1 Introduction and main results

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 \(n\geq 1\) set \(X_{n}=\sum_{i=-\infty }^{\infty }a_{i}Y_{i+n}\). The limit properties of the moving average process \(\{X_{n},n\geq 1\}\) have been extensively investigated by many authors. For example, Burton and Dehling [1] obtained a large deviation principle, Ibragimov [2] established the central limit theorem, Račkauskas and Suquet [3] proved the functional central limit theorems for self-normalized partial sums of linear processes, and An [4], Chen et al. [5], Kim and Ko [6], Li et al. [7], Li and Zhang [8], Wang and Hu [9], Yang and Hu [10], Zhang [11], Zhou [12], Zhou and Lin [13], Zhang [14], Zhang and Ding [15], Song and Zhu [16, 17] got the complete (moment) convergence of moving average process based on a sequence of different dependent (or mixing) random variables, respectively. But few results for moving average process based on m-WOD random variables are known. Firstly, we introduce some definitions.

Definition 1.1

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

$$ P\bigl\{ \vert Y_{i} \vert >x\bigr\} \leq CP\bigl\{ \vert Y \vert >x\bigr\} ,\quad x\geq 0, -\infty < i< \infty . $$

Definition 1.2

A real-valued function \(l(x)\), positive and measurable on \([a,\infty )\), \(a>0\), is said to be slowly varying at infinity if, for each \(\lambda >0\), \(\lim_{x\to \infty }\frac{l(\lambda x)}{l(x)}=1\).

The concept of widely orthant dependence structure was introduced by Wang et al. [18] as follows.

Definition 1.3

For the random variables \(\{X_{n},n\geq 1\}\), if there exists a finite positive sequence \(\{g_{U}(n),n\geq 1\}\) satisfying, for each \(n\geq 1\) and for all \(x_{i}\in R\), \(1\leq i\leq n\),

$$\begin{aligned} P(X_{1}>x_{1},X_{2}>x_{2}, \ldots ,X_{n}>x_{n})\leq g_{U}(n)\prod _{i=1}^{n} P(X_{i}>x_{i}), \end{aligned}$$
(1.1)

then we say that the random variables \(\{X_{n},n\geq 1\}\) are widely upper orthant dependent (WUOD, for short); if there exists a finite positive sequence \(\{g_{L}(n),n\geq 1\}\) satisfying, for each \(n\geq 1\) and for all \(x_{i}\in R\), \(1\leq i\leq n\),

$$\begin{aligned} P(X_{1}< x_{1},X_{2}< x_{2}, \ldots ,X_{n}< x_{n})\leq g_{L}(n)\prod _{i=1}^{n} P(X_{i}< x_{i}), \end{aligned}$$
(1.2)

then we say that the random variables \(\{X_{n},n\geq 1\}\) are widely lower orthant dependent (WLOD, for short); if they are both WUOD and WLOD, then we say that the random variables \(\{X_{n},n\geq 1\}\) are widely orthant dependent (WOD, for short), and \(g_{U}(n)\), \(g_{L}(n)\), \(n\geq 1\), are called dominated coefficients.

Inspired by WOD and m-NA, Fang et al. [19] introduced the following notion.

Definition 1.4

Let \(m\geq 1\) be a fixed integer. A sequence of random variables \(\{X_{n},n\geq 1\}\) is said to be m-WOD if, for any \(n\geq 2\) and \(i_{1},i_{2},\ldots ,i_{n}\) such that \(|i_{k}-i_{j}|\geq m\) for all \(1\leq k\neq j\leq n\), we have that \(X_{i_{1}},X_{i_{2}},\ldots ,X_{i_{n}}\) are WOD.

By (1.1) and (1.2), we can see that \(g_{U}(n)\geq 1\) and \(g_{L}(n)\geq 1\). Recall that when \(g_{U}(n)=g_{L}(n)=M\) for some positive constant M and any \(n\geq 1\), then the random variables \(\{X_{n},n\geq 1\}\) are called extended negatively dependent (END, for short). The definition of END was introduced by Liu [20]. If both (1.1) and (1.2) hold for \(g_{U}(n)=g_{L}(n)=1\) for any \(n\geq 1\), then the random variables \(\{X_{n},n\geq 1\}\) are called negatively orthant dependent (NOD, for short), which was introduced by Ebrahimi and Ghosh [21]. It is well known that negatively associated (NA, for short) random variables are NOD. Hu [22] pointed out that negatively superadditive dependent (NSD, for short) random variables are NOD. Hence, the class of m-WOD random variables includes independent sequence, m-NA sequence, NSD sequence, m-NOD sequence, and m-END sequence as special cases. Studying the probability limit theory and its applications for m-WOD random variables is of great interest. But there are few results on the complete moment convergence of moving average process based on an m-WOD sequence. Therefore, in this paper, we establish some results on the complete moment convergence for partial sums for moving average process.

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, \([x]\) denotes the integer part of x, \(X^{+}=\max \{X,0\}\), \(X^{-}=\max \{-X,0\}\).

2 Preliminary lemmas

In this section, we give some lemmas which will be useful to prove our main results.

Lemma 2.1

(Fang et al. [19])

Let \(\{X_{n},n\geq 1\}\) be a sequence of m-WOD random variables with dominating coefficients \(g(n)=\max \{g_{L}(n),g_{U}(n)\}\)). If \(\{f_{n}(\cdot ),n\geq 1\}\) are all nondecreasing (or nonincreasing), then \(\{f_{n}(X_{n}),n\geq 1\}\) are still m-WOD with dominating coefficients \(\{g(n),n\geq 1\}\).

