Convergence properties of the maximum partial sums for moving average process under \(\rho^{ }\) mixing assumption

R E S E A R C H

Open Access

Convergence properties of the maximum

partial sums for moving average process

under

ρ

–

-mixing assumption

Mingzhu Song

1

_{and Quanxin Zhu}

2,3*

*_{Correspondence:[email protected]}

2_{Key Laboratory of HPC-SIP (MOE),}

College of Mathematics and Statistics, Hunan Normal University, Changsha, China

3_{School of Mathematical Sciences,}

Qufu Normal University, Qufu, China Full list of author information is available at the end of the article

Abstract

In this paper, we assume that the sequence{Yn, –∞<n< +∞}is a

ρ

–-mixing

random variables which are stochastically dominated by a random variableY. Moreover, a real number sequence{an, –∞<n< +∞}is assumed to be absolute

summable. Then, complete convergence and complete

γ

-order moment convergence of the maximum partial sums for the moving average process {Xn=

+∞

j=–∞ajYn+j,n≥1}are obtained. The results in this paper extend and improve

the corresponding ones under NA and

ρ

-mixing conditions in the literature. MSC: 60F15; 62G05

Keywords: Moving average process;

ρ

–_{-mixing sequence; Complete convergence;}

Maximum partial sums; Complete

γ

-order moment convergence

1 Introduction

Let the{Yn, –∞<n< +∞}be a random variable sequence and assume that a real number

sequence{an, –∞<n< +∞}is absolute summable, i.e., +∞

j=–∞|aj|<∞. Then{Xn,n≥1}

is called a moving average process of the sequence{Yn, –∞<n< +∞}, if

Xn=

+∞

j=–∞

ajYj+n, n≥1, (1.1)

andSn= n

i=1Xi,n≥1 are the partial sums of{Xn,n≥1}.

The complete convergence plays a very important role in the probability theory and mathematical statistics, which was introduced by Hsu and Robbins [1]. A sequence

{Xn,n≥1}of random variables is said to converge completely to a constantθ, if

∞

n=1

P|Xn–θ|>ε

<∞, ∀ε> 0.

In view of the Borel–Cantelli lemma, the complete convergence implies thatXn→θ

almost surely. Therefore, complete convergence is a very important tool in establishing almost sure convergence for partial sums of random variables, as well as weighted sums of random variables.

Let{Xn,n≥1}be a sequence of random variables andan> 0,bn> 0,γ > 0. If

∞

n=1

anE

b–1_n |Xn|–ε γ

+<∞, ∀ε> 0,

then the above result was called the complete γ-order moment convergence by Chow [2]. We observe that the complete γ-order moment convergence is more general than complete convergence.

After the moving average process was defined, many conclusions about its convergence properties have been obtained. When the sequence{Yn, –∞<n< +∞}is identically

dis-tributed, many results about the moving average process{Xn,n≥1}are known. For

exam-ple, Ibragimov [3] obtained several limit theorems, Burton and Dehling [4] established a large deviation result, Li et al. [5] got the complete convergence property, Chen and Wang [6] obtained convergence rates of moderate deviations, and so on. Recently, several results have been obtained under the assumption that the sequence{Yn, –∞<n< +∞}is

depen-dent. For example, Zhang [7] and Chen et al. [8] discussed the limiting behavior of moving average processes underϕ-mixing assumptions; Baek et al. [9] and Yu and Wang [10] es-tablished complete convergence under NA assumptions; Li and Zhang [11], Kim and Ko [12–14] and Zhou [15,16] discussed the complete moment convergence under NA,ϕ -mixing andρ-mixing random variables, and Qiu and Chen [17] obtained the complete moment convergence under END.

In this paper, we aim to study some convergence properties of the maximum partial sums for a moving average process underρ–_{-mixing sequence assumption.}

Let T andS be two nonempty disjoint sets of integers, and then definedist(T,S) =:

min{|j–k|;j∈T,k ∈S}, and let the σ-field σ(T) (σ(S)) be generated by {Yk,k ∈T}

({Yk,k∈S}).

Definition 1.1 A sequence{Yn, –∞<n< +∞}is said to beρ–-mixing, if

ρ–(s) =supρ–(T,S);T,S⊂Z,dist(T,S)≥s→0, ass→ ∞,

ρ–(T,S) = 0∨supcorrf(Yi,i∈T),g(Yj,j∈S)

,

where the supremum is taken over all increasing real functionsf onRT_andgonRS_.

Definition 1.2 A sequence{Yn, –∞<n< +∞}is said to beρ-mixing, if

ρ(s) =supρ(T,S);T,S⊂Z,dist(T,S)≥s→0, ass→ ∞,

where

ρ(T,S) =supcorr(f,g);f ∈L2

σ(T),f ∈L2

σ(S).

Definition 1.3 A sequence{Yn, –∞<n< +∞}is said to be negatively associated (NA), if

Covf(Yi,i∈S),g(Yj,j∈T)

holds for every pair of disjoint subsetsT,S⊂Z, and coordinatewise increasing functions

f onRS_andgonRT_.

