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

R E S E A R C H

Open Access

Complete moment convergence for

moving average process generated by

ρ

–

-mixing random variables

Yong Zhang

*_{Correspondence:}

[email protected] College of Mathematics, Jilin University, Changchun, 130012, P.R. China

Abstract

Let{Yi, –∞<i<∞}be a sequence of

ρ

assumption of identical distributions, and{ai, –∞<i<∞}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{Xn=

_∞

i=–∞aiYi+n,n≥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

ρ

Keywords: complete moment convergence; moving average process;

ρ

Marcinkiewicz-Zygmund strong law of large numbers

1 Introduction

Let{Yi, –∞<i<∞}be a sequence of random variables and{ai, –∞<i<∞}be an abso-lutely summable sequence of real numbers, and forn≥ setXn=

_∞

i=–∞aiYi+n. The limit

behavior of the moving average process{Xn,n≥}has been extensively investigated by

many authors. For example, Baeket al.[] have obtained the convergence of moving

aver-age processes, Burton and Dehling [] have obtained a large deviation principle, Ibragimov [] has established the central limit theorem, Račkauskas and Suquet [] have proved the functional central limit theorems for self-normalized partial sums of linear processes, and Chenet al.[], Guo [], Kimet al.[, ], Koet al.[], Liet al.[], Li and Zhang [], Qiuet al.[], Wang and Hu [], Yang and Hu [], Zhang [], Zhenet al.[], Zhouet al.[],

Zhou and Lin [], Shenet al.[] have obtained the complete (moment) convergence

of moving average process based on a sequence of dependent (or mixing) random vari-ables, respectively. But very few results for moving average process based on aρ–-mixing random variables are known. Firstly, we recall some definitions.

For two nonempty disjoint setsS,Tof real numbers, we definedist(S,T) =min{|j–k|;j∈ S,k∈T}. Letσ(S) be theσ-field generated by{Yk,k∈S}, and defineσ(T) similarly.

Definition . A sequence{Yi, –∞<i<∞}is calledρ–-mixing, if

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

where

ρ–(S,T) = ∨sup{corrf(Xi,i∈S),g(Xj,j∈T),

where the supremum is taken over all coordinatewise increasing real functionsf onRS

andgonRT_.

Definition . A sequence{Yi, –∞<i<∞}is calledρ∗-mixing if

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

where

ρ(S,T) =supcorr(f,g);f ∈L

σ(S),g∈L

σ(T).

Definition . A sequence{Yi,i∈Z}is called negatively associated (NA) if for every pair of disjoint subsetsS,TofZand any real coordinatewise increasing functionsf onRSand

gonRT

Covf(Yi,i∈S),g(Yj,j∈T)≤.

Definition . A sequence{Yi, –∞<i<∞}of random variables is said to be stochasti-cally dominated by a random variableY if there exists a constantCsuch that

P|Yi|>x

≤CP|Y|>x, x≥, –∞<i<∞.

Definition . A real valued functionl(x), positive and measurable on [,∞), is said to be slowly varying at infinity if for eachλ> ,lim_x→∞l(λx)

l(x) = .

Li and Zhang [] obtained the following complete moment convergence of moving av-erage processes under NA assumptions.

Theorem A Suppose that{Xn=

∞

i=–∞aiεi+n,n≥},where{ai, –∞<i<∞}is a sequence

of real numbers with∞_i=–∞|ai|<∞and{εi, –∞<i<∞}is a sequence of identically

dis-tributed NA random variables with Eε= ,Eε<∞.Let h be a function slowly varying at

infinity, ≤q< ,r>  +q/.Then E|ε|rh(|ε|q) <∞implies

∞

n=

nr/q––/qh(n)E

n

j=

Xj

–εn/q

+

<∞

for allε> .

Chenet al.[] also established the following results for moving average processes under NA assumptions.

