Convergence rates in the law of large numbers for long-range dependent linear processes

Abstract

Baum and Katz (Trans. Am. Math. Soc. 120:108-123, 1965) obtained convergence rates in the Marcinkiewicz-Zygmund law of large numbers. Their result has already been extended to the short-range dependent linear processes by many authors. In this paper, we extend the result of Baum and Katz to the long-range dependent linear processes. As a corollary, we obtain convergence rates in the Marcinkiewicz-Zygmund law of large numbers for short-range dependent linear processes.

Introduction

There are many literature works concerning the convergence rates in the Marcinkiewicz-Zygmund law of large numbers. One can refer to Alf [2], Alsmeyer [3], Baum and Katz [1], Heyde and Rohatgi [4], Hu and Weber [5], Rohatgi [6], and so on.

Baum and Katz [1] obtained the following convergence rates in the Marcinkiewicz-Zygmund law of large numbers.

Theorem 1.1

Baum and Katz [1]

Let $r\ge 1$, $1\le p<2$ and $\{X,X_n,n\ge 1\}$ be a sequence of independent and identically distributed (i.i.d.) random variables. Then $EX=0$ and $E|X{|}^{rp}<\mathrm{\infty }$ imply

$$\sum _{n=1}^{\mathrm{\infty }}{n}^{r-2}P\left(\right|\sum _{k=1}^n{X}_k|>n^{1/p}\epsilon )<\mathrm{\infty }\mathit{\text{for all }}\epsilon >0.$$

When $r=2$, the cases of $p=1$ and $1\le p<2$ have already been proved by Hsu and Robbins [7] and Katz [8], respectively.

Let $\{{\zeta }_i,i\in \mathbb{Z}\}$ be a sequence of i.i.d. random variables and $\{a_i,i\in \mathbb{Z}\}$ be a sequence of real numbers. Here and in the following, $\mathbb{Z}$ denotes the set of all integers. Then $\{X_n,n\ge 1\}$ is called a linear process or an infinite order moving average process if $X_n$ is defined by

$$X_n=\sum _{i=-\mathrm{\infty }}^{\mathrm{\infty }}{a}_{i+n}{\zeta }_i\text{for }n\ge 1\text{.}$$ 1.1

If ${\sum }_{i=-\mathrm{\infty }}^{\mathrm{\infty }}|a_i|<\mathrm{\infty }$, then $\{X_n,n\ge 1\}$ has short memory or is short-range dependent. If ${\sum }_{i=-\mathrm{\infty }}^{\mathrm{\infty }}|a_i|=\mathrm{\infty }$, then $\{X_n,n\ge 1\}$ has long memory or is long-range dependent (see Chapter 3 in Giraitis et al. [9]).

In the short-range dependent case, Koopmans [10] showed that if ${\zeta }_0$ has the moment generating function, then the strong law of large numbers for the linear process holds with exponential convergence rate. Hanson and Koopmans [11] generalized this result to a class of linear processes of independent but non-identically distributed random variables $\{{\zeta }_i,i\in \mathbb{Z}\}$ and to arbitrary subsequences of $\{X_n,n\ge 1\}$. Li et al. [12] extended Katz [8] theorem to the setting of short-range dependent linear processes.

Theorem 1.2

Li et al. [12]

Let $1\le p<2$. Let $\{a_i,i\in \mathbb{Z}\}$ be an absolutely summable sequence of real numbers. Suppose that $\{X_n,n\ge 1\}$ is the linear process of a sequence $\{{\zeta }_i,i\in \mathbb{Z}\}$ of i.i.d. random variables with mean zero and $E|{\zeta }_0{|}^{2p}<\mathrm{\infty }$. Then

$$\sum _{n=1}^{\mathrm{\infty }}P\left(\right|\sum _{k=1}^n{X}_k|>n^{1/p}\epsilon )<\mathrm{\infty }\mathit{\text{for all }}\epsilon >0.$$

Note that Theorem 1.2 corresponds to Theorem 1.1 with $r=2$. Zhang [13] extended Theorem 1.1 with $r>1$ to the short-range dependent linear process of a sequence of identically distributed φ-mixing random variables. Since independent random variables are also φ-mixing, it follows by Zhang [13] theorem that Theorem 1.2 also holds for $r>1$.

In this paper, we obtain convergence rates in the Marcinkiewicz-Zygmund law of large numbers for long-range dependent linear processes of i.i.d. random variables. For convenience of notation, let

$$W_n(t)={(\sum _{i=-\mathrm{\infty }}^{\mathrm{\infty }}|{\omega }_{ni}{|}^t)}^{1/t}\text{for }n\ge 1\text{ and }t>0,$$

where ${\omega }_{ni}={\sum }_{k=1}^n{a}_{i+k}$. In the long-range dependent case, Characiejus and Račkauskas [14] obtained the convergence rate in the Marcinkiewicz-Zygmund law of large numbers for the linear process $\{Y_n,n\ge 1\}$ which is slightly different from (1.1) and defined by

$$Y_n=\sum _{i=0}^{\mathrm{\infty }}{a}_i{\zeta }_{n-i}\text{for }n\ge 1\text{,}$$ 1.2

where $a_i=0$ if $i<0$.

Theorem 1.3

Characiejus and Račkauskas [14]

Let $\{Y_n,n\ge 1\}$ be defined as above and $1<p<2$. Let $\{a_i,i\in \mathbb{Z}\}$ be a sequence of real numbers such that

$$\sum _{i=-\mathrm{\infty }}^{\mathrm{\infty }}|a_i{|}^p<\mathrm{\infty },$$

where $a_i=0$ if $i<0$. Assume that

$$W_n(q)/W_n(p)=O\left(n^{1/q-1/p}\right)\mathit{\text{for some }}q\in (p,2].$$

If $E{\zeta }_0=0$ and $E[|{\zeta }_0{|}^{p}log(1+|{\zeta }_0|)]<\mathrm{\infty }$, then

$$\sum _{n=1}^{\mathrm{\infty }}{n}^{-1}P\left(\right|\sum _{k=1}^n{Y}_k|>W_n(p)\epsilon )<\mathrm{\infty }\mathit{\text{for all }}\epsilon >0.$$ 1.3

The above theorem shows a convergence rate in the Marcinkiewicz-Zygmund weak law of large numbers with the norming sequence $W_n(p)$.