From the above definitions, we see thatρ–-mixing includes bothρ-mixing and the NA sequence. Recently many scholars have focused on the convergence properties of moving average processes underρ-mixing and NA condition. For example, Wang and Lu [18] ob-tained weak convergence and Rosenthal-type moment inequality; Budsaba et al. [19,20] and Zhang [21] got complete convergence for the moving average process underρ-mixing andρ–-mixing conditions; Tan et al. [22] gained the central limit theorem of partial sums forρ-mixing sequences. However, to the best of our knowledge, there have been few re-sults about the completeγ-order moment convergence for the moving average process underρ–-mixing condition. The main aim of this paper is to present some results about complete convergence and completeγ-order moment convergence for the maximum par-tial sums of{Xn,n≥1}underρ–-mixing condition.

Definition 1.4 The sequence{Yn, –∞<n< +∞}is called be stochastically dominated by

a random variableY, if for allx> 0

P|Yn|>x

≤DP|Y|>x, –∞<n< +∞,

where the constantD> 0; we denote{Yn, –∞<n< +∞} ≺Y.

Throughout this paper,I(A) denotes the indicator function of an eventA, the symbolC

represents a positive constant, which can take different values in different places, even in the same formula.

2 Several lemmas and main results

Lemma 2.1(Wang and Lu [18]) Let p≥2and consider aρ–-mixing random variables sequence of {Yn, –∞<n< +∞}.For every j≥1,assume EYj= 0and E|Yj|p <∞.Then,

there exists a positive constant C=C(q,ρ–(·))such that

E max

1≤k≤n k

j=1

Yj

p ≤C

_n

j=1

E|Yj|p+ n

j=1

EY_j2

p/2

. (2.1)

Lemma 2.2(Wang and Lu [18]) Let the sequence{Yn, –∞<n< +∞}beρ–-mixing.If the

functions of the sequence{fn, –∞<n< +∞}are all non-decreasing(non-increasing),then

the sequence{fn(Yn), –∞<n< +∞}is stillρ–-mixing.

Lemma 2.3(Wu [23]) Let{Yn, –∞<n< +∞} ≺Y.Then for all constants s,t> 0,there

exist positive constants C1,C2such that the following inequalities are true:

E|Yn|sI

|Yn| ≤t

≤C1

E|Y|sI|Y| ≤t+tsP|Y|>t,

E|Yn|sI

|Yn|>t

≤C2E|Y|sI

|Y|>t.

Theorem 2.1 Let1≤p< 2,α> 1.Assume that{Xn,n≥1}is a moving average process of

aρ–-mixing random variable sequence{Yn, –∞<n< +∞},and{Yn, –∞<n< +∞} ≺Y.

If EYj= 0for every j and E|Y|ap<∞,then

∞

n=1

nα–2_Pmax

1≤k≤n|Sk| ≥εn

1/p_{<∞ ∀ε> 0. (2.2)}

Ifα= 1, we have the following result.

Theorem 2.2 Let1≤p< 2.Suppose the+_i=–∞_∞|ai|θ<∞,whereθ∈(0, 1)if p= 1,andθ=

1if1 <p< 2.Let{Xn,n≥1}be a moving average process of aρ–-mixing random variable

sequence{Yn, –∞<n< +∞},and{Yn, –∞<n< +∞} ≺Y.If EYj= 0for every j and E|Y|p< ∞,then

∞

n=1

n–1P

max

1≤k≤n|Sk| ≥εn

1/p_{<∞ ∀ε> 0. (2.3)}

Theorem 2.3 Letγ > 0, 1≤p< 2,α> 1.Assume that{Xn,n≥1} is a moving average

process of aρ–-mixing random variable sequence{Yn, –∞<n< +∞},and{Yn, –∞<n<

+∞} ≺Y.If EYj= 0for every j,and

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

E|Y|αp_{<∞, ifγ<αp,}

E|Y|αp_{log(1 +|Y|) <∞, ifγ=αp,}

E|Y|γ_{<∞, ifγ>αp,}

then

∞

n=1

nα–2–γ/pE

max

1≤k≤n|Sk|–εn

1/pγ

+<∞ ∀ε> 0 (2.4)

and

∞

n=1

nα–2E

sup

k≥n

k–1/pSk–ε γ

+<∞ ∀ε> 0. (2.5)

Ifα= 1, we have the following result.

Theorem 2.4 Letγ > 0, 1≤p< 2,Suppose+_i=–∞_∞|ai|θ<∞,as well asθ∈(0, 1)if p= 1,

andθ= 1if1 <p< 2.Let{Xn,n≥1}be a moving average process of aρ–-mixing random

variable sequence{Yn, –∞<n< +∞},and{Yn, –∞<n< +∞} ≺Y.If EYj= 0for every j,

and

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

E|Y|p<∞, ifγ <p,

E|Y|p_{log(1 +|Y|) <∞, ifγ =p,}

then

∞

n=1

n–1–γ/pE

max

1≤k≤n|Sk|–εn

1/pγ

+<∞ ∀ε> 0. (2.6)

Remark2.1 Theorems2.1and2.2give complete convergence of the moving average

pro-cesses{Xn,n≥1}forρ–-mixing random variable sequences. It is worth pointing out that

slowly varying functions are not needed in this paper, so our Theorems2.1and2.2extend and improve those in Chen [8], Budsaba [19,20] and Zhang [21].