Theorem B Let q> , ≤p< ,r≥,rp= .Suppose that{Xn=

∞

i=–∞aiεi+n,n≥},

∞}is a sequence of identically distributed NA random variables.If Eε= and E|ε|rp<∞,

then

∞

n=

nr–P max

≤k≤n

k

j=

Xj

≥εn/p

<∞

for allε> .Furthermore if Eε= and E|ε|rp<∞for q<rp,E|ε|rplog( +|ε|) <∞for

q=rp,E|ε|q<∞for q>rp,then

∞

n=

nr––q/pE max

≤k≤n

k

j=

Xj

–εn/q

+q <∞

for allε> .

Recently, Zhou and Lin [] obtained the following complete moment convergence of

moving average processes underρ-mixing assumptions.

Theorem C Let h be a function slowly varying at infinity,p≥,pα> andα> /. Sup-pose that{Xn,n≥}is a moving average process based on a sequence{Yi, –∞<i<∞}of identically distributedρ-mixing random variables.If EY= and E|Y|p+δh(|Y|/α) <∞

for someδ> ,then for allε> ,

∞

n=

npα––α_{h(n)E max}

≤k≤n

k

j=

Xj

–εnα

+

<∞

and

∞

n=

npα–h(n)E sup

k≥n

k–α

k

j=

Xj

–ε

+

<∞.

Obviously,ρ–-mixing random variables include NA andρ∗-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 [] for a Rosenthal-type

moment inequality and weak convergence, Budsabaet al.[, ] for complete

conver-gence for moving average process based on aρ–-mixing sequence, Tanet al.[] for the almost sure central limit theorem. But there are few results on the complete moment

con-vergence of moving average process based on aρ–-mixing sequence. Therefore, in this

paper, we establish some results on the complete moment convergence for maximum par-tial sums with less restrictions. Throughout the sequel,Crepresents a positive constant although its value may change from one appearance to the next,I{A}denotes the indicator function of the setA.

2 Preliminary lemmas

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

() m_n=ns_l(n)≤Cms+_{l(m)fors> –and positive integerm,}

() ∞_n=mns_l(n)≤Cms+_{l(m)fors< –and positive integerm.}

Lemma .(Wang and Lu []) For a positive real number q≥,if{Xn,n≥}is a

se-quence ofρ–-mixing random variables,with EXi= ,E|Xi|q<∞for every i≥,then for

all n≥,there is a positive constant C=C(q,ρ–(·))such that

E

max

≤j≤n

j

i=

Xi

q

≤C

n

i=

E|Xi|q+

_n

i=

EX_i q



.

Lemma .(Wanget al.[]) Let{Xn,n≥}be a sequence of random variables which is stochastically dominated by a random variable X.Then for any a> and b> ,

E|Xn|aI

|Xn| ≤b

≤CE|X|aI|X| ≤b+baP|X|>b,

E|Xn|aI

|Xn|>b

≤CE|X|aI|X|>b.

3 Main results and proofs

Theorem . Let l be a function slowly varying at infinity, p≥,α > /,αp> . As-sume that{ai, –∞<i<∞} is an absolutely summable sequence of real numbers.

Sup-pose that{Xn=

∞

i=–∞aiYi+n,n≥}is a moving average process generated by a sequence {Yi, –∞<i<∞}ofρ–-mixing random variables which is stochastically dominated by a random variable Y.If EYi= for/ <α≤,E|Y|pl(|Y|/α) <∞for p> and E|Y|+δ<∞

for p= and someδ> ,then for anyε> 

∞

n=

nαp––α_{l(n)E max}

≤k≤n

k

j=

Xj

–εnα

+

<∞ (.)

and

∞

n=

nαp–l(n)E sup

k≥n

k–α

k

j=

Xj

–ε

+

<∞. (.)

