Complete convergence of randomly weighted END sequences and its application

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R E S E A R C H

Open Access

Complete convergence of randomly

weighted END sequences and its application

Penghua Li

1

_{, Xiaoqin Li}

2*

_{and Kehan Wu}

3

*_{Correspondence:}

[email protected]

2_{School of Mathematical Sciences,}

Anhui University, Hefei, 230601, China

Full list of author information is available at the end of the article

Abstract

We investigate the complete convergence of partial sums of randomly weighted extended negatively dependent (END) random variables. Some results of complete moment convergence, complete convergence and the strong law of large numbers for this dependent structure are obtained. As an application, we study the

convergence of the state observers of linear-time-invariant systems. Our results extend the corresponding earlier ones.

MSC: 60E15; 60F15

Keywords: complete convergence; randomly weighted; END sequences; strong law of large numbers

1 Introduction

Let us recall the concept of extended negatively dependent (END) random variables which was introduced by Liu [].

Deﬁnition . We call random variables{Xn,n≥}to be END if there exists a positive constantMsuch that both

P(Xi>xi,i= , , . . . ,n)≤M n

i=

P(Xi>xi)

and

P(Xi≤xi,i= , , . . . ,n)≤M n

i=

P(Xi≤xi)

hold for eachn≥ and all real numbersx,x, . . . ,xn.

Obviously, for all ≤i≤n, letxi= –∞orxi= +∞in Definition ., it is easy to see that the dominating coefficientM≥. If the dominating coefficientMis , then END random variables reduce to NOD random variables which contain NA random variables and NSD random variables (see Joag-Dev and Proschan [], Hu [] and Wanget al.[]). Various examples of NA random variables and related fields can be found in Bulinski and Shaskin [], Prakasa Rao [], Oliveira [] and the references therein. In view of the importance of

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END random variables, many researchers pay attention to the study of END. For example, Liu [, ] studied the precise large deviations and moderate deviations of END sequence with heavy tails; Chenet al.[] obtained strong law of large numbers of END sequence and gave some applications to the risk theory and renewal theory; Shen [] obtained some moment inequalities of END sequence; Wanget al.[] and Huet al.[] investigated the complete convergence for END sequences; Wanget al.[] and Yanget al.[] investigated the nonparametric regression model under END errors; Yanget al.[] obtained some large deviation results of nonlinear regression models under END errors; Wanget al.[] established some exponential inequality for m-END sequence; Denget al.[] studied the Hajek-Renyi-type inequality and strong law of large numbers for END sequences, etc. Furthermore, there are many works on the negatively dependent random variables. For example, Wanget al.[] studied the complete convergence for WOD random variables and gave the application to the estimator of nonparametric regression models; Wuet al.

[] obtained some results of complete convergence and complete moment convergence for weighted sums ofm-NA random variables; Yang and Hu [] investigated the complete moment convergence of pairwise NQD random variables; Shenet al.[] investigated the strong law of large numbers for NOD random variables; Liet al.[] obtained some results of inverse moment for WOD random variables, etc.

In addition, there are many researchers paying attention to the study of the properties of partial sums of randomly weighted random variables. For example, Thanh and Yin [] established the complete convergence of partial sums of randomly weighted independent sequences in Banach spaces; Thanhet al.[] investigated complete convergence of par-tial sums of randomly weightedρ˜-mixing sequences; Cabreraet al.[] and Shenet al.

[] investigated the conditional convergence for partial sums of randomly weighted de-pendent random variables; Yanget al.[] and Yao and Lin [] obtained the complete convergence and moment of maximum normed based on the randomly weighted mar-tingale diﬀerences; Han and Xiang [] obtained the complete moment convergence of double-indexed randomly weighted sums ofρ˜-mixing sequences; Liet al.[] studied the convergence of partial sums of randomly weighted pairwise NQD sequences, etc.

We aim to investigate the complete convergence of partial sums of randomly weighted END sequences. Some results of complete moment convergence, complete convergence and strong law of large numbers for this dependent structure are obtained. As an appli-cation, we study the convergence of state observers of linear-time-invariant systems. We extend some results of Thanhet al.[], Wanget al.[] and Yanget al.[] to the case of randomly weighted END sequences. For the details, see our results in Sections  and , and the conclusions in Section . Some lemmas and proofs of main results are presented in Sections  and , respectively.

