Open Access

Complete moment convergence of moving average process generated by a class of random variables

Journal of Inequalities and Applications20152015:225

https://doi.org/10.1186/s13660-015-0745-x

Received: 13 April 2015

Accepted: 29 June 2015

Published: 17 July 2015

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.

Keywords

complete moment convergence moving average process Rosenthal-type maximal inequality weak mean domination slowly varying

MSC

60F15

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
$$ X_{n}=\sum_{i=-\infty}^{\infty}a_{i} Y_{i+n},\quad n\geq1. $$
(1.1)

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
$$ \frac{1}{n} \sum_{k=1}^{n} P\bigl(|Y_{k}|>x\bigr)\leq CP\bigl(|Y|>x\bigr) $$
(1.2)
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
$$ E\Biggl(\max_{1\leq j \leq n}\Biggl|\sum_{i=1}^{j} (Y_{i}-EY_{i})\Biggr|\Biggr)^{q}\leq C_{q} \Biggl\{ \sum_{i=1}^{n} E|Y_{i}|^{q}+ \Biggl(\sum_{i=1}^{n} EY_{i}^{2} \Biggr)^{q/2}\Biggr\} . $$
(1.3)
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 \(\lambda>0\)
$$\lim_{x\rightarrow\infty}\frac{h(\lambda x)}{h(x)}=1. $$

Lemma 2.1

(Zhou [4])

If h is a slowly varying function at infinity and m a positive integer, then
  1. (1)

    \(\sum_{n=1}^{m} n^{t} h(n)\leq C m^{t+1} h(m)\) for \(t>-1\),

     
  2. (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
$$\frac{1}{n}\sum_{i=1}^{n} P\bigl(|X_{i}|>x\bigr)\leq C P\bigl(|X|>x\bigr) \quad\textit{for all } x>0 \textit{ and all }n \geq1. $$
Let \(r>0\) and for some \(A>0\)
$$\begin{aligned}& X_{i}^{\prime}=X_{i} I\bigl(|X_{i}|\leq A\bigr),\qquad X_{i}^{\prime\prime}=X_{i} I\bigl(|X_{i}|>A\bigr),\\& X_{i}^{*}=X_{i} I\bigl(|X_{i}|\leq A\bigr)-AI(X_{i}< -A)+AI(X_{i}>A), \end{aligned}$$
and
$$\begin{aligned}& X^{\prime}=X I\bigl(|X|\leq A\bigr),\qquad X^{\prime\prime}=X I\bigl(|X|>A\bigr),\\& X^{*}=X I\bigl(|X|\leq A\bigr)-AI(X< -A)+AI(X>A) . \end{aligned}$$
Then for some \(C>0\)
  1. (1)

    if \(E|X|^{r}<\infty\), then \((n^{-1})\sum_{i=1}^{n} E|X_{i}|^{r}\leq CE|X|^{r}\),

     
  2. (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. (3)

    \((n^{-1})\sum_{i=1}^{n} E|X_{i}^{\prime\prime}|^{r}\leq CE|X^{\prime\prime}|^{r}\) for any \(A>0\),

     
  4. (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
$$\frac{1}{n}\sum_{i=j+1}^{j+n} P\bigl(|Y_{i}|>x\bigr)\leq C P\bigl(|Y|>x\bigr) \quad\textit{for all } x>0, -\infty< j< \infty $$
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
$$ E\Biggl(\max_{1\leq i \leq n}\Biggl|\sum_{j=1}^{i} (Y_{xj}-EY_{xj})\Biggr|^{q}\Biggr)\leq C_{q} \Biggl\{ \sum_{j=1}^{n} E|Y_{xj}|^{q}+ \Biggl(\sum_{j=1}^{n} EY_{xj}^{2} \Biggr)^{q/2}\Biggr\} , $$
(3.1)
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 \(\epsilon>0\)
$$ \sum_{n=1}^{\infty}n^{\alpha p-2-\alpha}h(n)E\Biggl\{ \max_{1\leq i\leq n}\Biggl|\sum_{j=1}^{i} X_{j}\Biggr|-\epsilon n^{\alpha}\Biggr\} ^{+}< \infty $$
(3.2)
and
$$ \sum_{n=1}^{\infty}n^{\alpha p-2}h(n)E\Biggl\{ \sup_{i\geq n}\Biggl|i^{-\alpha}\sum_{j=1}^{i} X_{j}\Biggr|-\epsilon\Biggr\} ^{+}< \infty. $$
(3.3)