Proof Firstly to prove (.). Letf(n) =nαp––α_{l(n) andY}()

xj = –xI{Yj< –x}+YjI{|Yj| ≤x}+

xI{Yj>x}andYxj()=Yj–Yxj()be the monotone truncations of{Yj, –∞<j<∞}forx> . Then by the property ofρ–_{-mixing random variables (cf.Property P in Wang and Lu []),}

{Y_xj()–EY_xj(), –∞<j<∞}and{Y_xj(), –∞<j<∞}are two sequences ofρ–-mixing random variables. Note thatn_k=Xk=

∞

i=–∞ai

i+n

j=i+Yj. Since

∞

i=–∞|ai|<∞, by Lemma ., we have forx>nα_{, ifα> }

x– E

∞

i=–∞

ai i+n

j=i+

Y_xj()

≤x– ∞

i=–∞

|ai| i+n

j=i+

E|Yj|I

|Yj| ≤x

+xP|Y|>x

If / <α≤, noteαp> , this meansp> . ByE|Y|p_l(|Y|/α_{) <∞andlis slowly varying at}

infinity, for any  <<p– /α, we haveE|Y|p–_{<∞. Then notingEY}

i= , by Lemma .

we have x– E ∞

i=–∞

ai i+n

j=i+

Y_xj() =x–

E

∞

i=–∞

ai i+n

j=i+

Y_xj()

≤Cx– ∞

i=–∞

|ai| i+n

j=i+

E|Yj|I

|Yj|>x

≤Cx–nE|Y|I|Y|>x

≤Cx/α–E|Y|I|Y|>x≤CE|Y|/αI|Y|>x

≤E|Y|p–I|Y|>x→, asx→ ∞.

Hence forx>nα_{large enough, we get}

x– E ∞

i=–∞

ai i+n

j=i+

Y_xj() <ε/. Therefore ∞ n=

f(n)E max

≤k≤n

k j= Xj –εnα

+

≤

∞

n=

f(n)

∞

εnα

P max

≤k≤n

k j= Xj ≥x dx ≤C ∞ n=

f(n)

∞

nα

P max

≤k≤n

k j= Xj ≥εx

dx ≤C ∞ n=

f(n)

∞

nα P

max

≤k≤n

∞

i=–∞ ai

i+k

j=i+

Y_xj() ≥εx/

dx +C ∞ n=

f(n)

∞

nα P

max

≤k≤n

∞

i=–∞ ai

i+k

j=i+

Y_xj()–EY_xj() ≥εx/

dx

=:I+I. (.)

Firstly we show I < ∞. Noting |Yxj()| < |Yj|I{|Yj| > x}, by Markov’s inequality and Lemma ., we have

I≤C

∞

n=

f(n)

∞

nα x

–_Emax ≤k≤n

∞

i=–∞

ai i+k

j=i+

Y_xj() dx ≤C ∞ n=

f(n)

∞

nα

x– ∞

i=–∞

|ai| i+n

j=i+

EY_xj()dx

≤C ∞

n=

nf(n)

∞

nα

=C ∞

n=

nf(n)

∞

m=n

(m+)α

mα x

–_E|Y|I|Y|>xdx

≤C ∞

n=

nf(n)

∞

m=n

m–E|Y|I|Y|>mα

=C

∞

m=

m–E|Y|I|Y|>mα

m

n=

nαp––αl(n).

Ifp> , thenαp–  –α> –, and, by Lemma ., we obtain

I≤C

∞

m=

mαp––αl(m)E|Y|I|Y|>mα

=C

∞

m=

mαp––αl(m)

∞

k=m

E|Y|Ikα<|Y| ≤(k+ )α

=C

∞

k=

E|Y|Ikα_{<|Y| ≤(k+ )}α

k

m=

mαp––α_l(m)

≤C ∞

k=

kαp–αl(k)E|Y|Ikα<|Y| ≤(k+ )α

≤CE|Y|pl|Y|/α_<∞.

Ifp= , notice thatE|Y|+δ_{<∞impliesE|Y|}+δ_l(|Y|/α_{) <∞for any  <δ<δ, then by}

Lemma ., we obtain

I≤C

∞

m=

m–E|Y|I|Y|>mα

m

n=

n–l(n)

≤C ∞

m=

m–E|Y|I|Y|>mα

m

n=

n–+αδ_l(n)

≤C ∞

m=

mαδ–l(m)E|Y|I|Y|>mα

≤CE|Y|+δl|Y|/α≤CE|Y|+δ<∞.