Remark 2.2 Theorems2.3 and2.4give completeγ-order moment convergence of the

moving average processes{Xn,n≥1}forρ–-mixing random variable sequences. Noting

that theρ–-mixing condition generalizes both NA andρ-mixing conditions, our Theo-rems2.3and2.4also extend and improve the corresponding results of Li and Zhang [11], Budsaba [19,20] and Tan [22].

Remark2.3 It should be mentioned that our condition{Yn, –∞<n< +∞} ≺Yis weaker

than the assumption of identical distribution for{Yn, –∞<n< +∞}. Thus, our results still

hold for identically distributed random variables.

Remark2.4 Many dependent random variables satisfy condition (2.1). For example, one

can consider NA,ρ-mixing andϕ-mixing random variable sequences. Therefore, our re-sults extend and improve the corresponding rere-sults of Zhou [15,16] and Qiu [17].

3 Proof of theorems

Proof of Theorem2.1 Write

Y_j= –n1/pIYj< –n1/p

+YjI

|Yj| ≤n1/p

+n1/pIYj>n1/p

,

Y_j=Yj–Yj=

Yj+n1/p

IYj< –n1/p

+Yj–n1/p

IYj>n1/p

.

Note that

n

k=1

Xk= n

k=1 +∞

j=–∞ ajYk+j=

+∞

j=–∞ aj

j+n

i=j+1

Yi.

Since+_j=–∞_∞|aj|<∞,EYi= 0, it follows from Lemma2.3that

n–1/p

E1max≤k≤n

+∞

j=–∞

aj j+k

i=j+1

Y_i

(sinceEYi= 0)

≤n–1/p

+∞

j=–∞

|aj| j+n

i=j+1

n1/pP|Yi|>n1/p

+E|Yi|I

|Yi|>n1/p

≤Cn1–1/pE|Y|I|Y|>n1/p

Therefore, for large enoughn, we obtain

n–1/p

E1max≤k≤n

+∞

j=–∞ aj

j+k

i=j+1

Y_i

<ε/4. (3.1)

By (3.1), we have

∞

n=1

nα–2P

max

1≤k≤n|Sk| ≥εn

1/p

=

∞

n=1

nα–2_P

max

1≤k≤n

+∞

j=–∞

aj j+k

i=j+1

Yi ≥εn1/p

≤

∞

n=1

nα–2P

max

1≤k≤n

+∞

j=–∞ aj

j+k

i=j+1

Y_i

≥εn1/p/2

+

∞

n=1

nα–2P

max

1≤k≤n

+∞

j=–∞ aj

j+k

i=j+1

Y_i–EY_i

≥εn1/p/4

=:I1+I2.

ForI1, notingαp> 1, it follows from Lemma2.3and the Markov inequality that

I1≤C

∞

n=1

nα–2–1/pE max

1≤k≤n

∞

j=–∞ aj

j+k

i=j+1

Y_i ≤C ∞ n=1

nα–2–1/p

+∞

j=–∞

|aj| j+n

i=j+1

E|Yi|I

|Yi|>n1/p

≤C ∞

n=1

nα–1–1/p_E|Y|I|Y|>n1/p

=C ∞

n=1

nα–1–1/p ∞

l=n

E|Y|Il1/p<|Y| ≤(l+ 1)1/p

≤C ∞

l=1

E|Y|Il1/p<|Y| ≤(l+ 1)1/p

l

n=1

nα–1–1/p

≤C ∞

l=1

lα–1/pE|Y|Il1/p<|Y| ≤(l+ 1)1/p≤E|Y|αp<∞. (3.2)

ForI2, from Lemma2.2, we see that{Yj–EYj}is still aρ–-mixing random variable

se-quence. By Lemma2.1, Markov and Hölder inequalities, we have that for anyv≥2,

I2≤C

∞

n=1

nα–2–v/p_E

max

1≤k≤n

+∞

j=–∞

aj j+k

i=j+1

Y_i–EY_i

v ≤C ∞ n=1

nα–2–v/pE

_+∞

j=–∞

|aj| max

1≤k≤n

j+k

i=j+1

Y_i–EY_i

=C ∞

n=1

nα–2–v/p_{E +∞}

j=–∞

|aj|1–1/v |aj|1/vmax

1≤k≤n

j+k

i=j+1

Y_i–EY_i

v

≤C ∞

n=1

nα–2–v/p

+∞

j=–∞

|aj|

v–1 _+∞

j=–∞

|aj|E max

1≤k≤n

j+k

i=j+1

Y_i–EY_i

v

≤C ∞

n=1

nα–2–v/p

+∞

j=–∞

|aj| _j+n

i=j+1

EY_iv+

j+n

i=j+1

EY_i2

v/2

=C ∞

n=1

nα–2–v/p

+∞

j=–∞

|aj| _j+n

i=j+1

E|Yi|vI

|Yi| ≤n1/p

+nv/pP|Yi|>n1/p

+C ∞

n=1

nα–2–v/p

+∞

j=–∞

|aj| _j+n

i=j+1

E|Yi|2I

|Yi| ≤n1/p

+n2/pP|Yi|>n1/p v/2

≤C ∞

n=1

nα–1–v/pE|Y|vI|Y| ≤n1/p+nv/pP|Y|>n1/p

+C ∞

n=1

nα–2–v/p+v/2E|Y|2I|Y| ≤n1/p+n2/pP|Y|>n1/pv/2

=:I21+I22.

