{j}\vert \leq x\}+xI\{Y_{j}> x\}\) and \(Y^{(2)}_{xj}=Y_{j}-Y^{(1)}_{xj}\) be the monotone truncations of \(\{Y_{j},-\infty< j<\infty\}\) for \(x>0\). Then by the property of \(\rho^{-}\)-mixing random variables (cf. Property P2 in Wang and Lu [21]), \(\{Y^{(1)}_{xj}-EY^{(1)}_{xj},-\infty< j<\infty\}\) and \(\{Y^{(2)}_{xj},-\infty< j<\infty\}\) are two sequences of \(\rho^{-}\)-mixing random variables. Note that \(\sum_{k=1}^{n}X_{k}=\sum_{i=-\infty}^{\infty}a_{i}\sum_{j=i+1}^{i+n}Y_{j}\). Since \(\sum_{i=-\infty}^{\infty} \vert a_{i}\vert <\infty\), by Lemma 2.3, we have for \(x>n^{\alpha}\), if \(\alpha>1\)
If \(1/2<\alpha\leq1\), note \(\alpha p> 1\), this means \(p>1\). By \(E\vert Y\vert ^{p}l(\vert Y\vert ^{1/{\alpha}})<\infty\) and l is slowly varying at infinity, for any \(0<\epsilon<p-1/{\alpha}\), we have \(E\vert Y\vert ^{p-\epsilon}<\infty\). Then noting \(EY_{i}=0\), by Lemma 2.3 we have
Hence for \(x>n^{\alpha}\) large enough, we get
Therefore
Firstly we show \(I_{1}<\infty\). Noting \(\vert Y^{(2)}_{xj}\vert <\vert Y_{j}\vert I\{\vert Y_{j}\vert > x\}\), by Markov’s inequality and Lemma 2.3, we have
If \(p>1\), then \(\alpha p-1-\alpha>-1\), and, by Lemma 2.1, we obtain
If \(p=1\), notice that \(E\vert Y\vert ^{1+\delta}<\infty\) implies \(E\vert Y\vert ^{1+\delta '}l(\vert Y\vert ^{1/{\alpha}})<\infty\) for any \(0<\delta'<\delta\), then by Lemma 2.1, we obtain
So, we get
Next we show \(I_{2}<\infty\). By Markov’s inequality, the Hölder inequality, and Lemma 2.2, we conclude
where \(r\geq2\) will be specialized later.
For \(I_{21}\), if \(p>1\), take \(r>\max\{2,p\}\), then by \(C_{r}\) inequality, Lemma 2.3, and Lemma 2.1, we get
For \(I_{21}\), if \(p=1\), take \(r>\max\{1+\delta',2\}\), where \(0<\delta '<\delta\), then by the same argument as above we have
For \(I_{22}\), if \(1\leq p<2\), take \(r>2\), note \(\alpha p+r/2-\alpha pr/2-1=(\alpha p-1)(1-r/2)<0\), by the \(C_{r}\) inequality, Lemma 2.3, and Lemma 2.1, we obtain
For \(I_{22}\), if \(p\geq2\), take \(r>(\alpha p-1)/({\alpha-1/2})>2\); we have \(\alpha(p-r)+r/2-2<-1\), and therefore one gets
Thus, (3.1) can be deduced by combining (3.3)-(3.9).
Now, we show (3.2). By Lemma 2.1 and (3.1) we have
Hence the proof of Theorem 3.1 is completed. □
The next theorem treats the case \(\alpha p=1\).
Theorem 3.2
Let l be a function slowly varying at infinity, \(1\leq p<2\). Assume that \(\sum_{i=-\infty}^{\infty} \vert a_{i}\vert ^{\theta}<\infty\), where θ belong to \((0,1)\) if \(p=1\) and \(\theta=1\) if \(1< p<2\). Suppose that \(\{X_{n}=\sum_{i=-\infty}^{\infty} a_{i}Y_{i+n}, n\geq1\}\) is a moving average process generated by a sequence \(\{Y_{i},-\infty< i<\infty\}\) of \(\rho^{-}\)-mixing random variables which is stochastically dominated by a random variable Y. If \(EY_{i}=0\) and \(E\vert Y\vert ^{p}l(\vert Y\vert ^{p})<\infty\), then for any \(\varepsilon>0\)
Proof
Let \(g(n)=n^{-1-1/p}l(n)\). Similarly to the proof of (3.3), we have
For \(J_{1}\), by Markov’s inequality, the \(C_{r}\) inequality, Lemma 2.3, and Lemma 2.1, one gets
For \(J_{2}\), similar to the proof of \(I_{2}\), take \(r=2\), by Lemma 2.2, Lemma 2.3, and Lemma 2.1, we conclude
Hence from (3.11)-(3.13), (3.10) holds. □
For the complete convergence and strong law of large numbers, we have the following corollary from the above theorems immediately.
Corollary 3.3
Under the assumptions of Theorem 3.1, for any \(\varepsilon>0\) we have
Under the assumptions of Theorem 3.2, for any \(\varepsilon>0\) we have
in particular, the assumptions \(EY_{i}=0\) and \(E\vert Y\vert ^{p}<\infty\) imply the following Marcinkiewicz-Zygmund strong law of large numbers:
Remark 3.4
Corollary 3.3 provides complete convergence for the maximum of partial sums, which extends the corresponding results of Budsaba et al. [22, 23] and Theorem 1 of Baek et al. [1] with less restrictions. Since \(\rho^{-}\)-mixing random variables include NA and \(\rho^{*}\)-mixing random variables, our results also hold for NA and \(\rho^{*}\)-mixing, and therefore Theorem 3.1 improves upon the above Theorem A from Li and Zhang [11] with less restrictions, and our results also extend and generalize the above Theorem B from Chen et al. [20] with \(q=1\) partly.