So, we get

I<∞. (.)

Next we showI<∞. By Markov’s inequality, the Hölder inequality, and Lemma ., we

conclude

I≤C

∞

n=

f(n)

∞

nα x

–r_Emax

≤k≤n

∞

i=–∞ ai

i+k

j=i+

Y_xj()–EY_xj()

r

dx

≤C ∞

n=

f(n)

∞

nα x

–r_E

_∞

i=–∞

|ai| r–

r

|ai|/r max

≤k≤n

i+k

j=i+

Y_xj()–EY_xj()

r

≤C ∞

n=

f(n)

∞

nα

x–r _∞

i=–∞

|ai|

r–_∞

i=–∞

|ai|Emax

≤k≤n

i+k

j=i+

Y_xj()–EY_xj()

r

dx

≤C ∞

n=

f(n)

∞

nα

x–r ∞

i=–∞

|ai| i+n

j=i+

EY_xj()–EY_xj()rdx

+C

∞

n=

f(n)

∞

nα x

–r

∞

i=–∞

|ai|

_i+n

j=i+

EY_xj()–EY_xj() r/

dx

=:I+I, (.)

wherer≥ will be specialized later.

ForI, ifp> , taker>max{,p}, then byCrinequality, Lemma ., and Lemma ., we

get

I≤C

∞

n=

f(n)

∞

nα

x–r ∞

i=–∞

|ai| i+n

j=i+

E|Yj|rI

|Yj| ≤x

+xrP|Yj|>x

dx

≤C ∞

n=

nf(n)

∞

nα

x–rE|Y|rI|Y| ≤x+xrP|Y|>xdx

≤C ∞

n=

nf(n)

∞

m=n

(m+)α

mα

x–rE|Y|rI|Y| ≤x+P|Y|>xdx

≤C ∞

n=

nf(n)

∞

m=n

mα(–r)–E|Y|rI|Y| ≤(m+ )α+mα–P|Y|>mα

=C

∞

m=

mα(–r)–_E|Y|r_{I|Y| ≤(m+ )}α_+mα–_P|Y|>mα

m

n=

nf(n)

≤C ∞

m=

mα(p–r)–l(m) m

k=

E|Y|rIkα<|Y| ≤(k+ )α

+C

∞

m=

mαp–l(m)

∞

k=m

EIkα<|Y| ≤(k+ )α

=C

∞

k=

E|Y|rIkα<|Y| ≤(k+ )α

∞

m=k

mα(p–r)–l(m)

+C

∞

k=

EIkα<|Y| ≤(k+ )α k

m=

mαp–l(m)

≤C ∞

k=

kα(p–r)l(k)E|Y|p|Y|r–pIkα<|Y| ≤(k+ )α

+C

∞

k=

kαpl(k)E|Y|p|Y|–pIkα<|Y| ≤(k+ )α

ForI, ifp= , taker>max{ +δ, }, where  <δ<δ, then by the same argument as above

we have

I≤C

∞

m=

mα(–r)–E|Y|rI|Y| ≤(m+ )α+mα–P|Y|>mα

m

n=

nf(n)

≤C ∞

m=

mα(–r)–_E|Y|r_{I|Y| ≤(m+ )}α_+mα–_P|Y|>mα

m

n=

n–+αδ_l(n)

≤C ∞

m=

mα(–r+δ)–l(m)E|Y|rI|Y| ≤(m+ )α

+mα(+δ)–l(m)EI|Y|>mα

≤CE|Y|+δ_l|Y|/α_≤CE|Y|+δ_{<∞. (.)}

ForI, if ≤p< , taker> , noteαp+r/ –αpr/ –  = (αp– )( –r/) < , by theCr inequality, Lemma ., and Lemma ., we obtain

I≤C

∞

n=

nr/f(n)

∞

nα

x–rE|Y|I|Y| ≤xr/+xrPr/|Y|>xdx

≤C ∞

n=

nr/f(n)

∞

m=n

(m+)α

mα

x–rE|Y|I|Y| ≤xr/+Pr/|Y|>xdx

≤C ∞

n=

nr/f(n)