Lemma 2.2

(Fang et al. [19])

For a positive real number \(q\geq 2\), if \(\{X_{n},n\geq 1\}\) is a sequence of mean zero m-WOD random variables with dominating coefficients \(g(n)=\max \{g_{L}(n),g_{U}(n)\}\). If \({E}|X_{i}|^{q}<\infty \) for every \(i \geq 1\), then for all \(n\geq 1\) there exist positive constants \(C_{1}(m,q)\) and \(C_{2}(m,q)\) depending on q and m such that

$$ {E}\Biggl( \Biggl\vert \sum_{i=1}^{n}X_{i} \Biggr\vert ^{q}\Biggr)\leq {C_{1}(m,q)}\sum _{i=1}^{n}{E} \vert X_{i} \vert ^{q}+C_{2}(m,q)g(n) \Biggl(\sum_{i=1}^{n}{E}X_{i}^{2} \Biggr)^{\frac{q}{2}}. $$

Lemma 2.3

(Zhou [12])

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 2.4

(Wang et al. [23])

Let \(\{X_{n}, n\geq 1\}\) be a sequence of random variables which is stochastically dominated by a random variable X. Then, for any \(a>0\) and \(b>0\),

$$\begin{aligned}& E \vert X_{n} \vert ^{a}I\bigl\{ \vert X_{n} \vert \leq b\bigr\} \leq C\bigl[E \vert X \vert ^{a}I\bigl\{ \vert X \vert \leq b\bigr\} +b^{a}P\bigl( \vert X \vert >b\bigr)\bigr], \\& E \vert X_{n} \vert ^{a}I\bigl\{ \vert X_{n} \vert > b\bigr\} \leq CE \vert X \vert ^{a}I\bigl\{ \vert X \vert > b\bigr\} . \end{aligned}$$

3 Main results and proofs

Theorem 3.1

Let l be a function slowly varying at infinity, \(p\geq 1\), \(\alpha >1/2\), \(\alpha p> 1\). Assume that \(\{a_{i},-\infty < i<\infty \}\) is an absolutely summable sequence of real numbers. Suppose that \(\{X_{n}=\sum_{i=-\infty }^{\infty } a_{i}Y_{i+n}, n\geq 1\}\) is a moving average process generated by a sequence \(\{Y_{i},-\infty < i<\infty \}\) of m-WOD random variables with dominating coefficients \(g(n)=O(n^{\delta })\) for some \(\delta \geq 0\) which is stochastically dominated by a random variable Y. If \(EY_{i}=0\) for \(1/2<\alpha \leq 1\), \(E|Y|^{p}l(|Y|^{1/{\alpha }})<\infty \) for \(p>1\), and \(E|Y|^{1+\lambda }<\infty \) for \(p=1\) and some \(\lambda >0\), then for any \(\varepsilon >0\)

$$\begin{aligned} \sum_{n=1}^{\infty }n^{\alpha p-2-\alpha }l(n) E\Biggl\{ \Biggl\vert \sum_{j=1}^{n}X_{j} \Biggr\vert - \varepsilon n^{\alpha }\Biggr\} ^{+} < \infty . \end{aligned}$$
(3.1)

Proof

Let \(f(n)=n^{\alpha p-2-\alpha }l(n)\) and \(Y^{(1)}_{xj}=-xI\{Y_{j}< -x\}+Y_{j}I\{|Y_{j}|\leq x\}+xI\{Y_{j}> x\}\) and \(Y^{(2)}_{xj}=Y_{j}-Y^{(1)}_{xj}\) be the monotone truncations of \(\{Y_{j},-\infty < j<\infty \}\) for \(x>0\). Then, by Lemma 2.1, it is easy to know that \(\{Y^{(1)}_{xj}-EY^{(1)}_{xj},-\infty < j<\infty \}\) and \(\{Y^{(2)}_{xj},-\infty < j<\infty \}\) are two sequences of m-WOD random variables. Note that \(\sum_{k=1}^{n}X_{k}=\sum_{i=-\infty }^{\infty }a_{i}\sum_{j=i+1}^{i+n}Y_{j}\) and \(\sum_{i=-\infty }^{\infty }|a_{i}|<\infty \), then by Lemma 2.4 we have, for \(x>n^{\alpha }\), if \(\alpha >1\)

$$\begin{aligned}& x^{-1} \Biggl\vert E\sum_{i=-\infty }^{\infty }a_{i} \sum_{j=i+1}^{i+n}Y^{(1)} _{xj} \Biggr\vert \\& \quad \leq x^{-1}\sum_{i=-\infty }^{\infty } \vert a_{i} \vert \sum_{j=i+1}^{i+n} \bigl[E \vert Y_{j} \vert I \bigl\{ \vert Y_{j} \vert \leq x\bigr\} +xP\bigl( \vert Y_{j} \vert >x\bigr)\bigr] \\& \quad \leq Cx^{-1}n\bigl[E \vert Y \vert I\bigl\{ \vert Y \vert \leq x\bigr\} +x P\bigl( \vert Y \vert >x\bigr)\bigr] \leq C n^{1-\alpha } \to 0, \quad \text{as } n\to \infty . \end{aligned}$$