Remark 3.5
Obviously, the assumption that \(\{Y_{i},-\infty< i<\infty\}\) is stochastically dominated by a random variable Y is weaker than the assumption of identical distribution of the random variables \(\{Y_{i},-\infty< i<\infty\}\), therefore the above results also hold for identically distributed random variables.
Remark 3.6
Let \(a_{0}=1\), \(a_{i}=0\), \(i\neq0\), then \(S_{n}=\sum_{k=1}^{n}X_{k}=\sum_{k=1}^{n}Y_{k}\). Hence the above results hold when \(\{X_{k},k\geq1\}\) is a sequence of \(\rho^{-}\)-mixing random variables which is stochastically dominated by a random variable Y.
References
Baek, JI, Kim, TS, Liang, HY: On the convergence of moving average processes under dependent conditions. Aust. N. Z. J. Stat. 45, 331-342 (2003)
Burton, RM, Dehling, H: Large deviations for some weakly dependent random processes. Stat. Probab. Lett. 9, 397-401 (1990)
Ibragimov, IA: Some limit theorem for stationary processes. Theory Probab. Appl. 7, 349-382 (1962)
Račkauskas, A, Suquet, C: Functional central limit theorems for self-normalized partial sums of linear processes. Lith. Math. J. 51(2), 251-259 (2011)
Chen, PY, Hu, TC, Volodin, A: Limiting behaviour of moving average processes under φ-mixing assumption. Stat. Probab. Lett. 79, 105-111 (2009)
Guo, ML: On complete moment convergence of weighted sums for arrays of row-wise negatively associated random variables. Stochastics 86(3), 415-428 (2014)
Kim, TS, Ko, MH: Complete moment convergence of moving average processes under dependence assumptions. Stat. Probab. Lett. 78(7), 839-846 (2008)
Kim, TS, Ko, MH, Choi, YK: Complete moment convergence of moving average processes with dependent innovations. J. Korean Math. Soc. 45(2), 355-365 (2008)
Ko, MH, Kim, TS, Ryu, DH: On the complete moment convergence of moving average processes generated by \(\rho^{\ast}\)-mixing sequences. Commun. Korean Math. Soc. 23(4), 597-606 (2008)
Li, DL, Rao, MB, Wang, XC: Complete convergence of moving average processes. Stat. Probab. Lett. 14, 111-114 (1992)
Li, YX, Zhang, LX: Complete moment convergence of moving average processes under dependence assumptions. Stat. Probab. Lett. 70, 191-197 (2004)
Qiu, DH, Liu, XD, Chen, PY: Complete moment convergence for maximal partial sums under NOD setup. J. Inequal. Appl. 2015, 58 (2015) 12 pp
Wang, XJ, Hu, SH: Complete convergence and complete moment convergence for martingale difference sequence. Acta Math. Sin. Engl. Ser. 30(1), 119-132 (2014)
Yang, WZ, Hu, SH: Complete moment convergence of pairwise NQD random variables. Stochastics 87(2), 199-208 (2015)
Zhang, LX: Complete convergence of moving average processes under dependence assumptions. Stat. Probab. Lett. 30, 165-170 (1996)
Zhen, X, Zhang, LL, Lei, YJ, Chen, ZG: Complete moment convergence for weighted sums of negatively superadditive dependent random variables. J. Inequal. Appl. 2015, 117 (2015)
Zhou, XC: Complete moment convergence of moving average processes under φ-mixing assumptions. Stat. Probab. Lett. 80, 285-292 (2010)
Zhou, XC, Lin, JG: Complete moment convergence of moving average processes under ρ-mixing assumption. Math. Slovaca 61(6), 979-992 (2011)
Shen, AT, Wang, XH, Li, XQ, Wang, XJ: On the rate of complete convergence for weighted sums of arrays of rowwise ϕ-mixing random variables. Commun. Stat., Theory Methods 43, 2714-2725 (2014)
Chen, PY, Hu, TC, Volodin, A: Limiting behaviour of moving average processes under negative association assumption. Theory Probab. Math. Stat. 77, 154-166 (2007)
Wang, JF, Lu, FB: Inequalities of maximum of partial sums and weak convergence for a class of weak dependent random variables. Acta Math. Sin. 22, 693-700 (2006)
Budsaba, K, Chen, PY, Volodin, A: Limiting behavior of moving average processes based on a sequence of \(\rho^{-}\) mixing random variables. Thail. Stat. 5, 69-80 (2007)
Budsaba, K, Chen, PY, Volodin, A: Limiting behavior of moving average processes based on a sequence of \(\rho^{-}\) mixing and NA random variables. Lobachevskii J. Math. 26, 17-25 (2007)
Tan, XL, Zhang, Y, Zhang, Y: An almost sure central limit theorem of products of partial sums for \(\rho^{-}\) mixing sequences. J. Inequal. Appl. 2012, 51 (2012). doi:10.1186/1029-242X-2012-51
Wang, XJ, Li, XQ, Yang, WZ, Hu, SH: On complete convergence for arrays of rowwise weakly dependent random variables. Appl. Math. Lett. 25, 1916-1920 (2012)
Acknowledgements
This work was supported by National Natural Science Foundation of China (Grant No. 11101180) and the Science and Technology Development Program of Jilin Province (Grant No. 20130522096JH).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The author declares to have no competing interests.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Zhang, Y. Complete moment convergence for moving average process generated by \(\rho^{-}\)-mixing random variables. J Inequal Appl 2015, 245 (2015). https://doi.org/10.1186/s13660-015-0766-5
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13660-015-0766-5