∞

m=n

mα(–r)–E|Y|I|Y| ≤(m+ )αr/+mα–Pr/|Y|>mα

=C

∞

m=

mα(–r)–_E|Y|_{I|Y| ≤(m+ )}αr/_+mα–_Pr/_|Y|>mα

m

n=

nr/f(n)

≤C ∞

m=

mα(p–r)+r/–l(m)E|Y|p|Y|–pI|Y| ≤(m+ )αr/

+C

∞

m=

mαp+r/–l(m)E|Y|p|Y|–pI|Y|>mαr/

≤C ∞

m=

mαp+r/–αpr/–_l(m)E|Y|pr/_{<∞. (.)}

ForI, ifp≥, taker> (αp– )/(α– /) > ; we haveα(p–r) +r/ –  < –, and therefore

one gets

I≤C

∞

m=

mα(–r)–E|Y|I|Y| ≤(m+ )αr/+mα–Pr/|Y|>mα

m

n=

nr/f(n)

≤C ∞

m=

mα(p–r)+r/–l(m)E|Y|I|Y| ≤(m+ )αr/

+C

∞

m=

≤C ∞

m=

mα(p–r)+r/–l(m)E|Y|r/<∞. (.)

Thus, (.) can be deduced by combining (.)-(.). Now, we show (.). By Lemma . and (.) we have

∞

n=

nαp–l(n)E sup

k≥n

k–α

k j= Xj –ε + = ∞ n=

nαp–l(n)

_∞



P sup

k≥n

k–α

k j= Xj >ε+t

dt = ∞ i= i_–

n=i–

nαp–l(n)

∞



P sup

k≥n

k–α

k j= Xj >ε+t

dt ≤C ∞ i= ∞  P sup

k≥i– k–α

k j= Xj >ε+t

dt

i–

n=i–

nαp–_l(n)

≤C ∞

i=

i(αp–)_li ∞



P sup

k≥i– k–α

k j= Xj >ε+t

dt ≤C ∞ i=

i(αp–)li

∞

l=i

∞



P max

l–_≤k<l

k–α

k j= Xj >ε+t

dt ≤C ∞ l= ∞  P max

l–_≤k<l

k–α

k j= Xj >ε+t

dt

l

i=

i(αp–)li

≤C ∞

l=

l(αp–)ll

∞



P max

l–_≤k<l

k j= Xj

> (ε+t)(l–)α dt ≤C ∞ l=

l(αp––α)_ll ∞



P max

≤k<l

k j= Xj

>ε(l–)α+y dy ≤C ∞ n=

nαp––α_l(n)

∞



P max

≤k<n

k j= Xj

>εnα–α+y dy =C ∞ n=

nαp––αl(n)E max

≤k<n

k j= Xj –εnα

+

<∞.

Hence the proof of Theorem . is completed.

The next theorem treats the caseαp= .

Theorem . Let l be a function slowly varying at infinity,  ≤p< . Assume that ∞

i=–∞|ai|θ<∞,whereθ belong to(, )if p= andθ= if <p< .Suppose that{Xn=

∞

i=–∞aiYi+n,n≥}is a moving average process generated by a sequence{Yi, –∞<i<∞}

If EYi= and E|Y|pl(|Y|p) <∞,then for anyε> 

∞

n=

n––/pl(n)E max

≤k≤n

k

j=

Xj

–εn/p

+

<∞. (.)

Proof Letg(n) =n––/p_{l(n). Similarly to the proof of (.), we have}

∞

n=

g(n)E max

≤k≤n

k

j=

Xj

–εn/p

+

≤C ∞

n=

g(n)

∞

n/pP

max

≤k≤n

∞

i=–∞ ai

i+k

j=i+

Y_xj() ≥εx/

dx

+C

∞

n=

g(n)

∞

n/pP

max

≤k≤n

∞

i=–∞ ai

i+k

j=i+

Y_xj()–EY_xj() ≥εx/

dx

=:J+J. (.)