If \(1/2<\alpha \leq 1\), note that \(\alpha p> 1\), this means \(p>1\). By \(E|Y|^{p}l(|Y|^{1/{\alpha }})<\infty \) and l is slowly varying at infinity, it is easy to conclude that, for any \(0<\epsilon <p-1/{\alpha }\), we have \(E|Y|^{p-\epsilon }<\infty \). Then, noting \(EY_{i}=0\), by Lemma 2.4 we can obtain

$$\begin{aligned} x^{-1} \Biggl\vert E\sum_{i=-\infty }^{\infty }a_{i} \sum_{j=i+1}^{i+n}Y^{(1)}_{xj} \Biggr\vert & = x^{-1} \Biggl\vert E\sum _{i=-\infty }^{\infty }a_{i}\sum _{j=i+1}^{i+n}Y^{(2)}_{xj} \Biggr\vert \\ & \leq C x^{-1}\sum_{i=-\infty }^{\infty } \vert a_{i} \vert \sum_{j=i+1}^{i+n}E \vert Y_{j} \vert I \bigl\{ \vert Y_{j} \vert > x \bigr\} \leq Cx^{-1} nE \vert Y \vert I\bigl\{ \vert Y \vert > x \bigr\} \\ & \leq Cx^{1/{\alpha }-1}E \vert Y \vert I\bigl\{ \vert Y \vert > x \bigr\} \leq C E \vert Y \vert ^{1/{\alpha }}I\bigl\{ \vert Y \vert > x \bigr\} \\ & \leq E \vert Y \vert ^{p-\epsilon }I\bigl\{ \vert Y \vert > x \bigr\} \to 0, \quad \text{as } x\to \infty . \end{aligned}$$

Therefore, by the above discussion, for \(x>n^{\alpha }\) large enough, we know

$$\begin{aligned} x^{-1} \Biggl\vert E\sum_{i=-\infty }^{\infty }a_{i} \sum_{j=i+1}^{i+n}Y^{(1)}_{xj} \Biggr\vert < \varepsilon /4. \end{aligned}$$

Then

$$\begin{aligned}& \sum_{n=1}^{\infty }f(n) E \Biggl\{ \Biggl\vert \sum_{j=1}^{n}X_{j} \Biggr\vert -\varepsilon n^{ \alpha }\Biggr\} ^{+} \\& \quad \leq \sum_{n=1}^{\infty }f(n) \int _{\varepsilon n^{\alpha }}^{ \infty } P\Biggl\{ \Biggl\vert \sum _{j=1}^{n}X_{j} \Biggr\vert \geq x\Biggr\} \,dx \\& \quad \leq C\sum_{n=1}^{\infty }f(n) \int _{n^{\alpha }}^{\infty } P\Biggl\{ \Biggl\vert \sum _{j=1}^{n}X_{j} \Biggr\vert \geq \varepsilon x\Biggr\} \,dx \\& \quad \leq C\sum_{n=1}^{\infty }f(n) \int _{n^{\alpha }}^{\infty } P\Biggl\{ \Biggl\vert \sum _{i=-\infty }^{\infty }a_{i}\sum _{j=i+1}^{i+n}Y^{(2)}_{xj} \Biggr\vert \geq \varepsilon x/2\Biggr\} \,dx \\& \qquad {}+C\sum_{n=1}^{\infty }f(n) \int _{n^{\alpha }}^{\infty } P\Biggl\{ \Biggl\vert \sum _{i=- \infty }^{\infty }a_{i}\sum _{j=i+1}^{i+n}\bigl(Y^{(1)}_{xj}-EY^{(1)}_{xj} \bigr) \Biggr\vert \geq \varepsilon x/4\Biggr\} \,dx \\& \quad = :I_{1}+I_{2}. \end{aligned}$$
(3.2)

Firstly we prove \(I_{1}<\infty \). Noting \(|Y^{(2)}_{xj}|<|Y_{j}|I\{|Y_{j}|> x\}\), then by Markov’s inequality and Lemma 2.4, we have

$$\begin{aligned} I_{1} \leq & C \sum_{n=1}^{\infty }f(n) \int _{n^{\alpha }}^{\infty } x^{-1}E \Biggl\vert \sum _{i=-\infty }^{\infty }a_{i}\sum _{j=i+1}^{i+n}Y^{(2)}_{xj} \Biggr\vert \,dx \\ \leq & C\sum_{n=1}^{\infty }f(n) \int _{n^{\alpha }}^{\infty } x^{-1} \sum _{i=-\infty }^{\infty } \vert a_{i} \vert \sum _{j=i+1}^{i+n}E \bigl\vert Y^{(2)}_{xj} \bigr\vert \,dx \\ \leq & C \sum_{n=1}^{\infty }nf(n) \int _{n^{\alpha }}^{\infty } x^{-1}E \vert Y \vert I \bigl\{ \vert Y \vert > x\bigr\} \,dx \\ =&C \sum_{n=1}^{\infty }nf(n)\sum _{m=n}^{\infty } \int _{m^{\alpha }}^{(m+1)^{ \alpha }} x^{-1}E \vert Y \vert I \bigl\{ \vert Y \vert > x\bigr\} \,dx \\ \leq & C \sum_{n=1}^{\infty }nf(n)\sum _{m=n}^{\infty } m^{-1}E \vert Y \vert I\bigl\{ \vert Y \vert > m^{\alpha }\bigr\} \\ =&C\sum_{m=1}^{\infty } m^{-1}E \vert Y \vert I\bigl\{ \vert Y \vert > m^{\alpha }\bigr\} \sum _{n=1}^{m} n^{\alpha p-1-\alpha }l(n). \end{aligned}$$