We now compare Theorem 1.3 with Theorem 1.1. Since Theorem 1.3 deals with only the case $r=1$, it is interesting to prove that Theorem 1.3 holds for the case $r>1$. When $r=1$, Theorem 1.1 requires a finite pth moment condition, but Theorem 1.3 requires more than finite pth moment. To apply Theorem 1.3, it is necessary to estimate $W_n(p)$. If $\{a_i,i\in \mathbb{Z}\}$ is an absolutely summable sequence, then we have, by the result of Burton and Dehling [15] (see also Lemma 2.4), that for any $t>0$

$$\frac{1}{n}{W}_n^t(t)\to \sum _{i=-\mathrm{\infty }}^{\mathrm{\infty }}{a}_i,$$

and hence (1.3) holds with $W_n(p)$ replaced by $n^{1/p}$. However, for the long-range dependent case, it is not easy to estimate $W_n(t)$.

In this paper, we extend Theorem 1.1 to the long-range dependent linear processes. As a corollary, we obtain a long-range dependent setting of Theorem 1.2. Further, we propose a method to estimate $W_n(t)$ for the long-range dependent case.

Throughout this paper, C denotes a positive constant which may vary at each occurrence. For events A and B, $I(A)$ denotes the indicator function of the event A, and $I(A,B)=I(A\cap B)$.

Convergence of long-range dependent linear processes

In this section, we extend Theorem 1.1 to the long-range dependent linear processes. To prove the main results, we need the following lemmas. The first one is the von Bahr-Esseen inequality (see von Bahr and Esseen [16]). The second is known as Fuk-Nagaev inequality (see Corollary 1.8 in Nagaev [17]).

Lemma 2.1

Let $\{{\zeta }_i,i\ge 1\}$ be a sequence of independent random variables with $E{\zeta }_i=0$ and $E|{\zeta }_i{|}^t<\mathrm{\infty }$ for some $1\le t\le 2$. Then, for all $n\ge 1$,

$$E|\sum _{i=1}^n{\zeta }_i{|}^t\le C_t\sum _{i=1}^{n}E|{\zeta }_i{|}^t,$$

where $C_t>0$ is a positive constant depending only on t.

Lemma 2.2

Let $\{{\zeta }_i,i\ge 1\}$ be a sequence of independent random variables with $E{\zeta }_i=0$. Then, for any $t\ge 2$ and $x>0$,

$$P\left(\right|\sum _{i=1}^n{\zeta }_i|>x)\le {(1+2/t)}^t{x}^{-t}\sum _{i=1}^{n}E|{\zeta }_i{|}^t+2exp\{-\frac{2x^2}{{(t+2)}^2{e}^t{\sum }_{i=1}^{n}Var({\zeta }_i)}\}.$$

The following lemma is well known and can be easily proved by using a standard method.

Lemma 2.3

Let $p>0$ and ζ be a random variable. Then the following statements hold.

• (i) If $0<\theta <p$, then ${\sum }_{n=1}^{\mathrm{\infty }}{n}^{-\theta /p}E|\zeta {|}^{\theta }I(|\zeta |>n^{1/p})\le CE|\zeta {|}^p$.
• (ii) If $p<q$, then ${\sum }_{n=1}^{\mathrm{\infty }}{n}^{-q/p}E|\zeta {|}^{q}I(|\zeta |\le n^{1/p})\le CE|\zeta {|}^p$.
• (iii) If $r>1$, then ${\sum }_{n=1}^{\mathrm{\infty }}{n}^{r-2}E|\zeta {|}^{p}I(|\zeta |>n^{1/p})\le CE|\zeta {|}^{rp}$.
• (iv) If $rp<q$, then ${\sum }_{n=1}^{\mathrm{\infty }}{n}^{r-1-q/p}E|\zeta {|}^{q}I(|\zeta |\le n^{1/p})\le CE|\zeta {|}^{rp}$.

The following lemma is useful to estimate $W_n(t)$ when the sequence $\{a_i,i\in \mathbb{Z}\}$ is absolutely summable. However, it is not applicable to the long-range dependent case.

Lemma 2.4

Burton and Dehling [15]

Let ${\sum }_{i=-\mathrm{\infty }}^{\mathrm{\infty }}{a}_i$ be an absolutely convergent series of real numbers with $a={\sum }_{i=-\mathrm{\infty }}^{\mathrm{\infty }}{a}_i$. Then, for any $t>0$,

$$\underset{n\to \mathrm{\infty }}{lim}\frac{1}{n}\sum _{i=-\mathrm{\infty }}^{\mathrm{\infty }}|{\omega }_{ni}{|}^t=|a{|}^t,$$

where ${\omega }_{ni}={\sum }_{k=1}^n{a}_{i+k}$.

We now state and prove our main results. The first theorem treats the case $r>1$.

Theorem 2.1

Let $r>1$ and $1\le p<2$. Let $\{a_i,i\in \mathbb{Z}\}$ be a sequence of real numbers with

$$\sum _{i=-\mathrm{\infty }}^{\mathrm{\infty }}|a_i{|}^p<\mathrm{\infty }.$$

Suppose that $\{X_n,n\ge 1\}$ is the linear process of a sequence $\{{\zeta }_i,i\in \mathbb{Z}\}$ of i.i.d. random variables with mean zero and $E|{\zeta }_0{|}^{rp}<\mathrm{\infty }$. Furthermore, assume that one of the following conditions holds.