ForI21, takingv≥αp, we have

I21 =C

∞

n=1

nα–1–v/pE|Y|vI|Y| ≤n1/p+C ∞

n=1

nα–1P|Y|>n1/p

=C ∞

n=1

nα–1–v/p n

l=1

E|Y|vI(l– 1)1/p<|Y| ≤l1/p

+C ∞

n=1

nα–1 ∞

l=n

Pl1/p<|Y| ≤(l+ 1)1/p

=C ∞

l=1

E|Y|vI(l– 1)1/p<|Y| ≤l1/p ∞

n=l

nα–1–v/p

+C ∞

l=1

lαPl1/p<|Y| ≤(l+ 1)1/p

≤C ∞

l=1

lα–v/pE|Y|vI(l– 1)1/p<|Y| ≤l1/p+CE|Y|αp

≤CE|Y|αp<∞. (3.3)

ForI22, we will prove the claim based on two cases.

Ifαp< 2, settingv= 2, similar to the proof ofI21, we obtain

I22=C

∞

n=1

Ifαp≥2, noting thatE|Y|2_{<∞, choosingv>max{2, (α– 1)2p/(2 –p)}, we have that}

α–v/p+v/2 < 1, and therefore

I22≤C

∞

n=1

nα–2–v/p+v/2E|Y|2v/2<∞. (3.5)

Thus, from (3.2)–(3.5), we see that (2.2) is satisfied.

Next, we prove Theorem2.2.

Proof of Theorem2.2 From the proof of Theorem2.1, we only need to proveI1<∞,I2<

∞.

ForI1, notingθ∈(0, 1) ifp= 1, andθ= 1 if 1 <p< 2, by the Markov andCr-inequalities,

we have

I1≤C

∞

n=1

n–1–θ/pE max

1≤k≤n

+∞

j=–∞

aj j+k

i=j+1

Y_i

θ

≤C ∞

n=1

n–1–θ/p

+∞

j=–∞

|aj|θ j+n

i=j+1

E|Yi|θI

|Yi|>n1/p

≤C ∞

n=1

n–θ/pE|Y|θI|Y|>n1/p

=C ∞

n=1

n–θ/p

∞

l=n

E|Y|θ_Il1/p_{<|Y| ≤(l+ 1)}1/p

=C ∞

l=1

E|Y|θIl1/p<|Y| ≤(l+ 1)1/p

l

n=1

n–θ/p

≤C ∞

l=1

l1–θ/p_E|Y|θ_Il1/p_{<|Y| ≤(l+ 1)}1/p_<CE|Y|p_{<∞. (3.6)}

ForI2, by Markov and Hölder inequalities, as well as Lemmas2.1–2.3, we obtain

I2 =

∞

n=1

n–1–2/pE

max

1≤k≤n

+∞

j=–∞

aj j+k

i=j+1

Y_i–EY_i

2

=

∞

n=1

n–1–2/pE

_+∞

j=–∞

|aj|1/2 |aj|1/2 max

1≤k≤n

j+k

i=j+1

Y_i–EY_i

2

≤

∞

n=1

n–1–2/p

+∞

j=–∞

|aj|

∞

j=1

|aj|E max

1≤k≤n

j+k

i=j+1

Y_i–EY_i

2

≤C ∞

n=1

n–1–2/p

+∞

j=–∞

|aj| j+n

i=j+1

EY_i2

≤C ∞

n=1

n–1–2/p

+∞

j=–∞

|aj| _j+n

i=j+1

E|Yi|2I

|Yi| ≤n1/p

+n2/pP|Yi|>n1/p

≤C ∞

n=1

n–2/pE|Y|2I|Y| ≤n1/p+n2/pP|Y|>n1/p

≤C ∞

n=1

n–2/p

n

l=1

E|Y|2I(l– 1)1/p<|Y| ≤l1/p+CE|Y|p

≤C ∞

l=1

l1–2/pE|Y|2I(l– 1)1/p<|Y|<l1/p+CE|Y|p

≤CE|Y|p<∞. (3.7)

Combing (3.6) and (3.7), we see that Theorem2.2holds.

Proof of Theorem2.3 Letx>nγ/p_{and consider}

Y_j= –x1/γIYj< –x1/γ

+YjI

|Yj| ≤x1/γ

+x1/γIYj>x1/γ

,

Y_j=Yj–Yj=

Yj+x1/γ

IYj< –x1/γ

+Yj–x1/γ

IYj>x1/γ

.