If \(p>1\), then \(\alpha p-1-\alpha >-1\), by Lemma 2.3, we can get

$$\begin{aligned} I_{1} \leq & C \sum_{m=1}^{\infty } m^{\alpha p-1-\alpha }l(m)E \vert Y \vert I\bigl\{ \vert Y \vert > m^{\alpha }\bigr\} \\ =&C \sum_{m=1}^{\infty } m^{\alpha p-1-\alpha }l(m) \sum_{k=m}^{ \infty }E \vert Y \vert I\bigl\{ k^{\alpha }< \vert Y \vert \leq {(k+1)}^{\alpha }\bigr\} \\ =&C \sum_{k=1}^{\infty }E \vert Y \vert I \bigl\{ k^{\alpha }< \vert Y \vert \leq {(k+1)}^{\alpha }\bigr\} \sum _{m=1}^{k} m^{\alpha p-1-\alpha }l(m) \\ \leq &C\sum_{k=1}^{\infty }k^{\alpha p-\alpha }l(k) E \vert Y \vert I\bigl\{ k^{\alpha }< \vert Y \vert \leq {(k+1)}^{\alpha }\bigr\} \\ \leq &C E \vert Y \vert ^{p}l\bigl( \vert Y \vert ^{1/{\alpha }}\bigr)< \infty . \end{aligned}$$

If \(p=1\), \(E|Y|^{1+\lambda }<\infty \) implies \(E|Y|^{1+\lambda '}l(|Y|^{1/{\alpha }})<\infty \) for any \(0<\lambda '<\lambda \), then by Lemma 2.3 we get

$$\begin{aligned} I_{1} \leq & C \sum_{m=1}^{\infty } m^{-1}E \vert Y \vert I\bigl\{ \vert Y \vert > m^{\alpha } \bigr\} \sum_{n=1}^{m} n^{-1}l(n) \\ \leq & C \sum_{m=1}^{\infty } m^{-1}E \vert Y \vert I\bigl\{ \vert Y \vert > m^{\alpha }\bigr\} \sum _{n=1}^{m} n^{-1+\alpha \lambda '}l(n) \\ \leq &C \sum_{m=1}^{\infty } m^{\alpha \lambda '-1}l(m)E \vert Y \vert I\bigl\{ \vert Y \vert > m^{ \alpha }\bigr\} \\ \leq &C E \vert Y \vert ^{1+\lambda '}l\bigl( \vert Y \vert ^{1/{\alpha }}\bigr) < \infty . \end{aligned}$$

So, we conclude

$$\begin{aligned} I_{1}< \infty . \end{aligned}$$
(3.3)

Next we show \(I_{2}<\infty \). By Markov’s inequality, Hőlder’s inequality, and Lemma 2.2, we can obtain

$$\begin{aligned} I_{2} \leq & C\sum _{n=1}^{\infty }f(n) \int _{n^{\alpha }}^{\infty } x^{-r}E \Biggl\vert \sum _{i=-\infty }^{\infty }a_{i}\sum _{j=i+1}^{i+n}\bigl(Y^{(1)}_{xj}-EY^{(1)}_{xj} \bigr) \Biggr\vert ^{r} \,dx \\ \leq & C\sum_{n=1}^{\infty }f(n) \int _{n^{\alpha }}^{\infty } x^{-r} E \Biggl[\sum _{i=-\infty }^{\infty }\bigl( \vert a_{i} \vert ^{\frac{r-1}{r}}\bigr) \Biggl( \vert a_{i} \vert ^{1/r} \Biggl\vert \sum_{j=i+1}^{i+n} \bigl(Y^{(1)}_{xj}-EY^{(1)}_{xj}\bigr) \Biggr\vert \Biggr) \Biggr]^{r}\,dx \\ \leq & C\sum_{n=1}^{\infty }f(n) \int _{n^{\alpha }}^{\infty } x^{-r} \Biggl(\sum _{i=-\infty }^{\infty } \vert a_{i} \vert \Biggr)^{r-1} \Biggl(\sum_{i=- \infty }^{\infty } \vert a_{i} \vert E \Biggl\vert \sum _{j=i+1}^{i+n}\bigl(Y^{(1)}_{xj}-EY^{(1)}_{xj} \bigr) \Biggr\vert ^{r} \Biggr)\,dx \\ \leq & C\sum_{n=1}^{\infty }f(n) \int _{n^{\alpha }}^{\infty } x^{-r} \sum _{i=-\infty }^{\infty } \vert a_{i} \vert \sum _{j=i+1}^{i+n}E \bigl\vert Y^{(1)}_{xj}-EY^{(1)}_{xj} \bigr\vert ^{r}\,dx \\ &{}+C\sum_{n=1}^{\infty }f(n)g(n) \int _{n^{\alpha }}^{\infty } x^{-r} \sum _{i=-\infty }^{\infty } \vert a_{i} \vert \Biggl( \sum_{j=i+1}^{i+n}E \bigl\vert Y^{(1)}_{xj}-EY^{(1)}_{xj} \bigr\vert ^{2} \Biggr)^{r/2}\,dx \\ =&:I_{21}+I_{22}, \end{aligned}$$
(3.4)

where \(r\geq 2\) will be given later.