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 \(x>n^{\alpha}\),
$$\begin{aligned} x^{-1}\Biggl|E\sum_{i=-\infty}^{\infty}a_{i} \sum_{j=i+1}^{i+n} Y_{xj}\Biggr| &\leq Cx^{-1} n\bigl\{ E|Y|I\bigl[|Y|\leq x\bigr]+xP\bigl(|Y|>x\bigr)\bigr\} \\ &\leq C n^{1-\alpha}\rightarrow0 \quad\mbox{as }n\rightarrow\infty. \end{aligned}$$
(3.4i)
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
$$\begin{aligned} x^{-1}\Biggl|E\sum_{i=-\infty}^{\infty}a_{i} \sum_{j=i+1}^{i+n} Y_{xj}\Biggr|&=x^{-1}\Biggl|E\sum_{i=-\infty}^{\infty}a_{i}\sum_{j=i+1}^{i+n}\tilde{ Y_{xj}}\Biggr| \\ &\leq Cx^{-1}\sum_{i=-\infty}^{\infty}|a_{i}| \sum_{j=i+1}^{i+n}E|Y_{j}|I\bigl[|Y_{j}|>x\bigr] \\ &\leq Cx^{-1} n E|Y|I\bigl[|Y|>x\bigr]\leq Cx^{\frac{1}{\alpha}-1}E|Y|I\bigl[|Y|>x\bigr] \\ &\leq CE|Y|^{p}I\bigl[|Y|>x\bigr]\rightarrow0\quad \mbox{as }x\rightarrow\infty. \end{aligned}$$
(3.4ii)
It follows from (3.4i) and (3.4ii) that for \(x>n^{\alpha}\) large enough,
$$ x^{-1}\Biggl|E\sum_{i=-\infty}^{\infty}a_{i} \sum_{j=i+1}^{i+n} Y_{xj}\Biggr|< \frac {\epsilon}{4}, $$
(3.5)
which yields
$$\begin{aligned} &\sum_{n=1}^{\infty}l(n) E\Biggl\{ \max_{1\leq k \leq n}\Biggl|\sum_{j=1}^{k} X_{j}\Biggr|-\epsilon n^{\alpha}\Biggr\} ^{+} \\ &\quad\leq\sum_{n=1}^{\infty}l(n)\int _{\epsilon n^{\alpha}}^{\infty}P\Biggl(\max_{1\leq k \leq n}\Biggl|\sum _{j=1}^{k} X_{j}\Biggr|\geq x\Biggr)\,dx \quad\bigl(\mbox{letting } x=\epsilon x^{\prime}\bigr) \\ &\quad\leq\epsilon\sum_{n=1}^{\infty}l(n)\int _{n^{\alpha}}^{\infty}P\Biggl(\max_{1\leq k \leq n}\Biggl|\sum _{j=1}^{k} X_{j}\Biggr|\geq \epsilon x^{\prime}\Biggr)\,dx^{\prime} \\ &\quad\leq C \sum_{n=1}^{\infty}l(n)\int _{n^{\alpha}}^{\infty}P\Biggl(\max_{1\leq k \leq n}\Biggl|\sum _{i=-\infty}^{\infty}a_{i}\sum _{j=i+1}^{i+k} \tilde {Y_{xj}}\Biggr|\geq \frac{\epsilon x}{2}\Biggr)\,dx \\ &\qquad{}+C\sum_{n=1}^{\infty}l(n)\int _{n^{\alpha}}^{\infty}P\Biggl(\max_{1\leq k \leq n}\Biggl|\sum _{i=-\infty}^{\infty}a_{i}\sum _{j=i+1}^{i+k}( Y_{xj}-E Y_{xj})\Biggr|\geq \frac{\epsilon x}{4}\Biggr)\,dx \\ &\quad=I_{1}+I_{2}. \end{aligned}$$
(3.6)
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
