Complete moment convergence for moving average process generated by $\rho^{-}$-mixing random variables

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

$$\rho^{-}(s)=\sup\bigl\{ \rho^{-}(S,T); S,T\subset{Z}, \operatorname{dist}(S,T)\geq{s}\bigr\} \rightarrow{0}\quad \mbox{as } s\rightarrow\infty, $$

where

$$\rho^{-}(S,T)=0\vee{\sup}\{\operatorname{corr}\bigl(f(X_{i},i \in{S}),g(X_{j},j\in{T})\bigr), $$

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

$$\rho^{*}(s)=\sup\bigl\{ \rho(S,T);S,T\subset Z, \operatorname{dist}(S,T)\geq s \bigr\} \rightarrow0\quad \mbox{as } s\rightarrow\infty, $$

where

$$\rho(S,T)=\sup\bigl\{ \bigl\vert \operatorname{corr}(f,g)\bigr\vert ; f\in L_{2}\bigl(\sigma(S)\bigr),g\in L_{2}\bigl(\sigma (T)\bigr) \bigr\} . $$

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 $R^{T}$

$$\operatorname{Cov}\bigl\{ f(Y_{i},i\in S),g(Y_{j},j\in T)\bigr\} \leq0. $$

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

$$P\bigl\{ \vert Y_{i}\vert >x\bigr\} \leq CP\bigl\{ \vert Y\vert >x\bigr\} ,\quad x\geq0, -\infty< i< \infty. $$

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

$$\begin{aligned} \sum_{n=1}^{\infty}n^{r/q-2-1/q}h(n) E \Biggl\{ \Biggl\vert \sum_{j=1}^{n}X_{j} \Biggr\vert -\varepsilon n^{1/q}\Biggr\} ^{+} < \infty \end{aligned}$$

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

$$\begin{aligned} \sum_{n=1}^{\infty}n^{r-2} P\Biggl\{ \max_{1\leq k\leq n}\Biggl\vert \sum_{j=1}^{k}X_{j} \Biggr\vert \geq\varepsilon n^{1/p}\Biggr\} < \infty \end{aligned}$$

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

$$\begin{aligned} \sum_{n=1}^{\infty}n^{r-2-q/p} E\Biggl( \Biggl\{ \max_{1\leq k\leq n}\Biggl\vert \sum _{j=1}^{k}X_{j}\Biggr\vert -\varepsilon n^{1/q}\Biggr\} ^{+}\Biggr)^{q} < \infty \end{aligned}$$

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 $\varepsilon>0$,

$$\begin{aligned} \sum_{n=1}^{\infty}n^{p\alpha-2-\alpha}h(n) E \Biggl\{ \max_{1\leq k\leq n}\Biggl\vert \sum _{j=1}^{k}X_{j}\Biggr\vert -\varepsilon n^{\alpha}\Biggr\} ^{+} < \infty \end{aligned}$$

and

$$\begin{aligned} \sum_{n=1}^{\infty}n^{p\alpha-2}h(n) E \Biggl\{ \sup_{k\geq n}\Biggl\vert k^{-\alpha}\sum _{j=1}^{k}X_{j}\Biggr\vert -\varepsilon \Biggr\} ^{+} < \infty. \end{aligned}$$

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

$${E}\Biggl(\max_{1\leq{j}\leq{n}}\Biggl\vert \sum _{i=1}^{j}X_{i}\Biggr\vert ^{q}\Biggr)\leq {C}\Biggl\{ \sum_{i=1}^{n}{E} \vert X_{i}\vert ^{q}+ \Biggl(\sum _{i=1}^{n}{E}X_{i}^{2} \Biggr)^{\frac{q}{2}}\Biggr\} . $$

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 $b>0$,

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

3 Main results and proofs

Theorem 3.1

Let l be a function slowly varying at infinity, $p\geq1$, $\alpha>1/2$, $\alpha p> 1$. 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 $\varepsilon>0$

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

(3.1)

and

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

(3.2)

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 $\alpha>1$

$$\begin{aligned} &x^{-1}\Biggl\vert E\sum_{i=-\infty}^{\infty}a_{i} \sum_{j=i+1}^{i+n}Y^{(1)} _{xj}\Biggr\vert \\ &\quad \leq x^{-1}\sum_{i=-\infty}^{\infty} \vert a_{i}\vert \sum_{j=i+1}^{i+n} \bigl[E\vert Y_{j}\vert I\bigl\{ \vert Y_{j}\vert \leq x\bigr\} +xP\bigl(\vert Y\vert >x\bigr)\bigr] \\ &\quad \leq Cx^{-1}n\bigl[E\vert Y\vert I\bigl\{ \vert Y\vert \leq x \bigr\} +x P\bigl(\vert Y\vert >x\bigr)\bigr] \leq C n^{1-\alpha}\to0, \quad \mbox{as } n\to\infty. \end{aligned}$$

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

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

Hence for $x>n^{\alpha}$ large enough, we get

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

Therefore

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

(3.3)

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

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

If $p>1$, then $\alpha p-1-\alpha>-1$, and, by Lemma 2.1, we obtain

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

If $p=1$, 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

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

So, we get

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

(3.4)

Next we show $I_{2}<\infty$. By Markov’s inequality, the Hölder inequality, and Lemma 2.2, we conclude

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

(3.5)

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

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

(3.6)

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

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

(3.7)

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

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

(3.8)

