Abstract
In this paper, we establish the complete moment convergence of a moving average process generated by the class of random variables satisfying a Rosenthal-type maximal inequality and a weak mean dominating condition with a mean dominating variable.
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1 Introduction
Let \(\{Y_{i}, -\infty< i<\infty\}\) be a doubly infinite sequence of random variables with zero means and finite variances and \(\{a_{i}, -\infty < i<\infty\}\) an absolutely summable sequence of real numbers. Define a moving average process \(\{X_{n}, n\geq1\}\) by
The concept of complete moment convergence is as follows: Let \(\{Y_{n}, n\geq1\}\) be a sequence of random variables and \(a_{n}>0\), \(b_{n}>0\). If \(\sum_{n=1}^{\infty}a_{n} E\{b_{n}^{-1}|Y_{n}|-\epsilon\}^{+}<\infty\) for all \(\epsilon>0\), then we call that \(\{Y_{n}, n\geq1\}\) satisfies the complete moment convergence. It is well known that the complete moment convergence can imply the complete convergence.
Chow [1] first showed the following complete moment convergence for a sequence of i.i.d. random variables by generalizing the result of Baum and Katz [2].
Theorem 1.1
Suppose that \(\{Y_{n}, n\geq1\}\) is a sequence of i.i.d. random variables with \(EY_{1}=0\). For \(1\leq p<2\) and \(r>p\), if \(E\{|Y_{1}|^{r}+|Y_{1}|\log(1+|Y_{1}|)\}<\infty\), then \(\sum_{n=1}^{\infty}n^{\frac{r}{p}-2-\frac{1}{p}}E(|\sum_{i=1}^{n} Y_{i}|-\epsilon n^{\frac{1}{p}})^{+}<\infty\) for any \(\epsilon>0\).
Recently, under dependence assumptions many authors studied extensively the complete moment convergence of a moving average process; see for example, Li and Zhang [3] for NA random variables, Zhou [4] for φ-mixing random variables, and Zhou and Lin [5] for ρ-mixing random variables.
We recall that a sequence \(\{Y_{n}, n\geq1\}\) of random variables satisfies a weak mean dominating condition with a mean dominating random variable Y if there is some positive constant C such that
for all \(x>0\) and all \(n\geq1\) (see Kuczmaszewska [6]).
One of the most interesting inequalities in probability theory and mathematical statistics is the Rosenthal-type maximal inequality. For a sequence \(\{Y_{i}, 1\leq i \leq n\}\) of i.i.d. random variables with \(E|Y_{1}|^{q}<\infty\) for \(q\geq2\) there exists a positive constant \(C_{q}\) depending only on q such that
The above inequality has been obtained for dependent random variables by many authors. See, for example, Peligrad [7] for a strong stationary ρ-mixing sequence, Peligrad and Gut [8] for a \(\rho^{*}\)-mixing sequence, Stoica [9] for a martingale difference sequence, and so forth.
In this paper we will establish the complete moment convergence for a moving average process generated by the class of random variables satisfying a Rosenthal-type maximal inequality and a weak mean dominating condition.
2 Some lemmas
The following lemmas will be useful to prove the main results.
Recall that a real valued function h, positive and measurable on \([0, \infty)\), is said to be slowly varying at infinity if for each \(\lambda>0\)
Lemma 2.1
(Zhou [4])
If h is a slowly varying function at infinity and m a positive integer, then
-
(1)
\(\sum_{n=1}^{m} n^{t} h(n)\leq C m^{t+1} h(m)\) for \(t>-1\),
-
(2)
\(\sum_{n=m}^{\infty}n^{t} h(n)\leq C m^{t+1} h(m)\) for \(t<-1\).
Lemma 2.2
(Gut [10])
Let \(\{X_{n}, n\geq1\}\) be a sequence of random variables satisfying a weak dominating condition with a mean dominating random variable X, i.e., there exists some positive constant C
Let \(r>0\) and for some \(A>0\)
and
Then for some \(C>0\)
-
(1)
if \(E|X|^{r}<\infty\), then \((n^{-1})\sum_{i=1}^{n} E|X_{i}|^{r}\leq CE|X|^{r}\),
-
(2)
\((n^{-1})\sum_{i=1}^{n} E|X_{i}^{\prime}|^{r}\leq C(E|X^{\prime}|^{r}+A^{r} P(|X|>A))\) for any \(A>0\),
-
(3)
\((n^{-1})\sum_{i=1}^{n} E|X_{i}^{\prime\prime}|^{r}\leq CE|X^{\prime\prime}|^{r}\) for any \(A>0\),
-
(4)
\((n^{-1})\sum_{i=1}^{n} E|X_{i}^{*}|^{r}\leq CE|X^{*}|^{r}\) for any \(A>0\).