$$\begin{aligned} I_{1} \leq&C\sum_{n=1}^{\infty}l(n) \int_{n^{\alpha}}^{\infty}x^{-1}E\max _{1\leq k\leq n}\Biggl|\sum_{i=-\infty}^{\infty}a_{i} \sum_{j=i+1}^{i+k}\tilde {Y_{xj}}\Biggr|\,dx \\ \leq&C\sum_{n=1}^{\infty}l(n) \int _{n^{\alpha}}^{\infty}x^{-1} \sum _{-\infty}^{\infty}|a_{i}| \sum _{j=i+1}^{i+n}E|\tilde{Y_{xj}}|\,dx \\ \leq&C\sum_{n=1}^{\infty}n l(n) \int _{n^{\alpha}}^{\infty}x^{-1} E|Y|I\bigl[|Y|>x\bigr]\,dx \\ =&C\sum_{n=1}^{\infty}n l(n)\sum _{m=n}^{\infty}\int_{m^{\alpha}}^{(m+1)^{\alpha}} x^{-1}E|Y|I\bigl[|Y|>x\bigr]\,dx \\ \leq&C\sum_{n=1}^{\infty}n l(n)\sum _{m=n}^{\infty}m^{-1}E|Y|I\bigl[|Y|>m^{\alpha}\bigr] \\ =&C\sum_{m=1}^{\infty}m^{-1}E|Y|I \bigl[|Y|>m^{\alpha}\bigr]\sum_{n=1}^{m} n^{\alpha p-1-\alpha}h(n). \end{aligned}$$
(3.7)
If \(p>1\), note that \(\alpha p-1-\alpha>-1\). By Lemma 2.1 and (3.7) we obtain
$$\begin{aligned} I_{1} \leq&C\sum_{m=1}^{\infty}m^{\alpha p-1-\alpha}h(m)E|Y|I\bigl[|Y|>m^{\alpha}\bigr] \\ =&C\sum_{m=1}^{\infty}m^{\alpha p-1-\alpha}h(m) \sum_{k=m}^{\infty}E|Y|I\bigl[k^{\alpha}< |Y| \leq(k+1)^{\alpha}\bigr] \\ =&C\sum_{k=1}^{\infty}E|Y|I \bigl[k^{\alpha}< |Y|\leq(k+1)^{\alpha}\bigr]\sum _{m=1}^{k} m^{\alpha p-1-\alpha}h(m) \\ \leq&C\sum_{k=1}^{\infty}k^{\alpha p-\alpha}h(k) E|Y|I\bigl[k^{\alpha}< |Y|\leq (k+1)^{\alpha}\bigr] \\ \leq&C E|Y|^{p} h\bigl(|Y|^{\frac{1}{\alpha}}\bigr)< \infty. \end{aligned}$$
(3.8)
If \(p=1\), by (3.7), we also obtain
$$\begin{aligned} I_{1} \leq&C\sum_{m=1}^{\infty}m^{-1}E|Y|I\bigl[|Y|>m^{\alpha}\bigr]\sum _{n=1}^{m} n^{-1}h(n) \\ \leq&C\sum_{m=1}^{\infty}m^{-1}E|Y|I \bigl[|Y|>m^{\alpha}\bigr]\sum_{n=1}^{m} n^{-1+\alpha\delta}h(n) \quad\mbox{for any }\delta>0 \\ \leq&C\sum_{m=1}^{\infty}m^{\alpha\delta -1}h(m)E|Y|I \bigl[|Y|>m^{\alpha}\bigr] \\ \leq&C E|Y|^{1+\delta} h\bigl(|Y|^{\frac{1}{\alpha}}\bigr)< \infty. \end{aligned}$$
(3.9)
So, by (3.8) and (3.9) we get
$$ I_{1}< \infty \quad\mbox{for } p\geq1. $$
(3.10)
For \(I_{2}\), by Markov’s inequality, Hölder’s inequality, and (3.1) we get for any \(q\geq2\)