Similarly to the proof of (3.1), for large enoughx, we get

x–1/γ

E1max≤k≤n

+∞

j=–∞ aj

j+k

i=j+1

Y_i

<ε/4. (3.8)

Then, we have

∞

n=1

nα–2–γ/pE

max

1≤k≤n|Sk|–εn

1/pγ

+

=

∞

n=1

nα–2–γ/p

_∞

0

P

max

1≤k≤n|Sk|>εn

1/p_+x1/γ dx

≤

∞

n=1

nα–2–γ/p

nγ/p

0

P

max

1≤k≤n|Sk|>εn

1/p_dx

+

∞

n=1

nα–2–γ/p ∞

nγ/p

Pmax

1≤k≤n|Sk|>x

1/γ_dx

≤

∞

n=1

nα–2P

max

1≤k≤n|Sk|>εn

1/p

+

∞

n=1

nα–2–γ/p

∞

nγ/pP

max

1≤k≤n|Sk|>x

1/γ dx.

It follows from Theorem2.1that

∞

n=1

nα–2_Pmax

1≤k≤n|Sk|>εn

To prove (2.4) of Theorem2.3, we only need to prove

I3=:

∞

n=1

nα–2–γ/p

_∞

nγ/pP

max

1≤k≤n|Sk|>x

1/γ dx<∞.

By (3.8), we get

I3≤C

∞

n=1

nα–2–γ/p

∞

nγ/pP

max

1≤k≤n

+∞

j=–∞ aj

j+k

i=j+1

Y_i

≥x1/γ/2 dx +C ∞ n=1

nα–2–γ/p

_∞

nγ/pP

max

1≤k≤n

+∞

j=–∞ aj

j+k

i=j+1

Y_i–EY_i

≥x1/γ/4

dx

=:I31+I32.

ForI31, by Markov inequality and Lemma2.3, we get

I31≤C

∞

n=1

nα–2–γ/p

∞

nγ/px

–1/γ

E max

1≤k≤n

+∞

j=–∞

aj j+k

i=j+1

Y_i dx ≤C ∞ n=1

nα–2–γ/p

∞

nγ/px

–1/γ

+∞

j=–∞

|aj| j+n

i=j+1

EY_idx

≤C ∞

n=1

nα–2–γ/p

∞

nγ/px

–1/γ

+∞

j=–∞

|aj| j+n

i=j+1

E|Yi|I

|Yi|>x1/γ dx ≤C ∞ n=1

nα–1–γ/p

_∞

nγ/px

–1/γ

E|Y|I|Y|>x1/γdx

=C ∞

n=1

nα–1–γ/p ∞

l=n

(l+1)γ/p

lγ/p x

–1/γ

E|Y|I|Y|>x1/γdx

≤C ∞

n=1

nα–1–γ/p ∞

l=n

lγ/p(1–1/γ)–1E|Y|I|Y|>l1/p

=C ∞

l=1

lγ/p–1/p–1E|Y|I|Y|>l1/p

l

n=1

nα–1–γ/p

≤ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

C∞_l=1lγ/p–1/p–1_E|Y|I(|Y|>l1/p_)lα–γ/p_{, ifγ <αp,}

C∞_l=1lγ/p–1/p–1_E|Y|I(|Y|>l1/p_{)log(l+ 1), ifγ =αp,}

C∞_l=1lγ/p–1/p–1_E|Y|I(|Y|>l1/p_{), ifγ >αp}

≤ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

C∞_l=1lα–1/p_E|Y|I(|Y|>l1/p_{), ifγ <αp,}

C∞_l=1lα–1/p_{log(l+ 1)E|Y|I(|Y|>l}1/p_{), ifγ =αp,}

C∞_l=1lγ/p–1/p_E|Y|I(|Y|>l1/p_{), ifγ >αp}

≤ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

CE|Y|αp_{<∞, ifγ<αp,}

CE|Y|αp_{log(|Y|+ 1) <∞, ifγ=αp,}

CE|Y|γ_{<∞, ifγ>αp.}

ForI32, from Lemmas2.1–2.3, Markov and Hölder inequalities, we have that for anyv≥2,