1. If $1<rp<2$, then

   $$W_n(q)/W_n(p)=O\left(n^{1/q-1/p}\right)\mathit{\text{for some }}q\in (rp,2).$$
2. If $rp\ge 2$, then

   $$W_n(q)/W_n(p)=O\left(n^{1/q-1/p}\right)\mathit{\text{for some }}q>rp$$

   and

   $$W_n(s)/W_n(p)=o\left({(logn)}^{-1/s}\right)\mathit{\text{for some }}s\in (p,2].$$

   Then

   $$\sum _{n=1}^{\mathrm{\infty }}{n}^{r-2}P\left(\right|\sum _{k=1}^n{X}_k|>W_n(p)\epsilon )<\mathrm{\infty }\mathit{\text{for all }}\epsilon >0.$$

Proof

(1) For each $n\ge 1$, we have

$$\begin{array}{rl}\sum _{k=1}^n{X}_k=& \sum _{i=-\mathrm{\infty }}^{\mathrm{\infty }}\sum _{k=1}^n{a}_{i+k}{\zeta }_i=\sum _{i=-\mathrm{\infty }}^{\mathrm{\infty }}{\omega }_{ni}{\zeta }_i\\ =& \sum _{i=-\mathrm{\infty }}^{\mathrm{\infty }}{\omega }_{ni}[{\zeta }_{i}I(|{\zeta }_i|>n^{1/p})-E{\zeta }_{i}I(|{\zeta }_i|>n^{1/p})]\\ & +\sum _{i=-\mathrm{\infty }}^{\mathrm{\infty }}{\omega }_{ni}[{\zeta }_{i}I(|{\zeta }_i|\le n^{1/p})-E{\zeta }_{i}I(|{\zeta }_i|\le n^{1/p})]\\ :=& S_n^{\prime }+S_n^{″}\end{array}$$

and hence,

$$\begin{array}{rl}& \sum _{n=1}^{\mathrm{\infty }}{n}^{r-2}P\left(\right|\sum _{k=1}^n{X}_k|>W_n(p)\epsilon )\\ & \le \sum _{n=1}^{\mathrm{\infty }}{n}^{r-2}P\left(\right|S_n^{\prime }|>W_n(p)\epsilon /2)+\sum _{n=1}^{\mathrm{\infty }}{n}^{r-2}P\left(\right|S_n^{″}|>W_n(p)\epsilon /2).\end{array}$$ 2.1

By the Markov inequality, Lemmas 2.1 and 2.3, we have

$$\begin{array}{rl}\sum _{n=1}^{\mathrm{\infty }}{n}^{r-2}P\left(\right|S_n^{\prime }|>W_n(p)\epsilon /2)& \le \sum _{n=1}^{\mathrm{\infty }}{n}^{r-2}\frac{2^{p}E|S_n^{\prime }{|}^p}{{\epsilon }^p{W}_n^p(p)}\\ & \le C\sum _{n=1}^{\mathrm{\infty }}{n}^{r-2}\frac{{\sum }_{i=-\mathrm{\infty }}^{\mathrm{\infty }}|{\omega }_{ni}{|}^{p}E|{\zeta }_0{|}^{p}I(|{\zeta }_0|>n^{1/p})}{{\sum }_{i=-\mathrm{\infty }}^{\mathrm{\infty }}|{\omega }_{ni}{|}^p}\\ & =C\sum _{n=1}^{\mathrm{\infty }}{n}^{r-2}E|{\zeta }_0{|}^{p}I(|{\zeta }_0|>n^{1/p})\\ & \le CE|{\zeta }_0{|}^{rp}<\mathrm{\infty }.\end{array}$$

Thus the first series on the right-hand side of (2.1) converges.

Similarly, by the Markov inequality, Lemmas 2.1 and 2.3, we have

$$\begin{array}{rl}\sum _{n=1}^{\mathrm{\infty }}{n}^{r-2}P\left(\right|S_n^{″}|>W_n(p)\epsilon /2)& \le \sum _{n=1}^{\mathrm{\infty }}{n}^{r-2}\frac{2^{q}E|S_n^{″}{|}^q}{{\epsilon }^q{W}_n^q(p)}\\ & \le C\sum _{n=1}^{\mathrm{\infty }}{n}^{r-2}\frac{{\sum }_{i=-\mathrm{\infty }}^{\mathrm{\infty }}|{\omega }_{ni}{|}^{q}E|{\zeta }_0{|}^{q}I(|{\zeta }_0|\le n^{1/p})}{W_n^q(p)}\\ & =C\sum _{n=1}^{\mathrm{\infty }}{n}^{r-2}{\left(\frac{W_n(q)}{W_n(p)}\right)}^{q}E|{\zeta }_0{|}^{q}I(|{\zeta }_0|\le n^{1/p})\\ & \le C\sum _{n=1}^{\mathrm{\infty }}{n}^{r-2}{\left(n^{1/q-1/p}\right)}^{q}E|{\zeta }_0{|}^{q}I(|{\zeta }_0|\le n^{1/p})\\ & =C\sum _{n=1}^{\mathrm{\infty }}{n}^{r-1-q/p}E|{\zeta }_0{|}^{q}I(|{\zeta }_0|\le n^{1/p})\\ & \le CE|{\zeta }_0{|}^{rp}<\mathrm{\infty }.\end{array}$$

Hence the second series on the right-hand side of (2.1) also converges.