For \(I_{21}\), if \(p>1\), taking \(r>\max \{2,p\}\), then by \(C_{r}\) inequality, Lemma 2.3, and Lemma 2.4, we know

$$\begin{aligned} I_{21} \leq & C\sum _{n=1}^{\infty }f(n) \int _{n^{\alpha }}^{\infty } x^{-r} \sum _{i=-\infty }^{\infty } \vert a_{i} \vert \sum _{j=i+1}^{i+n}\bigl[E \vert Y_{j} \vert ^{r}I\bigl\{ \vert Y_{j} \vert \leq x\bigr\} +x^{r}P\bigl( \vert Y_{j} \vert >x\bigr)\bigr]\,dx \\ \leq & C\sum_{n=1}^{\infty }nf(n) \int _{n^{\alpha }}^{\infty } x^{-r} \bigl[E \vert Y \vert ^{r}I \bigl\{ \vert Y \vert \leq x\bigr\} +x^{r}P \bigl( \vert Y \vert >x\bigr)\bigr]\,dx \\ \leq & C\sum_{n=1}^{\infty }nf(n) \sum _{m=n}^{\infty } \int _{m^{ \alpha }}^{(m+1)^{\alpha }} \bigl[x^{-r}E \vert Y \vert ^{r}I\bigl\{ \vert Y \vert \leq x\bigr\} +P\bigl( \vert Y \vert >x\bigr)\bigr]\,dx \\ \leq & C\sum_{n=1}^{\infty }nf(n) \sum _{m=n}^{\infty } \bigl[m^{\alpha (1-r)-1}E \vert Y \vert ^{r}I \bigl\{ \vert Y \vert \leq (m+1)^{\alpha }\bigr\} + m^{\alpha -1}P\bigl( \vert Y \vert >m^{\alpha }\bigr)\bigr] \\ =&C\sum_{m=1}^{\infty } \bigl[m^{\alpha (1-r)-1}E \vert Y \vert ^{r}I\bigl\{ \vert Y \vert \leq (m+1)^{ \alpha }\bigr\} + m^{\alpha -1}P\bigl( \vert Y \vert >m^{\alpha }\bigr)\bigr] \sum_{n=1}^{m}nf(n) \\ \leq & C\sum_{m=1}^{\infty }m^{\alpha (p-r)-1}l(m) \sum_{k=1}^{m} E \vert Y \vert ^{r}I \bigl\{ k^{\alpha }< \vert Y \vert \leq (k+1)^{\alpha }\bigr\} \\ &{}+C\sum_{m=1}^{\infty }m^{\alpha p-1}l(m) \sum_{k=m}^{\infty }E I\bigl\{ k^{ \alpha }< \vert Y \vert \leq (k+1)^{\alpha }\bigr\} \\ =& C\sum_{k=1}^{\infty }E \vert Y \vert ^{r}I\bigl\{ k^{\alpha }< \vert Y \vert \leq (k+1)^{\alpha } \bigr\} \sum_{m=k}^{\infty }m^{\alpha (p-r)-1}l(m) \\ &{}+C\sum_{k=1}^{\infty } E I\bigl\{ k^{\alpha }< \vert Y \vert \leq (k+1)^{\alpha }\bigr\} \sum _{m=1}^{k} m^{\alpha p-1}l(m) \\ \leq &C\sum_{k=1}^{\infty }k^{\alpha (p-r)}l(k) E \vert Y \vert ^{p} \vert Y \vert ^{r-p}I\bigl\{ k^{ \alpha }< \vert Y \vert \leq (k+1)^{\alpha }\bigr\} \\ &{}+C\sum_{k=1}^{\infty }k^{\alpha p}l(k) E \vert Y \vert ^{p} \vert Y \vert ^{-p}I\bigl\{ k^{\alpha }< \vert Y \vert \leq (k+1)^{\alpha }\bigr\} \\ \leq & CE \vert Y \vert ^{p}l\bigl( \vert Y \vert ^{1/{\alpha }}\bigr) < \infty . \end{aligned}$$
(3.5)

For \(I_{21}\), if \(p=1\), taking \(r>\max \{1+\lambda ',2\}\), where \(0<\lambda '<\lambda \), then by the same argument as above we know

$$\begin{aligned} I_{21} \leq &C\sum _{m=1}^{\infty }\bigl[m^{\alpha (1-r)-1}E \vert Y \vert ^{r}I\bigl\{ \vert Y \vert \leq (m+1)^{ \alpha }\bigr\} + m^{\alpha -1}P\bigl( \vert Y \vert >m^{\alpha }\bigr)\bigr] \sum _{n=1}^{m}nf(n) \\ \leq & C\sum_{m=1}^{\infty } \bigl[m^{\alpha (1-r)-1}E \vert Y \vert ^{r}I\bigl\{ \vert Y \vert \leq (m+1)^{ \alpha }\bigr\} + m^{\alpha -1}P\bigl( \vert Y \vert >m^{\alpha }\bigr)\bigr] \sum_{n=1}^{m}n^{-1+ \alpha \lambda '}l(n) \\ \leq & C\sum_{m=1}^{\infty }m^{\alpha (1-r+\lambda ')-1}l(m) E \vert Y \vert ^{r}I \bigl\{ \vert Y \vert \leq (m+1)^{\alpha }\bigr\} \\ &{}+m^{\alpha (1+\lambda ')-1}l(m)E I\bigl\{ \vert Y \vert >m^{\alpha }\bigr\} \\ \leq & CE \vert Y \vert ^{1+\lambda '}l\bigl( \vert Y \vert ^{1/{\alpha }}\bigr) < \infty . \end{aligned}$$
(3.6)