I32≤C

∞

n=1

nα–2–γ/p

_∞

nγ/px

–v/γ E

max

1≤k≤n

+∞

j=–∞ aj

j+k

i=j+1

Y_i–EY_i

v dx ≤C ∞ n=1

nα–2–γ/p ∞

nγ/px

–v/γ_E

_+∞

j=–∞

|aj|max

1≤k≤n

j+k

i=j+1

Y_i–EY_i

v dx ≤C ∞ n=1

nα–2–γ/p

∞

nγ/px

–v/γ E

_+∞

j=–∞

|aj|1–1/v |aj|1/v max

1≤k≤n

j+k

i=j+1

Y_i–EY_i

v dx ≤C ∞ n=1

nα–2–γ/p

∞

nγ/px

–v/γ

+∞

j=–∞

|aj| v–1

×

+∞

j=–∞

|aj|E max

1≤k≤n

j+k

i=j+1

Y_i–EY_i

v dx ≤C ∞ n=1

nα–2–γ/p

∞

nγ/px

–v/γ

+∞

j=–∞

|aj| _j+n

i=j+1

EY_iv+

j+n

i=j+1

EY_j2

v/2 dx ≤C ∞ n=1

nα–2–γ/p ∞

nγ/p

x–v/γ

+∞

j=–∞

|aj|

× _j+n

i=j+1

E|Yi|vI

|Yi| ≤x1/γ

+xv/γ_P|Y

i|>x1/γ

dx +C +∞ n=1

nα–2–γ/p ∞

nγ/px

–v/γ

+∞

j=–∞

|aj|

× _j+n

i=j+1

E|Yi|2I

|Yi| ≤x1/γ

+x2/γP|Yi|>x1/γ v/2 dx ≤C ∞ n=1

nα–1–γ/p

∞

nγ/px

–v/γ

E|Y|vI|Y| ≤x1/γ+xv/γP|Y|>x1/γdx

+C ∞

n=1

nα–2–γ/p+v/2

_∞

nγ/px

–v/γ

E|Y|2I|Y| ≤x1/γ+x2/γP|Y|>x1/γv/2dx

=:I321+I322.

ForI321, takingv>αp, we have

I321 =C

∞

n=1

nα–1–γ/p ∞

l=n

(l+1)γ/p

lγ/p

x–v/γE|Y|vI|Y| ≤x1/γ+P|Y|>x1/γdx

≤C ∞

n=1

nα–1–γ/p

∞

l=n

l(γ/p(1–v/γ)–1)_E|Y|v_{I|Y| ≤(l+ 1)}1/p_+lγ/p–1_P|Y|>l1/p

≤C ∞

l=1

l(γ/p–v/p–1)E|Y|vI|Y| ≤(l+ 1)1/p+lγ/p–1P|Y|>l1/p

l

n=1

≤ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

C∞_l=1[lα–v/p–1_E|Y|v_{I(|Y| ≤(l+ 1)}1/p_{) +l}α–1_P(|Y|>l1/p_)],

ifγ <αp,

C∞_l=1[lα–v/p–1_E|Y|v_{I(|Y| ≤(l+ 1)}1/p_{) +l}α–1_P(|Y|>l1/p_{)]log(l+ 1),}

ifγ =αp,

C∞_l=1[lγ/p–v/p–1_E|Y|v_{I(|Y| ≤(l+ 1)}1/p_{) +l}γ/p–1_P(|Y|>l1/p_)],

ifγ >αp

≤ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

C∞_k=1kα–v/p_E|Y|v_{I((k– 1)}1/p_{<|Y| ≤k}1/p_{) +CE|Y|}αp_,

ifγ <αp,

C∞_k=1kα–v/p_E|Y|v_{I((k– 1)}1/p_{<|Y| ≤k}1/p_{)log(k+ 1) +CE|Y|}αp_{log(|Y|+ 1),}

ifγ =αp,

C∞_k=1kγ/p–v/p_E|Y|v_{I((k– 1)}1/p_{<|Y| ≤k}1/p_{) +CE|Y|}γ_,

ifγ >αp

≤ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

CE|Y|αp_{<∞, ifγ <αp,}

CE|Y|αp_{log(|Y|+ 1) <∞, ifγ =αp,}

CE|Y|γ_{<∞, ifγ >αp.}

(3.11)

ForI322, we prove the claim by considering two cases.

Ifαp< 2, settingv= 2, we have

I322=C

∞

n=1

nr–1–γ/p ∞

nγ/p

x–2/γ_E|Y|2_{I|Y| ≤x}1/γ_+x2/γ_P|Y|>x1/γ_dx.

Similarly to the proof ofI321, we have

I322<∞. (3.12)

Ifαp≥2, noting thatE|Y|2_{<∞, choosingv>max{2,γ, (α– 1)2p/(2 –p)}, we obtain}

α–v/p+v/2 < 1. Therefore, we have

I322≤C

∞

n=1

nα–2–γ/p+v/2

∞

nγ/px

–v/γ

E|Y|2v/2dx

≤C ∞

n=1

nα–2+v/2–v/p_E|Y|2v/2_{<∞. (3.13)}

Then, by (3.6)–(3.13), we see that (2.4) is true. Next, we prove (2.5).

∞

n=1

nα–2E

sup

k≥n

k–1/pSk–ε γ

+

=

∞

n=1

nα–2

∞

0

P

sup

k≥n

k–1/pSk>ε+x1/γ

=

∞

j=1 2j–1

n=2j–1 nα–2

∞

0

P

sup

k≥n _k–1/p_S

k>ε+x1/γ

dx

≤C ∞

j=1

∞

0

P

sup

k≥2j–1

k–1/pSk>ε+x1/γ

dx

2j–1

n=2j–1 2j(α–2)

≤C ∞

j=1

2j(α–1)

∞

0

P

sup

k≥2j–1

k–1/pSk>ε+x1/γ

dx

≤C ∞

j=1

2j(α–1)

∞

l=j _∞

0

P

max

2l–1_≤k≤2l

k–1/pSk>ε+x1/γ

dx

≤C ∞

l=1

∞

0

P

max

2l–1_≤k≤2l

_k–1/p_S

k>ε+x1/γ

dx

l

j=1

2j(α–1)