(2) For each $n\ge 1$, we have

$$\begin{array}{rl}\sum _{k=1}^n{X}_k=& \sum _{i=-\mathrm{\infty }}^{\mathrm{\infty }}\sum _{k=1}^n{a}_{i+k}{\zeta }_i=\sum _{i=-\mathrm{\infty }}^{\mathrm{\infty }}{\omega }_{ni}{\zeta }_i\\ =& \sum _{i=-\mathrm{\infty }}^{\mathrm{\infty }}[{\omega }_{ni}{\zeta }_{i}I(|{\omega }_{ni}{\zeta }_i|>W_n(p))-E{\omega }_{ni}{\zeta }_{i}I(|{\omega }_{ni}{\zeta }_i|>W_n(p))]\\ & +\sum _{i=-\mathrm{\infty }}^{\mathrm{\infty }}[{\omega }_{ni}{\zeta }_{i}I(|{\omega }_{ni}{\zeta }_i|\le W_n(p))-E{\omega }_{ni}{\zeta }_{i}I(|{\omega }_{ni}{\zeta }_i|\le W_n(p))]\\ :=& T_n^{\prime }+T_n^{″}\end{array}$$

and hence,

$$\begin{array}{rl}& \sum _{n=1}^{\mathrm{\infty }}{n}^{r-2}P\left(\right|\sum _{k=1}^n{X}_k|>W_n(p)\epsilon )\\ & \le \sum _{n=1}^{\mathrm{\infty }}{n}^{r-2}P\left(\right|T_n^{\prime }|>W_n(p)\epsilon /2)+\sum _{n=1}^{\mathrm{\infty }}{n}^{r-2}P\left(\right|T_n^{″}|>W_n(p)\epsilon /2).\end{array}$$ 2.2

By the Markov inequality, Lemmas 2.1 and 2.3, we have

$$\begin{array}{rl}& \sum _{n=1}^{\mathrm{\infty }}{n}^{r-2}P\left(\right|T_n^{\prime }|>W_n(p)\epsilon /2)\\ & \le \sum _{n=1}^{\mathrm{\infty }}{n}^{r-2}\frac{2^{p}E|T_n^{\prime }{|}^p}{{\epsilon }^p{W}_n^p(p)}\\ & \le C\sum _{n=1}^{\mathrm{\infty }}{n}^{r-2}\frac{{\sum }_{i=-\mathrm{\infty }}^{\mathrm{\infty }}E|{\omega }_{ni}{\zeta }_i{|}^{p}I(|{\omega }_{ni}{\zeta }_i|>W_n(p))}{W_n^p(p)}\\ & =C\sum _{n=1}^{\mathrm{\infty }}{n}^{r-2}\frac{{\sum }_{i=-\mathrm{\infty }}^{\mathrm{\infty }}E|{\omega }_{ni}{\zeta }_i{|}^{p}I(|{\omega }_{ni}{\zeta }_i|>W_n(p),|{\zeta }_i|>n^{1/p})}{W_n^p(p)}\\ & +C\sum _{n=1}^{\mathrm{\infty }}{n}^{r-2}\frac{{\sum }_{i=-\mathrm{\infty }}^{\mathrm{\infty }}E|{\omega }_{ni}{\zeta }_i{|}^{p}I(|{\omega }_{ni}{\zeta }_i|>W_n(p),|{\zeta }_i|\le n^{1/p})}{W_n^p(p)}\\ & \le C\sum _{n=1}^{\mathrm{\infty }}{n}^{r-2}\frac{{\sum }_{i=-\mathrm{\infty }}^{\mathrm{\infty }}|{\omega }_{ni}{|}^{p}E|{\zeta }_i{|}^{p}I(|{\zeta }_i|>n^{1/p})}{W_n^p(p)}\\ & +C\sum _{n=1}^{\mathrm{\infty }}{n}^{r-2}\frac{{\sum }_{i=-\mathrm{\infty }}^{\mathrm{\infty }}E[|{\omega }_{ni}{\zeta }_i{|}^{p-q}|{\omega }_{ni}{\zeta }_i{|}^{q}I(|{\omega }_{ni}{\zeta }_i|>W_n(p),|{\zeta }_i|\le n^{1/p})]}{W_n^p(p)}\\ & \le C\sum _{n=1}^{\mathrm{\infty }}{n}^{r-2}\frac{{\sum }_{i=-\mathrm{\infty }}^{\mathrm{\infty }}|{\omega }_{ni}{|}^{p}E|{\zeta }_0{|}^{p}I(|{\zeta }_0|>n^{1/p})}{W_n^p(p)}\\ & +C\sum _{n=1}^{\mathrm{\infty }}{n}^{r-2}\frac{{(W_n(p))}^{p-q}{\sum }_{i=-\mathrm{\infty }}^{\mathrm{\infty }}|{\omega }_{ni}{|}^{q}E|{\zeta }_0{|}^{q}I(|{\zeta }_0|\le n^{1/p})}{W_n^p(p)}\\ & =C\sum _{n=1}^{\mathrm{\infty }}{n}^{r-2}E|{\zeta }_0{|}^{p}I(|{\zeta }_0|>n^{1/p})\\ & +C\sum _{n=1}^{\mathrm{\infty }}{n}^{r-2}{\left(\frac{W_n(q)}{W_n(p)}\right)}^{q}E|{\zeta }_0{|}^{q}I(|{\zeta }_0|\le n^{1/p})\\ & \le C\sum _{n=1}^{\mathrm{\infty }}{n}^{r-2}E|{\zeta }_0{|}^{p}I(|{\zeta }_0|>n^{1/p})+C\sum _{n=1}^{\mathrm{\infty }}{n}^{r-1-q/p}E|{\zeta }_0{|}^{q}I(|{\zeta }_0|\le n^{1/p})\\ & \le CE|{\zeta }_0{|}^{rp}<\mathrm{\infty }.\end{array}$$

Thus the first series on the right-hand side of (2.2) converges.

We next prove that the second series on the right-hand side of (2.2) converges. We have by Lemma 2.2 that for $t>2$,

$$\begin{array}{rl}& \sum _{n=1}^{\mathrm{\infty }}{n}^{r-2}P\left(\right|T_n^{″}|>W_n(p)\epsilon /2)\\ & \le C\sum _{n=1}^{\mathrm{\infty }}{n}^{r-2}\frac{{\sum }_{i=-\mathrm{\infty }}^{\mathrm{\infty }}E|{\omega }_{ni}{\zeta }_i{|}^{t}I(|{\omega }_{ni}{\zeta }_i|\le W_n(p))}{W_n^t(p)}\\ & +C\sum _{n=1}^{\mathrm{\infty }}{n}^{r-2}exp\{-\frac{{\epsilon }^2{W}_n^2(p)}{2{(t+2)}^2{e}^t{\sum }_{i=-\mathrm{\infty }}^{\mathrm{\infty }}Var({\omega }_{ni}{\zeta }_{i}I(|{\omega }_{ni}{\zeta }_i|\le W_n(p)))}\}.\end{array}$$ 2.3

Hence it is enough to show that two series on the right-hand side of (2.3) converge.