For \(I_{22}\), if \(1\leq p<2\), noting that \(g(n)=O(n^{\delta })\), taking \(r>2\) such that \(\alpha p+r/2-\alpha pr/2-1+\delta =(\alpha p-1)(1-r/2)+\delta <0\), then by \(C_{r}\) inequality, Lemma 2.3, and Lemma 2.4, we obtain

$$\begin{aligned} I_{22} \leq & C\sum _{n=1}^{\infty }n^{r/2}f(n)g(n) \int _{n^{\alpha }}^{ \infty } x^{-r} \bigl[\bigl(E \vert Y \vert ^{2}I\bigl\{ \vert Y \vert \leq x\bigr\} \bigr)^{r/2}+x^{r}P^{r/2}\bigl( \vert Y \vert >x \bigr)\bigr]\,dx \\ \leq & C\sum_{n=1}^{\infty }n^{r/2}f(n) g(n)\sum_{m=n}^{\infty } \int _{m^{\alpha }}^{(m+1)^{\alpha }} \bigl[x^{-r}\bigl(E \vert Y \vert ^{2}I\bigl\{ \vert Y \vert \leq x\bigr\} \bigr)^{r/2}+P^{r/2}\bigl( \vert Y \vert >x\bigr)\bigr]\,dx \\ \leq & C\sum_{n=1}^{\infty }n^{r/2}f(n) g(n)\sum_{m=n}^{\infty } \bigl[m^{ \alpha (1-r)-1} \bigl(E \vert Y \vert ^{2}I\bigl\{ \vert Y \vert \leq (m+1)^{\alpha }\bigr\} \bigr)^{r/2}\\ &{} + m^{ \alpha -1}P^{r/2} \bigl( \vert Y \vert >m^{\alpha }\bigr)\bigr] \\ =&C\sum_{m=1}^{\infty } \bigl[m^{\alpha (1-r)-1}\bigl(E \vert Y \vert ^{2}I\bigl\{ \vert Y \vert \leq (m+1)^{ \alpha }\bigr\} \bigr)^{r/2} \\ &{}+ m^{\alpha -1}P^{r/2}\bigl( \vert Y \vert >m^{\alpha }\bigr) \bigr] \sum_{n=1}^{m}n^{r/2}f(n)g(n) \\ \leq & C\sum_{m=1}^{\infty }m^{\alpha (p-r)+r/2+\delta -2}l(m) \bigl(E \vert Y \vert ^{p} \vert Y \vert ^{2-p}I \bigl\{ \vert Y \vert \leq (m+1)^{\alpha }\bigr\} \bigr)^{r/2} \\ &{}+C\sum_{m=1}^{\infty }m^{\alpha p+r/2+\delta -2}l(m) \bigl(E \vert Y \vert ^{p} \vert Y \vert ^{-p}I \bigl\{ \vert Y \vert >m^{\alpha }\bigr\} \bigr)^{r/2} \\ \leq & C\sum_{m=1}^{\infty }m^{\alpha p+r/2-\alpha pr/2+\delta -2}l(m) \bigl(E \vert Y \vert ^{p}\bigr)^{r/2} < \infty . \end{aligned}$$
(3.7)

For \(I_{22}\), if \(p\geq 2\), noting that \(g(n)=O(n^{\delta })\), taking \(r>(\alpha p-1)/({\alpha -1/2})\geq p\) such that \(\alpha (p-r)+r/2+\delta -1<0\), then by \(C_{r}\) inequality, Lemma 2.3, and Lemma 2.4, similar to the proof of (3.7), one gets

$$\begin{aligned} I_{22} \leq &C\sum _{m=1}^{\infty }\bigl[m^{\alpha (1-r)-1}\bigl(E \vert Y \vert ^{2}I\bigl\{ \vert Y \vert \leq (m+1)^{ \alpha }\bigr\} \bigr)^{r/2}\\ &{} + m^{\alpha -1}P^{r/2}\bigl( \vert Y \vert >m^{\alpha }\bigr)\bigr] \sum_{n=1}^{m}n^{r/2}f(n)g(n) \\ \leq & C\sum_{m=1}^{\infty }m^{\alpha (p-r)+r/2+\delta -2}l(m) \bigl(E \vert Y \vert ^{2}I \bigl\{ \vert Y \vert \leq (m+1)^{\alpha }\bigr\} \bigr)^{r/2} \\ &{}+C\sum_{m=1}^{\infty }m^{\alpha p+r/2+\delta -2}l(m) \bigl(E \vert Y \vert ^{2} \vert Y \vert ^{-2}I \bigl\{ \vert Y \vert >m^{\alpha }\bigr\} \bigr)^{r/2} \\ \leq & C\sum_{m=1}^{\infty }m^{\alpha (p-r)+r/2+\delta -2}l(m) \bigl(E \vert Y \vert ^{2}\bigr)^{r/2} < \infty . \end{aligned}$$
(3.8)

Thus, (3.1) can be deduced immediately by combining (3.2)–(3.8). □

The next theorem will discuss the case \(\alpha p=1\).

Theorem 3.2

Let l be a function slowly varying at infinity, \(1\leq p<2\). Assume that \(\sum_{i=-\infty }^{\infty }|a_{i}|^{\theta }<\infty \), where θ belongs to \((0,1)\) if \(p=1\) and \(\theta =1\) if \(1< p<2\). Suppose that \(\{X_{n}=\sum_{i=-\infty }^{\infty } a_{i}Y_{i+n}, n\geq 1\}\) is a moving average process generated by a sequence \(\{Y_{i},-\infty < i<\infty \}\) of m-WOD random variables with dominating coefficients \(g(n)=O(n^{\delta })\) for some \(0\leq \delta <(2-p)/p\) which is stochastically dominated by a random variable Y. If \(EY_{i}=0\) and \(E|Y|^{p(1+\delta )}l(|Y|^{p})<\infty \), then for any \(\varepsilon >0\)

$$\begin{aligned} \sum_{n=1}^{\infty }n^{-1-1/p}l(n) E\Biggl\{ \Biggl\vert \sum_{j=1}^{k}X_{j} \Biggr\vert - \varepsilon n^{1/p}\Biggr\} ^{+} < \infty . \end{aligned}$$
(3.9)