≤C ∞

l=1

2l(α–1)

∞

0

P max

2l–1_≤k≤2l|Sk|>

ε+x1/γ₂(l–1)/p)_dx

Lety= 2(l–1)γ/p_x

≤C ∞

l=1

2l(α–1–γ/p)

_∞

0

P

max

1≤k≤2l|Sk|>ε2

(l–1)/p_+y1/γ dy

≤C ∞

n=1

n(α–2–γ/p)

∞

0

P

max

1≤k≤n|Sk|>εn

1/p₂–1/p_+y1/γ dy

Letε0= 2–1/pε

≤C ∞

n=1

n(α–2–γ/p)E

max

1≤k≤n|Sk|–ε0n

1/pγ

+<∞.

Thus, (2.5) holds.

Next, we prove Theorem2.4.

Proof of Theorem2.4 Since the proof of Theorem2.4is similar to that of Theorem2.3, we only give the proof outline as follows.

By (3.1), we have

∞

n=1

n–1–γ/p_Emax

1≤k≤n|Sk|–εn

1/pγ

+

≤

∞

n=1

n–1P

max

1≤k≤n|Sk|>εn

1/p₊

∞

n=1

n–1–γ/p

∞

nγ/pP

max

1≤k≤n|Sk|>x

1/γ dx

=:J1+J2.

ForJ1, by Theorem2.2, we have

J1=

∞

n=1

n–1P

max

1≤k≤n|Sk|>εn

ForJ2, similar to the proof ofI3, we have

J2≤C

∞

n=1

n–1–γ/p ∞

nγ/pP

max

1≤k≤n

+∞

j=–∞

aj j+k

i=j+1

Y_i

≥x1/γ/2 dx +C ∞ n=1

n–1–γ/p

∞

nγ/pP

max

1≤k≤n

+∞

j=–∞ aj

j+k

i=j+1

Y_i–EY_i

≥x1/γ/4

dx

=:J21+J22.

ForJ21, by Lemma2.3, Markov andCrinequalities, similarly to the proof ofI31, we get

J21≤C

∞

n=1

n–1–γ/p

∞

nγ/px

–θ/γ

E max

1≤k≤n

+∞

j=–∞ aj

j+k

i=j+1

Y_i θ dx ≤C ∞ n=1

n–γ/p

∞

l=n

(l+1)γ/p

lγ/p

x–θ/γ_E|Y|θ_I|Y|>x1/γ_dx

≤C ∞

n=1

n–γ/p

∞

l=n

lγ/p(1–θ/γ)–1_E|Y|θ_I|Y|>l/p

≤ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

C∞_l=1l–θ/p_E|Y|θ_I(|Y|>l1/p_{), ifγ <p,}

C∞_l=1l–θ/p_E|Y|θ_I(|Y|>l1/p_{)log(l+ 1), ifγ =p,}

C∞_l=1lγ/p–θ/p–1_E|Y|θ_I(|Y|>l1/p_{), ifγ >p}

≤ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

CE|Y|p_{<∞, ifγ <p,}

CE|Y|p_{log(|Y|+ 1) <∞, ifγ =p,}

CE|Y|γ _{<∞, ifγ >p.}

(3.15)

ForJ22, similar to the proof ofI32, takingv= 2, we obtain

J22≤C

∞

n=1

n–1–γ/p ∞

nγ/p

x–2/γ_E

max

1≤k≤n

+∞

j=–∞

aj j+k

i=j+1

Y_i–EY_i

2 dx ≤C ∞ n=1

n–1–γ/p

∞

nγ/px

–2/γ E

_+∞

j=–∞

|aj|1/2 |aj|1/2 max

1≤k≤n

j+k

i=j+1

Y_i–EY_i

2 dx ≤C ∞ n=1

n–1–γ/p

∞

nγ/px

–2/γ

+∞

j=–∞

|aj| i+n

i=j+1

EY_i2dx

≤C ∞

n=1

n–γ/p ∞

nγ/p

x–2/γ_E|Y|2_{I|Y| ≤x}1/γ_+x2/γ_P|Y|>x1/γ_dx

=C ∞

n=1

n–γ/p

∞

l=n

(l+1)γ/p

lγ/p

x–2/γ_E|Y|2_{I|Y| ≤x}1/γ_+P|Y|>x1/γ_dx

≤C ∞

l=1

l(γ/p–2/p–1)E|Y|2I|Y| ≤(l+ 1)1/p+lγ/p–1P|Y|>l1/p

l

n=1

≤ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

C∞_k=1[k1–2/p_E|Y|2_{I((k– 1)}1/p_{<|Y| ≤k}1/p_{)] +CE|Y|}p_,

ifγ <p,

C∞_k=1[k1–2/pE|Y|2I((k– 1)1/p<|Y| ≤k1/p)]log(l+ 1) +CE|Y|plog(|Y|+ 1),

ifγ =p,

C∞_k=1[kγ/p–2/p_E|Y|2_{I((k– 1)}1/p_{<|Y| ≤k}1/p_{)] +CE|Y|}γ_,

ifγ >p

≤ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

CE|Y|p_{<∞, ifγ <p,}

CE|Y|p_{log(|Y|+ 1) <∞, ifγ =p,}

CE|Y|γ _{<∞, ifγ >p.}

(3.16)

Therefore, from (3.15)–(3.16), we see that (2.6) holds.