If we take $t>q$, then we have by Lemma 2.3 that

$$\begin{array}{rl}& \sum _{n=1}^{\mathrm{\infty }}{n}^{r-2}\frac{{\sum }_{i=-\mathrm{\infty }}^{\mathrm{\infty }}E|{\omega }_{ni}{\zeta }_i{|}^{t}I(|{\omega }_{ni}{\zeta }_i|\le W_n(p))}{W_n^t(p)}\\ & =\sum _{n=1}^{\mathrm{\infty }}{n}^{r-2}\frac{{\sum }_{i=-\mathrm{\infty }}^{\mathrm{\infty }}E|{\omega }_{ni}{\zeta }_i{|}^{t}I(|{\omega }_{ni}{\zeta }_i|\le W_n(p),|{\zeta }_i|>n^{1/p})}{W_n^t(p)}\\ & +\sum _{n=1}^{\mathrm{\infty }}{n}^{r-2}\frac{{\sum }_{i=-\mathrm{\infty }}^{\mathrm{\infty }}E|{\omega }_{ni}{\zeta }_i{|}^{t}I(|{\omega }_{ni}{\zeta }_i|\le W_n(p),|{\zeta }_i|\le n^{1/p})}{W_n^t(p)}\\ & =\sum _{n=1}^{\mathrm{\infty }}{n}^{r-2}\frac{{\sum }_{i=-\mathrm{\infty }}^{\mathrm{\infty }}E[|{\omega }_{ni}{\zeta }_i{|}^{t-p}|{\omega }_{ni}{\zeta }_i{|}^{p}I(|{\omega }_{ni}{\zeta }_i|\le W_n(p),|{\zeta }_i|>n^{1/p})]}{W_n^t(p)}\\ & +\sum _{n=1}^{\mathrm{\infty }}{n}^{r-2}\frac{{\sum }_{i=-\mathrm{\infty }}^{\mathrm{\infty }}E[|{\omega }_{ni}{\zeta }_i{|}^{t-q}|{\omega }_{ni}{\zeta }_i{|}^{q}I(|{\omega }_{ni}{\zeta }_i|\le W_n(p),|{\zeta }_i|\le n^{1/p})]}{W_n^t(p)}\\ & \le \sum _{n=1}^{\mathrm{\infty }}{n}^{r-2}\frac{{(W_n(p))}^{t-p}{\sum }_{i=-\mathrm{\infty }}^{\mathrm{\infty }}|{\omega }_{ni}{|}^{p}E|{\zeta }_0{|}^{p}I(|{\zeta }_0|>n^{1/p})}{W_n^t(p)}\\ & +\sum _{n=1}^{\mathrm{\infty }}{n}^{r-2}\frac{{(W_n(p))}^{t-q}{\sum }_{i=-\mathrm{\infty }}^{\mathrm{\infty }}|{\omega }_{ni}{|}^{q}E|{\zeta }_0{|}^{q}I(|{\zeta }_0|\le n^{1/p})}{W_n^t(p)}\\ & =\sum _{n=1}^{\mathrm{\infty }}{n}^{r-2}E|{\zeta }_0{|}^{p}I(|{\zeta }_0|>n^{1/p})+\sum _{n=1}^{\mathrm{\infty }}{n}^{r-2}{\left(\frac{W_n(q)}{W_n(p)}\right)}^{q}E|{\zeta }_0{|}^{q}I(|{\zeta }_0|\le n^{1/p})\\ & \le \sum _{n=1}^{\mathrm{\infty }}{n}^{r-2}E|{\zeta }_0{|}^{p}I(|{\zeta }_0|>n^{1/p})+\sum _{n=1}^{\mathrm{\infty }}{n}^{r-1-q/p}E|{\zeta }_0{|}^{q}I(|{\zeta }_0|\le n^{1/p})\\ & \le CE|{\zeta }_0{|}^{rp}<\mathrm{\infty }.\end{array}$$

Hence the first series on the right-hand side of (2.3) converges.

Finally, we show that the second series on the right-hand side of (2.3) converges. Since $p<s\le 2$, we have that

$$\begin{array}{rl}& \frac{{\sum }_{i=-\mathrm{\infty }}^{\mathrm{\infty }}Var({\omega }_{ni}{\zeta }_{i}I(|{\omega }_{ni}{\zeta }_i|\le W_n(p)))}{W_n^2(p)}\\ & \le \frac{{\sum }_{i=-\mathrm{\infty }}^{\mathrm{\infty }}E|{\omega }_{ni}{\zeta }_i{|}^{2}I(|{\omega }_{ni}{\zeta }_i|\le W_n(p))}{W_n^2(p)}\\ & =\frac{{\sum }_{i=-\mathrm{\infty }}^{\mathrm{\infty }}E|{\omega }_{ni}{\zeta }_i{|}^{s+2-s}I(|{\omega }_{ni}{\zeta }_i|\le W_n(p))}{W_n^2(p)}\\ & \le \frac{{(W_n(p))}^{2-s}{\sum }_{i=-\mathrm{\infty }}^{\mathrm{\infty }}E|{\omega }_{ni}{\zeta }_i{|}^s}{W_n^2(p)}\\ & =\frac{{\sum }_{i=-\mathrm{\infty }}^{\mathrm{\infty }}|{\omega }_{ni}{|}^{s}E|{\zeta }_0{|}^s}{W_n^s(p)}\\ & ={\left(\frac{W_n(s)}{W_n(p)}\right)}^{s}E|{\zeta }_0{|}^s\\ & =o(1/logn),\end{array}$$

which implies that

$$\begin{array}{rl}& \sum _{n=1}^{\mathrm{\infty }}{n}^{r-2}\{-\frac{{\epsilon }^2{W}_n^2(p)}{2{(t+2)}^2{e}^t{\sum }_{i=-\mathrm{\infty }}^{\mathrm{\infty }}Var({\omega }_{ni}{\zeta }_{i}I(|{\omega }_{ni}{\zeta }_i|\le W_n(p)))}\}\\ & \le C\sum _{n=1}^{\mathrm{\infty }}{n}^{r-2}\{-\frac{{\epsilon }^{2}logn}{2{(t+2)}^2{e}^{t}o(1)}\}<\mathrm{\infty }.\end{array}$$

 □

The next theorem treats the case $r=1$.