Proof

Let \(h(n)=n^{-1-1/p}l(n)\). Similar to the proof of (3.2), we obtain

$$\begin{aligned}& \sum_{n=1}^{\infty }h(n) E \Biggl\{ \Biggl\vert \sum_{j=1}^{n}X_{j} \Biggr\vert -\varepsilon n^{1/p} \Biggr\} ^{+} \\& \quad \leq C\sum_{n=1}^{\infty }h(n) \int _{n^{1/p}}^{\infty } P\Biggl\{ \Biggl\vert \sum _{i=- \infty }^{\infty }a_{i} \sum _{j=i+1}^{i+n}Y_{xj}^{(2)} \Biggr\vert \geq \varepsilon x/2\Biggr\} \,dx \\& \qquad {}+C\sum_{n=1}^{\infty }h(n) \int _{n^{1/p}}^{\infty } P\Biggl\{ \Biggl\vert \sum _{i=- \infty }^{\infty }a_{i} \sum _{j=i+1}^{i+n}\bigl(Y_{xj}^{(1)}-EY_{xj}^{(1)} \bigr) \Biggr\vert \geq \varepsilon x/4\Biggr\} \,dx \\& \quad =:J_{1}+J_{2}. \end{aligned}$$
(3.10)

For \(J_{1}\), by Markov’s inequality, \(C_{r}\) inequality, Lemma 2.3, and Lemma 2.4, one gets

$$\begin{aligned} J_{1} \leq & C\sum _{n=1}^{\infty }h(n) \int _{n^{1/p}}^{\infty }x^{-\theta } E \Biggl\vert \sum _{i=-\infty }^{\infty }a_{i} \sum _{j=i+1}^{i+n}Y_{xj}^{(2)} \Biggr\vert ^{ \theta }\,dx \\ \leq & C\sum_{n=1}^{\infty }n h(n) \int _{n^{1/p}}^{\infty }x^{-\theta } E \vert Y \vert ^{\theta }I\bigl\{ \vert Y \vert >x\bigr\} \,dx \\ =&C\sum_{n=1}^{\infty }n h(n)\sum _{m=n}^{\infty } \int _{m^{1/p}}^{(m+1)^{1/p}}x^{- \theta } E \vert Y \vert ^{\theta }I\bigl\{ \vert Y \vert >x\bigr\} \,dx \\ \leq & C\sum_{n=1}^{\infty }n h(n)\sum _{m=n}^{\infty } m^{(1-\theta )/p-1}E \vert Y \vert ^{ \theta }I\bigl\{ \vert Y \vert >m^{1/p}\bigr\} \\ =&C\sum_{m=1}^{\infty }m^{(1-\theta )/p-1}E \vert Y \vert ^{\theta }I\bigl\{ \vert Y \vert >m^{1/p} \bigr\} \sum_{n=1}^{m}n h(n) \\ \leq &C\sum_{m=1}^{\infty }m^{-\theta /p}l(m)E \vert Y \vert ^{\theta }I\bigl\{ \vert Y \vert >m^{1/p} \bigr\} \\ =&C\sum_{m=1}^{\infty }m^{-\theta /p}l(m) \sum_{k=m}^{\infty } E \vert Y \vert ^{ \theta }I\bigl\{ k^{1/p}< \vert Y \vert < (k+1)^{1/p} \bigr\} \\ =&C\sum_{k=1}^{\infty }E \vert Y \vert ^{\theta }I\bigl\{ k^{1/p}< \vert Y \vert < (k+1)^{1/p}\bigr\} \sum_{m=1}^{k}m^{-\theta /p}l(m) \\ \leq &C\sum_{k=1}^{\infty }k^{1-\theta /p}l(k)E \vert Y \vert ^{\theta }I\bigl\{ k^{1/p}< \vert Y \vert < (k+1)^{1/p} \bigr\} \\ \leq &CE \vert Y \vert ^{p}l\bigl( \vert Y \vert ^{p}\bigr)< \infty . \end{aligned}$$
(3.11)

For \(J_{2}\), as the same argument of \(I_{2}\), noting that \(g(n)=O(n^{\delta })\) for some \(0\leq \delta <(2-p)/p\), taking \(r=2\), by Lemma 2.2, Lemma 2.3, and Lemma 2.4, we conclude