3 Main result
Theorem 3.1
Let h be a function slowly varying at infinity, \(p\geq1\), \(\alpha>\frac{1}{2}\) and \(\alpha p>1\). Assume that \(\{a_{i}, -\infty< i<\infty\}\) is an absolutely summable sequence of real numbers and that \(\{Y_{i}, -\infty< i<\infty\}\) is a sequence of mean zero random variables satisfying a weak mean dominating condition with a mean dominating random variable Y, i.e. there exists some positive constant C
and all \(n\geq1\) and \(E|Y|^{p} h(|Y|^{\frac{1}{\alpha}})<\infty\).
Suppose that \(\{X_{n}, n\geq1\}\) is a moving average process, where \(X_{n}=\sum_{i=-\infty}^{\infty}a_{i} Y_{i+n}\), \(n\geq1\) is defined as (1.1).
Assume that for any \(q\geq2\), there exists a positive \(C_{q}\) depending only on q such that
where \(Y_{xj}=-xI(Y_{j}<-x)+Y_{j}I(|Y_{j}|\leq x)+xI(Y_{j}>x)\) for all \(x>0\).
Then for all \(\epsilon>0\)
and
Proof of (3.2)
Let \(\tilde{Y_{xj}}=Y_{j}-Y_{xj}\) and \(l(n)=n^{\alpha p-2-\alpha}h(n)\).
Recall that \(\sum_{k=1}^{n} X_{k}=\sum_{k=1}^{n} \sum_{i=-\infty}^{\infty}a_{i} Y_{i+k}=\sum_{i=-\infty}^{\infty}a_{i} \sum_{j=i+1}^{i+n}Y_{j}\) by (1.1).
If \(\alpha>1\), by the assumption that \(\sum_{i=-\infty}^{\infty}|a_{i}|<\infty\) and Lemma 2.2 we have, for \(x>n^{\alpha}\),
If \(\frac{1}{2}<\alpha\leq1\), \(\alpha p>1\) implies \(p>1\). By the assumption \(EY_{i}=0\) for all \(-\infty< i<\infty\) and Lemma 2.2 we obtain
It follows from (3.4i) and (3.4ii) that for \(x>n^{\alpha}\) large enough,
which yields
Now we will by an estimate show that \(I_{1}<\infty\). It is clear that \(|\tilde{Y_{xj}}|\leq|Y_{j}|I[|Y_{j}|>x]\). Hence for \(I_{1}\), by Markov’s inequality and Lemma 2.2, we have
If \(p>1\), note that \(\alpha p-1-\alpha>-1\). By Lemma 2.1 and (3.7) we obtain
If \(p=1\), by (3.7), we also obtain
For \(I_{2}\), by Markov’s inequality, Hölder’s inequality, and (3.1) we get for any \(q\geq2\)
For \(I_{21}\), we consider the following two cases.
If \(p>1\), take \(q>\max\{2,p\}\), then by the assumption that \(\sum_{i=-\infty}^{\infty}|a_{i}|<\infty\), \(C_{r}\) inequality and Lemmas 2.1 and 2.2 we get
For \(I_{21}\), if \(p=1\), take \(q>\max\{1+\delta, 2\}\) by the same argument as above one gets for any \(\delta>0\)
It follows from (3.12) and (3.13) that, for \(p\geq1\),
It remains to estimate \(I_{22}<\infty\).
For \(I_{22}\), we consider the following two cases. If \(1\leq p<2\), take \(q>2\), note that \(\alpha p+\frac{q}{2}-\frac {\alpha p q}{2}-1=(\alpha p-1)(1-\frac{q}{2})<0\). Then by \(C_{r}\) inequality and Lemma 2.2, we obtain
If \(p\geq2\), take \(q>\frac{p \alpha-1}{\alpha-\frac{1}{2}}>2\), which yields \(\alpha(p-q)+\frac{q}{2}-2<-1\). Then we get
Hence, by (3.15) and (3.16) we get
Moreover, by (3.14) and (3.17), we also get
The proof of (3.2) is completed by (3.6), (3.10), and (3.18). □
Proof of (3.3)
By Lemma 2.1 and (3.2), we have
where \(\epsilon^{\prime}=\epsilon2^{-\alpha}\). Hence the proof of (3.3) is completed. □
Remark
There are many sequences of dependent random variables satisfying (3.1) for all \(q\geq2\).
Examples include sequences of NA random variables (see Shao [11]), \(\rho^{*}\)-mixing random variables (see Utev and Peligrad [12]), φ-mixing random variables (see Zhou [4]), and ρ-mixing random variables (see Zhou and Lin [5]).
Corollary 3.2
Under the assumptions of Theorem 3.1 for any \(\epsilon>0\)
Proof
As in Remark 1.2 of Li and Zhang [3] we can obtain (3.19). □
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Acknowledgements
This paper was supported by Wonkwang University in 2015.
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Ko, MH. Complete moment convergence of moving average process generated by a class of random variables. J Inequal Appl 2015, 225 (2015). https://doi.org/10.1186/s13660-015-0745-x
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DOI: https://doi.org/10.1186/s13660-015-0745-x