Theorem 2.2

Let $1\le p<2$. Let $\{a_i,i\in \mathbb{Z}\}$ be a sequence of real numbers with

$$\sum _{i=-\mathrm{\infty }}^{\mathrm{\infty }}|a_i{|}^{\theta }<\mathrm{\infty }\mathit{\text{for some }}0<\theta <p.$$

Suppose that $\{X_n,n\ge 1\}$ is the linear process of a sequence $\{{\zeta }_i,i\in \mathbb{Z}\}$ of i.i.d. random variables with mean zero and $E|{\zeta }_0{|}^p<\mathrm{\infty }$. Furthermore, assume that

$$W_n(\theta )/W_n(p)=O\left(n^{1/\theta -1/p}\right)$$

and

$$W_n(q)/W_n(p)=O\left(n^{1/q-1/p}\right)\mathit{\text{for some }}q\in (p,2).$$

Then

$$\sum _{n=1}^{\mathrm{\infty }}{n}^{-1}P\left(\right|\sum _{k=1}^n{X}_k|>W_n(p)\epsilon )<\mathrm{\infty }\mathit{\text{for all }}\epsilon >0.$$

Proof

The proof is similar to that of Theorem 2.1(1). We proceed with two cases $1\le \theta <p$ and $0<\theta <1$.

For the case $1\le \theta <p$, we have by Lemmas 2.1 and 2.3 that

$$\begin{array}{rl}\sum _{n=1}^{\mathrm{\infty }}{n}^{-1}P\left(\right|S_n^{\prime }|>W_n(p)\epsilon /2)& \le \sum _{n=1}^{\mathrm{\infty }}{n}^{-1}\frac{2^{\theta }E|S_n^{\prime }{|}^{\theta }}{{\epsilon }^{\theta }{W}_n^{\theta }(p)}\\ & \le C\sum _{n=1}^{\mathrm{\infty }}{n}^{-1}\frac{{\sum }_{i=-\mathrm{\infty }}^{\mathrm{\infty }}|{\omega }_{ni}{|}^{\theta }E|{\zeta }_0{|}^{\theta }I(|{\zeta }_0|>n^{1/p})}{W_n^{\theta }(p)}\\ & =C\sum _{n=1}^{\mathrm{\infty }}{n}^{-1}{\left(\frac{W_n(\theta )}{W_n(p)}\right)}^{\theta }E|{\zeta }_0{|}^{p}I(|{\zeta }_0|>n^{1/p})\\ & \le C\sum _{n=1}^{\mathrm{\infty }}{n}^{-1}{n}^{(1/\theta -1/p)\theta }E|{\zeta }_0{|}^{p}I(|{\zeta }_0|>n^{1/p})\\ & \le CE|{\zeta }_0{|}^p<\mathrm{\infty }.\end{array}$$

As in the proof of Theorem 2.1(1), we have that

$$\sum _{n=1}^{\mathrm{\infty }}{n}^{-1}P\left(\right|S_n^{″}|>W_n(p)\epsilon /2)\le CE|{\zeta }_0{|}^p<\mathrm{\infty }.$$

For the case $0<\theta <1$, we rewrite ${\sum }_{k=1}^n{X}_k$ as

$$\begin{array}{rl}\sum _{k=1}^n{X}_k=& \sum _{i=-\mathrm{\infty }}^{\mathrm{\infty }}{\omega }_{ni}{\zeta }_{i}I(|{\zeta }_i|>n^{1/p})+\sum _{i=-\mathrm{\infty }}^{\mathrm{\infty }}{\omega }_{ni}[{\zeta }_{i}I(|{\zeta }_i|\le n^{1/p})-E{\zeta }_{i}I(|{\zeta }_i|\le n^{1/p})]\\ & -\sum _{i=-\mathrm{\infty }}^{\mathrm{\infty }}{\omega }_{ni}E{\zeta }_{i}I(|{\zeta }_i|>n^{1/p})\\ :=& S_n^{\prime }+S_n^{″}-S_n^{‴}.\end{array}$$

If $0<\theta <1$, then ${\sum }_{n=1}^{\mathrm{\infty }}|a_n|\le {({\sum }_{n=1}^{\mathrm{\infty }}|a_n{|}^{\theta })}^{1/\theta }<\mathrm{\infty }$. It follows by Lemma 2.4 that

$$\begin{array}{rl}{W}_n^{-1}(p)\left|S_n^{‴}\right|& \le W_n^{-1}(p)\sum _{i=-\mathrm{\infty }}^{\mathrm{\infty }}|{\omega }_{ni}|E|{\zeta }_0|I(|{\zeta }_0|>n^{1/p})\\ & \le C{n}^{1-1/p}E|{\zeta }_0|I(|{\zeta }_0|>n^{1/p})\\ & \le CE|{\zeta }_0{|}^{p}I(|{\zeta }_0|>n^{1/p})\to 0\end{array}$$

as $n\to \mathrm{\infty }$. Hence

$$\begin{array}{rl}& \sum _{n=1}^{\mathrm{\infty }}{n}^{-1}P\left(\right|\sum _{k=1}^n{X}_k|>W_n(p)\epsilon )\\ & \le C\sum _{n=1}^{\mathrm{\infty }}{n}^{-1}P\left(\right|S_n^{\prime }|>W_n(p)\epsilon /3)+C\sum _{n=1}^{\mathrm{\infty }}{n}^{-1}P\left(\right|S_n^{″}|>W_n(p)\epsilon /3).\end{array}$$

The rest of the proof is the same as that of the previous case and is omitted. □

The following corollary extends Theorem 1.1 to the short-range dependent linear processes.