Acknowledgements

The authors would like to thank the referees for their valuable comments and suggestions, which improved the manuscript.

Funding

This work was jointly supported by the National Natural Science Foundation of China (No. 61773217), the Scientific Research Fund of Hunan Provincial Education Department (18A013), the Construct Program of the Key Discipline in Hunan Province, the Key University Science Research Project of Anhui Province (No. KJ2016A705), the Key Program in the Youth Talent Support Plan in Universities of Anhui Province (No. gxyqZD2016317).

Availability of data and materials

Not applicable.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors contributed equally and significantly in writing this paper. Both authors read and approved the final manuscript.

Author details

1_{Department of Mathematics and Computer Science, Tongling University, Tongling, China.}2_{Key Laboratory of HPC-SIP}

(MOE), College of Mathematics and Statistics, Hunan Normal University, Changsha, China.3_{School of Mathematical} Sciences, Qufu Normal University, Qufu, China.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Received: 20 February 2019 Accepted: 21 March 2019

References

1. Hsu, P., Robbins, H.: Complete convergence and the law of large numbers. Proc. Natl. Acad. Sci. USA33, 25–31 (1947) 2. Chow, Y.S.: On the rate of moment complete convergence of sample sums and extremes. Bull. Inst. Math. Acad. Sin.

16, 177–201 (1988)

3. Ibragimov, I.A.: Some limit theorem for stationary processes. Theory Probab. Appl.7, 349–382 (1962) 4. Burton, R.M., Dehling, H.: Large deviations for some weakly dependent random processes. Stat. Probab. Lett.9,

397–401 (1990)

5. Li, D., Bhaskara Rao, M., Wang, X.: Complete convergence of moving average processes. Stat. Probab. Lett.14, 111–114 (1992)

6. Chen, P.-Y., Wang, D.-C.: Convergence rates for probabilities of moderate deviations for moving average process. Acta Math. Sin. Engl. Ser.24(4), 611–622 (2008)

7. Zhang, L.: Complete convergence of moving average processes under dependence assumptions. Stat. Probab. Lett.

30, 165–170 (1996)

8. Chen, P.-Y., Hu, T.-C., Volodin, A.: Limiting behaviour of moving average processes underϕ-mixing assumption. Stat. Probab. Lett.75, 105–111 (2009)

9. Baek, J.I., Kim, T.S., Liang, H.Y.: On the convergence of moving average processes under dependent conditions. Aust. N. Z. J. Stat.45(3), 331–342 (2003)

11. Li, Y.-X., Zhang, L.-X.: Complete moment convergence of moving average processes under dependence assumptions. Stat. Probab. Lett.70(3), 191–197 (2004)

12. Kim, T.S., Ko, M.H.: Complete moment convergence of moving average processes under dependence assumptions. Stat. Probab. Lett.78(7), 839–846 (2008)

13. Kim, T.S., Ko, M.H., Choi, Y.K.: Complete moment convergence of moving average processes with dependent innovations. J. Korean Math. Soc.45(2), 355–365 (2008)

14. Ko, M.H., Kim, T.S., Ryu, D.H.: On the complete moment convergence of moving average processes generated by

ρ-mixing sequences. Commun. Korean Math. Soc.23(4), 597–606 (2008)

15. Zhou, X.-C.: Complete moment convergence of moving average processes underϕ-mixing assumptions. Stat. Probab. Lett.80, 285–292 (2010)

16. Zhou, X.-C., Lin, J.-G.: Complete moment convergence of moving average processes underρ-mixing assumption. Math. Slovaca61(6), 979–992 (2011)

17. Qiu, D.-H., Chen, P.-Y.: Convergence for moving average processes under END set-up. Acta Math. Sci. Ser. A35(4), 756–768 (2018)

18. Wang, J.-F., Lu, F.-B.: Inequalities of maximum of partial sums and weak convergence for a class of weak dependent random variables. Acta Math. Sin.22, 693–700 (2006)

19. Budsaba, K., Chen, P.-Y., Volodin, A.: Limiting behavior of moving average processes based on a sequence ofρ-mixing random variables. Thail. Stat.5, 69–80 (2007)

20. Budsaba, K., Chen, P.-Y., Volodin, A.: Limiting behavior of moving average processes based on a sequence ofρ-mixing and NA random variables. Lobachevskii J. Math.26, 17–25 (2007)

21. Zhang, Y.: Complete moment convergence for moving average processes generatedρ–_{-mixing random variable.} J. Inequal. Appl. (2015).https://doi.org/10.1186/s13660-015-0766-5

22. Tan, X.-L., Zhang, Y.: An almost sure central limit theorem of products of partial sums forρ-mixing sequences. J. Inequal. Appl. (2012).https://doi.org/10.1186/1029-242X-2012-51