ForJ, by Markov’s inequality, theCrinequality, Lemma ., and Lemma ., one gets

J≤C

∞

n=

g(n)

∞

n/px

–θ Emax

≤k≤n

∞

i=–∞ ai

i+k

j=i+

Y_xj()

θ

dx

≤C ∞

n=

ng(n)

∞

n/px

–θ

E|Y|θI|Y|>xdx

=C

∞

n=

ng(n)

∞

m=n

(m+)/p

m/p x

–θ

E|Y|θI|Y|>xdx

≤C ∞

n=

ng(n)

∞

m=n

m(–θ)/p–E|Y|θI|Y|>m/p

=C

∞

m=

m(–θ)/p–_E|Y|θ_I|Y|>m/p m

n=

ng(n)

≤C ∞

m=

m–θ/pl(m)E|Y|θI|Y|>m/p

=C

∞

m=

m–θ/pl(m)

∞

k=m

E|Y|θIk/p<|Y|< (k+ )/p

=C

∞

k=

E|Y|θ_Ik/p_{<|Y|< (k+ )}/p k

m=

m–θ/p_l(m)

≤C ∞

k=

k–θ/p_l(k)E|Y|θ_Ik/p_{<|Y|< (k+ )}/p

ForJ, similar to the proof ofI, taker= , by Lemma ., Lemma ., and Lemma ., we

conclude

J≤C

∞

n=

g(n)

_∞

n/px

–_E|max ≤k≤n

∞

i=–∞ ai

i+k

j=i+

Y_xj()–EY_xj()



dx

≤C ∞

n=

ng(n)

∞

n/px

–_E|Y|_{I|Y| ≤x+x}_P|Y|>xdx

=C

∞

n=

ng(n)

∞

m=n

(m+)/p

m/p x

–_E|Y|_{I|Y| ≤x+x}_P|Y|>xdx

≤C ∞

n=

ng(n)

∞

m=n

m––/pE|Y|I|Y| ≤(m+ )/p+m/p–P|Y|>m/p

=C

∞

m=

m––/pE|Y|I|Y| ≤(m+ )/p+m/p–P|Y|>m/p

m

n=

ng(n)

≤C ∞

m=

m–/pl(m)E|Y|I|Y| ≤(m+ )/p+l(m)P|Y|>m/p

≤CE|Y|pl|Y|p<∞. (.)

Hence from (.)-(.), (.) holds.

For the complete convergence and strong law of large numbers, we have the following corollary from the above theorems immediately.

Corollary . Under the assumptions of Theorem.,for anyε> we have

∞

n=

nαp–_{l(n)P max} ≤k≤n

k

j=

Xj

>εnα

<∞. (.)

Under the assumptions of Theorem.,for anyε> we have

∞

n=

n–l(n)P max

≤k≤n

k

j=

Xj

>εn/p

<∞; (.)

in particular,the assumptions EYi= and E|Y|p<∞imply the following

Marcinkiewicz-Zygmund strong law of large numbers:

lim

n→∞



n/p n

j=

Xj=  a.s. (.)

Remark . Corollary . provides complete convergence for the maximum of partial

sums, which extends the corresponding results of Budsabaet al.[, ] and Theorem 

Theorem . improves upon the above Theorem A from Li and Zhang [] with less

re-strictions, and our results also extend and generalize the above Theorem B from Chenet

al.[] withq=  partly.

Remark . Obviously, the assumption that{Yi, –∞<i<∞}is stochastically dominated

by a random variableY is weaker than the assumption of identical distribution of the

random variables{Yi, –∞<i<∞}, therefore the above results also hold for identically distributed random variables.

Remark . Leta= ,ai= ,i= , thenSn=

n k=Xk=

n

k=Yk. Hence the above results

hold when{Xk,k≥}is a sequence ofρ–-mixing random variables which is stochastically

dominated by a random variableY.

Competing interests

The author declares to have no competing interests.

Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant No. 11101180) and the Science and Technology Development Program of Jilin Province (Grant No. 20130522096JH).

Received: 17 April 2015 Accepted: 23 July 2015

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