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

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

(3.9)

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

$$\begin{aligned} &\sum_{n=1}^{\infty}n^{\alpha p-2}l(n) E \Biggl\{ \sup_{ k\geq n}\Biggl\vert k^{-\alpha}\sum _{j=1}^{k}X_{j}\Biggr\vert -\varepsilon \Biggr\} ^{+} \\ &\quad = \sum_{n=1}^{\infty}n^{\alpha p-2}l(n) \int_{0}^{\infty} P\Biggl\{ \sup_{ k\geq n} \Biggl\vert k^{-\alpha}\sum_{j=1}^{k}X_{j} \Biggr\vert >\varepsilon+t \Biggr\} \,dt \\ &\quad = \sum_{i=1}^{\infty}\sum _{n=2^{i-1}}^{2^{i}-1}n^{\alpha p-2}l(n)\int _{0}^{\infty} P\Biggl\{ \sup_{ k\geq n} \Biggl\vert k^{-\alpha}\sum_{j=1}^{k}X_{j} \Biggr\vert >\varepsilon+t \Biggr\} \,dt \\ &\quad \leq C\sum_{i=1}^{\infty}\int _{0}^{\infty} P\Biggl\{ \sup_{ k\geq 2^{i-1}} \Biggl\vert k^{-\alpha}\sum_{j=1}^{k}X_{j} \Biggr\vert >\varepsilon+t \Biggr\} \,dt \sum_{n=2^{i-1}}^{2^{i}-1}n^{\alpha p-2}l(n) \\ &\quad \leq C\sum_{i=1}^{\infty}2^{i(\alpha p-1)}l \bigl(2^{i}\bigr)\int_{0}^{\infty} P\Biggl\{ \sup_{ k\geq2^{i-1}}\Biggl\vert k^{-\alpha}\sum _{j=1}^{k}X_{j}\Biggr\vert > \varepsilon+t \Biggr\} \,dt \\ &\quad \leq C\sum_{i=1}^{\infty}2^{i(\alpha p-1)}l \bigl(2^{i}\bigr)\sum_{l=i}^{\infty} \int_{0}^{\infty} P\Biggl\{ \max_{ 2^{l-1}\leq k< 2^{l}} \Biggl\vert k^{-\alpha}\sum_{j=1}^{k}X_{j} \Biggr\vert >\varepsilon +t \Biggr\} \,dt \\ &\quad \leq C\sum_{l=1}^{\infty}\int _{0}^{\infty} P\Biggl\{ \max_{ 2^{l-1}\leq k< 2^{l}} \Biggl\vert k^{-\alpha}\sum_{j=1}^{k}X_{j} \Biggr\vert >\varepsilon+t \Biggr\} \,dt \sum_{i=1}^{l}2^{i(\alpha p-1)}l \bigl(2^{i}\bigr) \\ &\quad \leq C\sum_{l=1}^{\infty}2^{l(\alpha p-1)}l \bigl(2^{l}\bigr)\int_{0}^{\infty} P\Biggl\{ \max_{ 2^{l-1}\leq k< 2^{l}}\Biggl\vert \sum_{j=1}^{k}X_{j} \Biggr\vert >(\varepsilon +t)2^{(l-1)\alpha} \Biggr\} \,dt \\ &\quad \leq C\sum_{l=1}^{\infty}2^{l(\alpha p-1-\alpha)}l \bigl(2^{l}\bigr)\int_{0}^{\infty} P\Biggl\{ \max_{ 1\leq k< 2^{l}}\Biggl\vert \sum_{j=1}^{k}X_{j} \Biggr\vert >\varepsilon2^{(l-1)\alpha}+y \Biggr\} \,dy \\ &\quad \leq C\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha}l(n) \int_{0}^{\infty} P\Biggl\{ \max_{ 1\leq k< n} \Biggl\vert \sum_{j=1}^{k}X_{j} \Biggr\vert >\varepsilon n^{\alpha}2^{-\alpha }+y \Biggr\} \,dy \\ &\quad = C\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha}l(n) E \Biggl\{ \max_{ 1\leq k< n}\Biggl\vert \sum _{j=1}^{k}X_{j}\Biggr\vert - \varepsilon_{0} n^{\alpha}\Biggr\} ^{+} < \infty. \end{aligned}$$

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 $\varepsilon>0$

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

(3.10)

Proof

Let $g(n)=n^{-1-1/p}l(n)$. Similarly to the proof of (3.3), we have

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

(3.11)

For $J_{1}$, by Markov’s inequality, the $C_{r}$ inequality, Lemma 2.3, and Lemma 2.1, one gets

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

(3.12)

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

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

(3.13)

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

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

(3.14)

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

$$\begin{aligned} \sum_{n=1}^{\infty}n^{-1}l(n) P\Biggl\{ \max_{1\leq k\leq n}\Biggl\vert \sum _{j=1}^{k}X_{j}\Biggr\vert >\varepsilon n^{1/p}\Biggr\} < \infty; \end{aligned}$$

(3.15)

in particular, the assumptions $EY_{i}=0$ and $E\vert Y\vert ^{p}<\infty$ imply the following Marcinkiewicz-Zygmund strong law of large numbers:

$$\begin{aligned} \lim_{n\to\infty}\frac{1}{n^{1/p}}\sum _{j=1}^{n}X_{j}=0\quad \textit{a.s.} \end{aligned}$$

(3.16)

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.