$$\begin{aligned} I_{2} \leq&C\sum_{n=1}^{\infty}l(n) \int_{n^{\alpha}}^{\infty}x^{-q}E\max _{1\leq k\leq n}\Biggl|\sum_{i=-\infty}^{\infty}a_{i} \sum_{j=i+1}^{i+k}(Y_{xj}-E Y_{xj})\Biggr|^{q}\,dx \\ \leq&C\sum_{n=1}^{\infty}l(n) \int _{n^{\alpha}}^{\infty}x^{-q} \\ &{} \times E\Biggl[\sum_{i=-\infty}^{\infty}\bigl(|a_{i}|^{1-\frac{1}{q}}\bigr) \Biggl(|a_{i}|^{\frac {1}{q}} \max_{1\leq k\leq n}\Biggl|\sum_{j=i+1}^{i+k}(Y_{xj}-E Y_{xj})\Biggr|\Biggr)\Biggr]^{q}\,dx \\ \leq&C\sum_{n=1}^{\infty}l(n) \int _{n^{\alpha}}^{\infty}x^{-q} \\ &{} \times\Biggl(\sum_{i=-\infty}^{\infty}|a_{i}| \Biggr)^{q-1}\Biggl(\sum_{i=-\infty}^{\infty}|a_{i}|E\max_{1\leq k\leq n}\biggl|\sum_{j=i+1}^{i+k}(Y_{xj}-E Y_{xj})\biggr|^{q}\Biggr)\,dx \\ \leq&C\sum_{n=1}^{\infty}l(n) \int _{n^{\alpha}}^{\infty}x^{-q}\sum _{i=-\infty}^{\infty}|a_{i}|\sum _{j=i+1}^{i+n}E|Y_{xj}-E Y_{xj}|^{q}\,dx \\ &{} +C\sum_{n=1}^{\infty}l(n) \int _{n^{\alpha}}^{\infty}x^{-q}\sum _{i=-\infty }^{\infty}|a_{i}|\Biggl(\sum _{j=i+1}^{i+n}E|Y_{xj}-E Y_{xj}|^{2} \Biggr)^{\frac{q}{2}}\,dx \\ =:&I_{21}+II_{22}. \end{aligned}$$
(3.11)
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
$$\begin{aligned} I_{21} \leq&C\sum_{n=1}^{\infty}n l(n) \int_{n^{\alpha}}^{\infty}x^{-q}\bigl\{ E|Y|^{q}I\bigl[|Y|\leq x\bigr]+ x^{q}P\bigl(|Y|>x\bigr)\bigr\} \,dx \\ \leq&C\sum_{n=1}^{\infty}n l(n) \sum _{m=n}^{\infty}\int_{m^{\alpha}}^{(m+1)^{\alpha}} \bigl\{ x^{-q}E|Y|^{q}I\bigl[|Y|\leq x\bigr]+ P\bigl(|Y|>x\bigr)\bigr\} \,dx \\ \leq&C\sum_{n=1}^{\infty}n l(n) \sum _{m=n}^{\infty}\bigl\{ m^{\alpha(1-q)-1} E|Y|^{q}I \bigl[|Y|\leq(m+1)^{\alpha}\bigr]+ m^{\alpha-1}P\bigl(|Y|>m^{\alpha}\bigr)\bigr\} \\ =&C\sum_{m=1}^{\infty}\bigl\{ m^{\alpha(1-q)-1} E|Y|^{q}I\bigl[|Y|\leq(m+1)^{\alpha}\bigr]+ m^{\alpha-1}P \bigl(|Y|>m^{\alpha}\bigr)\bigr\} \sum_{n=1}^{m} n l(n) \\ \leq&C\sum_{m=1}^{\infty}m^{\alpha(p-q)-1}h(m) \sum_{k=1}^{m} E|Y|^{q}I \bigl[k^{\alpha}< |Y|\leq(k+1)^{\alpha}\bigr] \\ &{} +C\sum_{m=1}^{\infty}m^{\alpha p-1}h(m) \sum_{k=m}^{\infty}EI\bigl[k^{\alpha}< |Y| \leq(k+1)^{\alpha}\bigr] \\ =&C\sum_{k=1}^{\infty}E|Y|^{q}I \bigl[k^{\alpha}< |Y|\leq(k+1)^{\alpha}\bigr]\sum _{m=k}^{\infty}m^{\alpha(p-q)-1}h(m) \\ &{} +C\sum_{k=1}^{\infty}EI \bigl[k^{\alpha}< |Y|\leq(k+1)^{\alpha}\bigr]\sum _{m=1}^{k} m^{\alpha p-1}h(m) \\ \leq&C\sum_{k=1}^{\infty}k^{\alpha(p-q)}h(k) E|Y|^{q}I\bigl[k^{\alpha}< |Y|\leq (k+1)^{\alpha}\bigr] \\ &{} +C\sum_{k=1}^{\infty}k^{\alpha p}h(k) EI\bigl[k^{\alpha}< |Y|\leq (k+1)^{\alpha}\bigr] \\ \leq&CE|Y|^{p} h\bigl(|Y|^{\frac{1}{\alpha}}\bigr)< \infty. \end{aligned}$$
(3.12)
For \(I_{21}\), if \(p=1\), take \(q>\max\{1+\delta, 2\}\) by the same argument as above one gets for any \(\delta>0\)
$$\begin{aligned} I_{21} \leq&C\sum_{m=1}^{\infty}\bigl\{ m^{\alpha(1-q)-1}E|Y|^{q}I\bigl[|Y|\leq (m+1)^{\alpha}\bigr]+m^{\alpha-1}P\bigl(|Y|>m^{\alpha}\bigr)\bigr\} \sum _{n=1}^{m} n l(n) \\ =&C\sum_{m=1}^{\infty}\bigl\{ m^{\alpha(1-q)-1}E|Y|^{q}I\bigl[|Y|\leq(m+1)^{\alpha}\bigr]+m^{\alpha-1}P\bigl(|Y|>m^{\alpha}\bigr)\bigr\} \sum _{n=1}^{m} n^{-1} l(n) \\ \leq&C\sum_{m=1}^{\infty}\bigl\{ m^{\alpha(1-q)-1}E|Y|^{q}I\bigl[|Y|\leq(m+1)^{\alpha}\bigr] +m^{\alpha-1}P\bigl(|Y|>m^{\alpha}\bigr)\bigr\} \sum _{n=1}^{m} n^{-1+\alpha\delta }h(n) \\ \leq&C\sum_{m=1}^{\infty}\bigl\{ m^{\alpha(1-q+\delta)-1}h(n)E|Y|^{q}I\bigl[|Y|\leq (m+1)^{\alpha}\bigr] +m^{\alpha(1+\delta)-1}h(x)EI\bigl[|Y|>m^{\alpha}\bigr]\bigr\} \\ \leq&CE|Y|^{1+\delta}h\bigl(|Y|^{\frac{1}{\alpha}}\bigr)< \infty. \end{aligned}$$
(3.13)
It follows from (3.12) and (3.13) that, for \(p\geq1\),
$$ I_{21}< \infty. $$
(3.14)
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
$$\begin{aligned} I_{22} \leq&C\sum_{n=1}^{\infty}n^{\frac{q}{2}} l(n) \int_{n^{\alpha}}^{\infty}x^{-q}\bigl\{ \bigl(E|Y|^{2}I[|Y|\leq x]\bigr)^{\frac {q}{2}}+x^{q} \bigl(P(|Y|>x)\bigr)^{\frac{q}{2}}\bigr\} \,dx \\ \leq&C\sum_{n=1}^{\infty}n^{\frac{q}{2}} l(n)\sum_{m=n}^{\infty}\int_{m^{\alpha}}^{(m+1)^{\alpha}} \bigl\{ x^{-q}\bigl(E|Y|^{2}I[|Y|\leq x]\bigr)^{\frac {q}{2}}+ \bigl(P(|Y|>x)\bigr)^{\frac{q}{2}}\bigr\} \,dx \\ \leq&C\sum_{n=1}^{\infty}n^{\frac{q}{2}} l(n)\sum_{m=n}^{\infty}\bigl\{ m^{\alpha(1-q)-1} \bigl(E|Y|^{2}I\bigl[|Y|\leq(m+1)^{2}\bigr] \bigr)^{\frac{q}{2}} +m^{\alpha-1}\bigl(P\bigl(|Y|>m^{\alpha}\bigr) \bigr)^{\frac{q}{2}}\bigr\} \\ =&C\sum_{m=1}^{\infty}\bigl\{ m^{\alpha(1-q)-1} \bigl(E|Y|^{2}I\bigl[|Y|\leq(m+1)^{\alpha}\bigr] \bigr)^{\frac{q}{2}} +m^{\alpha-1}\bigl(P\bigl(|Y|>m^{\alpha}\bigr) \bigr)^{\frac{q}{2}}\bigr\} \sum_{n=1}^{m} n^{\frac {q}{2}}l(n) \\ \leq&C\sum_{m=1}^{\infty}m^{\alpha(p-q)+\frac {q}{2}-2}h(m) \bigl(E|Y|^{2}I\bigl[|Y|\leq(m+1)^{\alpha}\bigr] \bigr)^{\frac{q}{2}} \\ &{} +C\sum_{m=1}^{\infty}m^{\alpha p+\frac{q}{2}-2}h(m) \bigl(EI\bigl[|Y|>m^{\alpha}\bigr]\bigr)^{\frac{q}{2}} \\ \leq&C\sum_{m=1}^{\infty}m^{\alpha p+\frac{q}{2}-\frac{\alpha p q}{2}-2}h(m) \bigl(E|Y|^{p}\bigr)^{\frac{q}{2}}< \infty. \end{aligned}$$
(3.15)
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
$$\begin{aligned} I_{22} \leq&C\sum_{m=1}^{\infty}\bigl\{ m^{\alpha(1-q)-1}\bigl(E|Y|^{2}I\bigl[|Y|\leq (m+1)^{\alpha}\bigr]\bigr)^{\frac{q}{2}} \\ &{} +m^{\alpha-1}\bigl(P\bigl(|Y|>m^{\alpha}\bigr) \bigr)^{\frac{q}{2}}\bigr\} \sum_{n=1}^{m} n^{\frac {q}{2}} l(n) \\ \leq&C\sum_{m=1}^{\infty}m^{\alpha(p-q)+\frac {q}{2}-2}h(m) \bigl(E|Y|^{2}I\bigl[|Y|\leq(m+1)^{\alpha}\bigr] \bigr)^{\frac{q}{2}} \\ &{} +C\sum_{m=1}^{\infty}m^{\alpha p+\frac{q}{2}-2}h(m) \bigl(EI\bigl[|Y|>m^{\alpha}\bigr]\bigr)^{\frac{q}{2}} \\ \leq&C\sum_{m=1}^{\infty}m^{\alpha(p-q)+\frac {q}{2}-2}h(m) \bigl(E|Y|^{2}\bigr)^{\frac{q}{2}}< \infty. \end{aligned}$$
(3.16)
Hence, by (3.15) and (3.16) we get
$$ I_{22}< \infty \quad \mbox{for } p\geq1. $$
(3.17)
Moreover, by (3.14) and (3.17), we also get
$$ I_{2}< \infty \quad\mbox{for } p\geq1. $$
(3.18)
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
$$\begin{aligned} &\sum_{n=1}^{\infty}n^{\alpha p-2}h(n)E \Biggl\{ \sup_{i\geq n}\Biggl|i^{-\alpha}\sum _{j=1}^{i} X_{j}\Biggr|-\epsilon\Biggr\} ^{+} \\ &\quad=\sum_{n=1}^{\infty}n^{\alpha p-2}h(n) \int_{0}^{\infty}P\Biggl(\sup_{i\geq n}\Biggl|i^{-\alpha} \sum_{j=1}^{i} X_{j}\Biggr|>\epsilon+x \Biggr)\,dx \\ &\quad=\sum_{k=1}^{\infty}\sum _{n=2^{k-1}}^{2^{k}-1} n^{\alpha p-2}h(n)\int _{0}^{\infty}P\Biggl(\sup_{i\geq n}\Biggl|i^{-\alpha} \sum_{j=1}^{i} X_{j}\Biggr|>\epsilon+x \Biggr)\,dx \\ &\quad\leq C\sum_{k=1}^{\infty}\int _{0}^{\infty}P\Biggl(\sup_{i\geq2^{k-1}}\Biggl|i^{-\alpha } \sum_{j=1}^{i} X_{j}\Biggr|>\epsilon+x \Biggr)\,dx\sum_{n=2^{k-1}}^{2^{k}-1} n^{\alpha p-2}h(n) \\ &\quad\leq C\sum_{k=1}^{\infty}2^{k(\alpha p-1)}h \bigl(2^{k}\bigr)\int_{0}^{\infty}P\Biggl( \sup_{i\geq2^{k-1}}\Biggl|i^{-\alpha}\sum_{j=1}^{i} X_{j}\Biggr|>\epsilon+x\Biggr)\,dx \\ &\quad\leq C\sum_{k=1}^{\infty}2^{k(\alpha p-1)}h \bigl(2^{k}\bigr)\sum_{m=k}^{\infty}\int_{0}^{\infty}P\Biggl(\max_{2^{m-1}\leq i< 2^{m}}\Biggl|i^{-\alpha} \sum_{j=1}^{i} X_{j}\Biggr|>\epsilon+x \Biggr)\,dx \\ &\quad\leq C\sum_{m=1}^{\infty}\int _{0}^{\infty}P\Biggl(\max_{2^{m-1}\leq i< 2^{m}}\Biggl|i^{-\alpha} \sum_{j=1}^{i} X_{j}\Biggr|>\epsilon+x \Biggr)\,dx \sum_{k=1}^{m} 2^{k(\alpha p-1)}h \bigl(2^{k}\bigr) \\ &\quad\leq C\sum_{m=1}^{\infty}2^{m(\alpha p-1)}h \bigl(2^{m}\bigr) \int_{0}^{\infty}P\Biggl( \max_{2^{m-1}\leq i< 2^{m}}\Biggl|\sum_{j=1}^{i} X_{j}\Biggr|>(\epsilon+x)2^{(m-1)\alpha}\Biggr)\,dx \\ &\qquad{} \bigl(\mbox{letting }y=2^{(m-1)\alpha}x\bigr) \\ &\quad\leq C\sum_{m=1}^{\infty}2^{m(\alpha p-1-\alpha)}h \bigl(2^{m}\bigr) \int_{0}^{\infty}P\Biggl( \max_{1\leq i< 2^{m}}\Biggl|\sum_{j=1}^{i} X_{j}\Biggr|>\epsilon2^{(m-1)\alpha}+y\Biggr)\,dy \\ &\quad\leq C\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha}h(n) \int_{0}^{\infty}P\Biggl(\max _{1\leq i< n}\Biggl|\sum_{j=1}^{i} X_{j}\Biggr|>\epsilon n^{\alpha}2^{-\alpha}+y\Biggr)\,dy \\ &\quad=C\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha}h(n) E\Biggl(\max_{1\leq i< n}\Biggl|\sum _{j=1}^{i} X_{j}\Biggr|-\epsilon^{\prime}n^{\alpha}\Biggr)^{+}< \infty, \end{aligned}$$
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 \(\epsilon>0\)
$$ \sum_{n=1}^{\infty}n^{\alpha p-2}h(n) P \Biggl(\max_{1\leq i \leq n}\Biggl|\sum_{j=1}^{i} X_{j}\Biggr|>\epsilon n^{\alpha}\Biggr)< \infty. $$
(3.19)

Proof

As in Remark 1.2 of Li and Zhang [3] we can obtain (3.19). □

Declarations

Acknowledgements

This paper was supported by Wonkwang University in 2015.

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.

Authors’ Affiliations

(1)
Division of Mathematics and Informational Statistics, Wonkwang University

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