Corollary 2.1

Let $r\ge 1$, $1\le p<2$, and $rp>1$. Let $\{a_i,i\in \mathbb{Z}\}$ be an absolutely summable sequence of real numbers. Suppose that $\{X_n,n\ge 1\}$ is the linear process of a sequence $\{{\zeta }_i,i\in \mathbb{Z}\}$ of i.i.d. random variables with mean zero and $E|{\zeta }_0{|}^{rp}<\mathrm{\infty }$. Then

$$\sum _{n=1}^{\mathrm{\infty }}{n}^{r-2}P\left(\right|\sum _{k=1}^n{X}_k|>n^{1/p}\epsilon )<\mathrm{\infty }\mathit{\text{for all }}\epsilon >0.$$

Proof

We first note that

$$\sum _{i=-\mathrm{\infty }}^{\mathrm{\infty }}|a_i{|}^p\le {(\sum _{i=-\mathrm{\infty }}^{\mathrm{\infty }}|a_i|)}^p<\mathrm{\infty }.$$

If $1<p<2$, then we take θ such that $1\le \theta <p$. Then

$$\sum _{i=-\mathrm{\infty }}^{\mathrm{\infty }}|a_i{|}^{\theta }\le {(\sum _{i=-\mathrm{\infty }}^{\mathrm{\infty }}|a_i|)}^{\theta }<\mathrm{\infty }.$$

By Lemma 2.4, for any $t>0$, there exist positive constants $C_1$ and $C_2$ independent of n such that

$$C_1{n}^{1/t}\le W_n(t)\le C_2{n}^{1/t}\text{for all }n\ge 1\text{.}$$

Then all conditions on $W_n(\cdot )$ in Theorems 2.1 and 2.2 are easily satisfied. Hence the proof follows from Theorems 2.1 and 2.2. □

Remark 2.1

In Corollary 2.1, the case $rp=1$ (i.e., $r=1$ and $p=1$) is not considered. In fact, Corollary 2.1 does not hold for this case (see Sung [18]).

An estimation of $W_n(t)$ for the long-range dependent case

As we have seen in Sections 1 and 2, it is easy to estimate $W_n(t)$ for the short-range dependent case. In this section, we propose a method to estimate $W_n(t)$ for the long-range dependent case. It is not easy to estimate $W_n(t)$ when the sequence $\{a_i,i\in \mathbb{Z}\}$ is not absolutely summable. For simplicity, we will consider non-increasing sequences of positive numbers. For the finiteness of $W_n(t)$, without loss of generality, it is necessary to assume that $a_i=0$ if $i\le 0$ and ${\sum }_{i=1}^{\mathrm{\infty }}{a}_i^t<\mathrm{\infty }$.

Lemma 3.1

Let $t>0$. Let $\{a_i,i\in \mathbb{Z}\}$ be a non-increasing sequence of positive real numbers satisfying $a_i=0$ if $i\le 0$ and ${\sum }_{i=1}^{\mathrm{\infty }}{a}_i^t<\mathrm{\infty }$. Then

$$\frac{n}{2}{(a_1+\cdots +a_{[n/2]})}^t+n^t\sum _{i=n}^{\mathrm{\infty }}{a}_i^t\le W_n^t(t)\le 2n{(a_1+\cdots +a_n)}^t+n^t\sum _{i=n}^{\mathrm{\infty }}{a}_i^t.$$

Proof

Since $a_i=0$ if $i\le 0$ and $0<a_i\downarrow$, we get that

$$\begin{array}{rl}{W}_n^t(t)& =\sum _{i=1}^n{\left(\sum _{j=1}^i{a}_j\right)}^t+\sum _{i=1}^n{\left(\sum _{j=1}^n{a}_{i+j}\right)}^t+\sum _{i=n+1}^{\mathrm{\infty }}{\left(\sum _{j=1}^n{a}_{i+j}\right)}^t\\ & \le 2n{(a_1+\cdots +a_n)}^t+n^t\sum _{i=n+1}^{\mathrm{\infty }}{a}_{i+1}^t\\ & \le 2n{(a_1+\cdots +a_n)}^t+n^t\sum _{i=n}^{\mathrm{\infty }}{a}_i^t.\end{array}$$

Similarly,

$$\begin{array}{rl}{W}_n^t(t)& =\sum _{i=1}^{n-1}{\left(\sum _{j=1}^i{a}_j\right)}^t+\sum _{i=1}^{\mathrm{\infty }}{\left(\sum _{j=0}^{n-1}{a}_{i+j}\right)}^t\\ & \ge \sum _{i=[n/2]}^{n-1}{\left(\sum _{j=1}^i{a}_j\right)}^t+n^t\sum _{i=n}^{\mathrm{\infty }}{a}_i^t\\ & \ge \frac{n}{2}{(a_1+\cdots +a_{[n/2]})}^t+n^t\sum _{i=n}^{\mathrm{\infty }}{a}_i^t.\end{array}$$

Thus the proof is completed. □

The following lemma can be found in Martikainen [19].

Lemma 3.2

Martikainen [19]

Let $\{b_n,n\ge 1\}$ be a non-decreasing sequence of positive real numbers. Then

$$\sum _{i=n}^{\mathrm{\infty }}\frac{1}{i{b}_i}=O\left(b_n^{-1}\right)⟺\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\frac{b_{rn}}{b_n}>1\mathit{\text{for some integer }}r\ge 2.$$

Similarly, we can obtain a counterpart of Lemma 3.2.