$$\begin{aligned} J_{2} \leq & C\sum _{n=1}^{\infty }h(n) \int _{n^{1/p}}^{\infty }x^{-2} E \Biggl\vert \sum _{i=-\infty }^{\infty }a_{i}\sum _{j=i+1}^{i+n}\bigl(Y_{xj}^{(1)}-EY_{xj}^{(1)} \bigr) \Biggr\vert ^{2}\,dx \\ \leq & C\sum_{n=1}^{\infty }n h(n) \bigl(1+g(n)\bigr) \int _{n^{1/p}}^{\infty }x^{-2} \bigl[E \vert Y \vert ^{2}I\bigl\{ \vert Y \vert \leq x\bigr\} +x^{2}P \bigl( \vert Y \vert >x\bigr)\bigr]\,dx \\ =& C\sum_{n=1}^{\infty }n h(n) \bigl(1+g(n)\bigr)\sum_{m=n}^{\infty } \int _{m^{1/p}}^{(m+1)^{1/p}}x^{-2} \bigl[E \vert Y \vert ^{2}I\bigl\{ \vert Y \vert \leq x\bigr\} +x^{2}P \bigl( \vert Y \vert >x\bigr)\bigr]\,dx \\ \leq & C\sum_{n=1}^{\infty }n h(n) \bigl(1+g(n)\bigr)\sum_{m=n}^{\infty } \bigl[m^{-1-1/p} E \vert Y \vert ^{2}I\bigl\{ \vert Y \vert \leq (m+1)^{1/p}\bigr\} \\ &{}+m^{1/p-1}P\bigl( \vert Y \vert >m^{1/p}\bigr)\bigr] \\ =& C\sum_{m=1}^{\infty }[m^{-1-1/p} \bigl[E \vert Y \vert ^{2}I\bigl\{ \vert Y \vert \leq (m+1)^{1/p}\bigr\} \\ &{}+m^{1/p-1}P\bigl( \vert Y \vert >m^{1/p}\bigr)\bigr] \sum_{n=1}^{m}n h(n) \bigl(1+g(n)\bigr) \\ \leq &C\sum_{m=1}^{\infty } \bigl[m^{-2/p+\delta }l(m)E \vert Y \vert ^{2}I\bigl\{ \vert Y \vert \leq (m+1)^{1/p} \bigr\} +m^{\delta }l(m)P\bigl( \vert Y \vert >m^{1/p}\bigr)\bigr] \\ \leq &C \sum_{m=1}^{\infty }m^{-2/p+\delta }l(m) \sum_{k=1}^{m}E \vert Y \vert ^{2}I \bigl\{ k^{1/p}< \vert Y \vert \leq (k+1)^{1/p}\bigr\} \\ &{}+C\sum_{m=1}^{\infty }m^{\delta }l(m) \sum_{k=m}^{\infty }EI\bigl\{ k^{1/p}< \vert Y \vert \leq (k+1)^{1/p}\bigr\} \\ \leq &C \sum_{k=1}^{\infty }E \vert Y \vert ^{2}I\bigl\{ k^{1/p}< \vert Y \vert \leq (k+1)^{1/p}\bigr\} \sum_{m=k}^{\infty } m^{-2/p+\delta }l(m) \\ &{}+C\sum_{k=1}^{\infty }EI\bigl\{ k^{1/p}< \vert Y \vert \leq (k+1)^{1/p}\bigr\} \sum _{m=1}^{k}m^{ \delta }l(m) \\ \leq &C \sum_{k=1}^{\infty }k^{-2/p+\delta +1}l(k)E \vert Y \vert ^{2}I\bigl\{ k^{1/p}< \vert Y \vert \leq (k+1)^{1/p}\bigr\} \\ &{}+C\sum_{k=1}^{\infty }k^{\delta +1}l(k)EI \bigl\{ k^{1/p}< \vert Y \vert \leq (k+1)^{1/p} \bigr\} \\ \leq &C \sum_{k=1}^{\infty }l(k)E \vert Y \vert ^{p(1+\delta )}I\bigl\{ k^{1/p}< \vert Y \vert \leq (k+1)^{1/p}\bigr\} \\ \leq &CE \vert Y \vert ^{p(1+\delta )}l\bigl( \vert Y \vert ^{p}\bigr)< \infty . \end{aligned}$$
(3.12)

Hence, by combining (3.10)–(3.12), (3.9) holds. □

For the complete convergence, 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

$$\begin{aligned} \sum_{n=1}^{\infty }n^{\alpha p-2}l(n) P\Biggl\{ \Biggl\vert \sum_{j=1}^{n}X_{j} \Biggr\vert > \varepsilon n^{\alpha }\Biggr\} < \infty . \end{aligned}$$
(3.13)

Under the assumptions of Theorem 3.2, for any \(\varepsilon >0\), we have

$$\begin{aligned} \sum_{n=1}^{\infty }n^{-1}l(n) P\Biggl\{ \Biggl\vert \sum_{j=1}^{n}X_{j} \Biggr\vert >\varepsilon n^{1/p} \Biggr\} < \infty . \end{aligned}$$
(3.14)

Remark 3.4

Since m-WOD random variables include independent, m-NA, NSD, WOD, m-NOD, and m-END random variables, so our results also hold for independent, m-NA, NSD, WOD, m-NOD, and m-END random variables, and therefore Theorem 3.1 and Theorem 3.2 improve upon the known results.

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 results of Theorem 3.1 and Theorem 3.2 also hold for identically distributed random variables.

Remark 3.6

Let \(a_{0}=1\), \(a_{i}=0\), \(i\neq 0\), then \(S_{n}=\sum_{k=1}^{n}X_{k}=\sum_{k=1}^{n}Y_{k}\). Hence the results of Theorem 3.1 and Theorem 3.2 also hold when \(\{X_{k},k\geq 1\}\) is a sequence of m-WOD random variables which is stochastically dominated by a random variable Y.

Remark 3.7

The results obtained by this paper and Fang et al. [19] are different. In our paper, we mainly discuss the complete moment convergence of moving average processes for an m-WOD sequence, Fang et al. [19] proved the asymptotic approximations of ratio moments based on the m-WOD sequence.

4 Conclusions

In this paper, using the moment inequality for m-WOD sequences and truncation method, the complete moment convergence for the partial sum of moving average processes \(\{X_{n}=\sum_{i=-\infty }^{\infty }a_{i}Y_{i+n},n\geq 1\}\) is established, where \(\{Y_{i},-\infty < i<\infty \}\) is a sequence of m-WOD random variables which is stochastically dominated by a random variable Y, and \(\{a_{i},-\infty < i<\infty \}\) is an absolutely summable sequence of real numbers. These conclusions obtained extend and improve the corresponding results from m-END sequences to m-WOD sequences.