Lemma 3.3

Let $\{b_n,n\ge 1\}$ be a non-decreasing sequence of positive real numbers. Then

$$\sum _{i=1}^n\frac{b_i}{i}=O(b_n)⟺\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\frac{b_{rn}}{b_n}>1\mathit{\text{for some integer }}r\ge 2.$$

Proof

The proof is similar to that of Lemma 3.2 and is omitted. □

Using Lemmas 3.2 and 3.3, we have the following lemma.

Lemma 3.4

Let $t>1$ and let $\{a_n,n\ge 1\}$ be a sequence of positive real numbers satisfying $n{a}_n\uparrow$, $n{a}_n^t\downarrow$, and

$$\frac{1}{r}<\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\frac{a_{rn}}{a_n}\le \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\frac{a_{rn}}{a_n}<{\left(\frac{1}{r}\right)}^{1/t}\mathit{\text{for some integer }}r\ge 2.$$

Then the following statements hold:

• (i) ${\sum }_{i=n}^{\mathrm{\infty }}{a}_i^t=O(n{a}_n^t)$.
• (ii) ${\sum }_{i=1}^n{a}_i=O(n{a}_n)$.

Proof

The proof of (i) follows from Lemma 3.2. The proof of (ii) follows from Lemma 3.3. □

Now we present a method to estimate $W_n(t)$ for the long-range dependent case.

Theorem 3.1

Let $t>1$, and let $\{a_n,n\ge 1\}$ be a sequence of positive real numbers satisfying the same conditions as in Lemma  3.4. Then there exist positive constants $C_1$ and $C_2$ independent of n such that

$$C_1{n}^{1+t}{a}_n^t\le W_n^t(t)\le C_2{n}^{1+t}{a}_n^t\mathit{\text{for all }}n\ge 1,$$

where $a_i=0$ if $i\le 0$.

Proof

By the condition $n{a}_n^t\downarrow$, we have ${(a_{n+1}/a_n)}^t\le n/(n+1)$, which implies $0<a_n\downarrow$. The upper bound of $W_n^t(t)$ follows by Lemmas 3.1 and 3.4. For the lower bound, we have by ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{a}_{rn}/a_n>1/r$ that

$$a_{rn}/a_n\ge 1/r\text{for all large }n\text{.}$$

It follows that for all large n

$$n^t\sum _{i=n}^{\mathrm{\infty }}{a}_i^t\ge n^t\sum _{i=n}^{rn}{a}_i^t\ge (r-1)n^{1+t}{a}_{rn}^t\ge (r-1)r^{-t}{n}^{1+t}{a}_n^t.$$

Since $0<a_n\downarrow$,

$$n{(a_1+\cdots +a_{[n/2]})}^t\ge n{[n/2]}^t{a}_{[n/2]}^t\ge n{[n/2]}^t{a}_n^t.$$

Hence the lower bound follows from Lemma 3.1. □

Finally, we give two long-range dependent linear processes.

Example 3.1

Let $a_i=1/i$ if $i\ge 1$ and $a_i=0$ if $i\le 0$. Then the series ${\sum }_{i=-\mathrm{\infty }}^{\mathrm{\infty }}{a}_i$ diverges, but ${\sum }_{i=-\mathrm{\infty }}^{\mathrm{\infty }}{a}_i^t$ converges if $t>1$. Observe that

$$ln(n+1)\le \sum _{i=1}^n{a}_i\le 1+lnn.$$

If $t>1$, then

$$\frac{1}{t-1}{n}^{-t+1}\le \sum _{i=n}^{\mathrm{\infty }}{a}_i^t\le n^{-t}+\frac{1}{t-1}{n}^{-t+1}.$$

By Lemma 3.1, for any $t>1$, there exist positive constants $C_1$ and $C_2$ independent of n such that

$$C_{1}n{(lnn)}^t\le W_n^t(t)\le C_{2}n{(lnn)}^t\text{for all }n\ge 2.$$

Let $X_n={\sum }_{i=-\mathrm{\infty }}^{\mathrm{\infty }}{a}_{i+n}{\zeta }_i$ be the long-range dependent linear process of a sequence $\{{\zeta }_i\}$ of i.i.d. random variables with mean zero and $E|{\zeta }_0{|}^{rp}<\mathrm{\infty }$, where $r>1$ and $1<p<2$. Then all conditions of Theorem 2.1 are easily satisfied. By Theorem 2.1,

$$\sum _{n=1}^{\mathrm{\infty }}{n}^{r-2}P\left(\right|\sum _{k=1}^n{X}_k|>n^{1/p}lnn\epsilon )<\mathrm{\infty }\text{for all }\epsilon >0.$$

Example 3.2

Let $1<p<2$. Let $a_i=1/i^d$ if $i\ge 1$ and $a_i=0$ if $i\le 0$, where $1/p<d<1$. Then the series ${\sum }_{i=-\mathrm{\infty }}^{\mathrm{\infty }}{a}_i$ diverges, but ${\sum }_{i=-\mathrm{\infty }}^{\mathrm{\infty }}{a}_i^t$ converges if $t>1/d$. Since $a_{2n}/a_n=2^{-d}$, we have by Theorem 3.1 that

$$C_1{n}^{1+t-dt}\le W_n^t(t)\le C_2{n}^{1+t-dt}\text{for all }n\ge 1\text{.}$$

Let $X_n={\sum }_{i=-\mathrm{\infty }}^{\mathrm{\infty }}{a}_{i+n}{\zeta }_i$ be the long-range dependent linear process of a sequence $\{{\zeta }_i\}$ of i.i.d. random variables with mean zero and $E|{\zeta }_0{|}^p<\mathrm{\infty }$. Take θ such that $1/d<\theta <p$. Then all conditions of Theorem 2.2 are easily satisfied. By Theorem 2.2,

$$\sum _{n=1}^{\mathrm{\infty }}{n}^{-1}P\left(\right|\sum _{k=1}^n{X}_k|>n^{1/p+1-d}\epsilon )<\mathrm{\infty }\text{for all }\epsilon >0.$$

Authors’ contributions

All authors read and approved the manuscript.

Competing interests

The authors declare that they have no competing interests.