2 Some lemmas

Lemma .(Liu []) Let the random variables{Xn,n≥}be a sequence of END random

variables.If{fn,n≥}is a sequence of all nondecreasing(or nonincreasing)functions,then

{fn(Xn),n≥}is also a sequence of END random variables.

Remark . Let{Xn,n≥}be an END sequence and{Yn,n≥}be a sequence of non-negative and independent random variables, which is independent of {Xn,n≥}. Let

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of{Yn}, we establish, for all real numbersz, . . . ,zn,

P(Z≤z, . . . ,Zn≤zn) =P(XY≤z, . . . ,XnYn≤zn)

=

· · ·

P(Xu≤z, . . . ,Xnun≤zn)dFY(u)· · ·dFYn(un)

≤M

· · · n

i=

P(Xiui≤zi)dFY(u)· · ·dFYn(un)

=M

n

i=

P(XiYi≤zi) =M n

i=

P(Zi≤zi),

by using the fact that uX,uX, . . . ,unXn are END random variables following from Lemma .. Similarly, for all real numbersz, . . . ,zn, one has

P(Z>z, . . . ,Zn>zn) =

· · · P(Xu>z, . . . ,Xnun>zn)dFY(u)· · ·dFYn(un)

≤M

· · · n

i=

P(Xiui>zi)dFY(u)· · ·dFYn(un)

=M

n

i=

P(XiYi>zi) =M n

i=

P(Zi>zi).

Therefore, it can be found that{Zn,n≥}is also an END sequence with the same domi-nating coeﬃcientM.

Lemma .(Shen []) Let p≥and{Xn,n≥}be an END sequence such that EXn= 

and E|Xn|p<∞for all n≥.Then there exists a positive constant Cpsuch that for all n≥

E

n

i=

Xi

p

≤Cp

_n

i=

E|Xi|p+

_n

i=

EX_i p/

.

Lemma .(Sung []) Let{Xn,n≥}and{Yn,n≥}be sequences of random variables.

Then,for any n≥,q> ,ε> and a> ,

E

n

i=

(Xi+Yi)

–εa

+

≤



εq + 

q– 



aq–E

n

i=

Xi

q

+E

n

i=

Yi

.

Lemma . (Adler and Rosalsky [] and Adler et al. []) Let {Xn,n≥} be a

se-quence of random variables which is stochastically dominated by a random variable X,

i.e.sup_n_≥_P(|Xn|>t)≤CP(|X|>t)for some positive constant Cand all t≥.Then,for

all n≥,α> andβ> ,the following two statements hold:

E|Xn|αI

|Xn| ≤β

≤C

E|X|α_I_|_X_{| ≤}_β₊_βα_P_|_X_|_>_β_,

E|Xn|αI

|Xn|>β

≤CE

|X|α_I_|_X_|_>_β_.

Consequently,one has E[|Xn|α]≤CE|X|αfor all n≥.Here C,C,Care some positive

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3 The complete convergence for partial sums of randomly weighted END sequences

In the following, we list two assumptions:

(A.) Let{Xn,n≥}be a mean zero sequence of END random variables stochastically

dominated by a random variablesX.

(A.) For everyn≥, let{Ani, ≤i≤n}be a sequence of independent random variables

satisfying that{Ani, ≤i≤n}is independent of{Xn,n≥}.

Theorem . Assume that(A.)and(A.)are satisﬁed.Letα> /,  <p< ,E|X|p_<_∞

andβ≥such that

n

i=

EA_ni=Onβ. (.)

Then,for everyε> ,

∞

n=

nαp––β–αE

n

i=

AniXi

–εnα

+

<∞. (.)

So one has

∞

n=

nαp––βP

n

i=

AniXi

>εnα

<∞. (.)

Theorem . Assume that(A.)and(A.)are satisﬁed.Letα> /,p≥,E|X|p_<_∞_and

β≥such that

n

i=

E|Ani|q=O

nβ _(.)

for some q> (αp– )/(α– ).Then we also obtain the results of(.)and(.).

For someβ≥ and / <α< ( +β)/, we takeαp=  +βin Theorem . and establish the following result.

Theorem . Suppose that(A.)and(A.)are fulﬁlled.Letβ≥, / <α< ( +β)/and E|X|(+β)/α_<_∞_._If

n

i=

E|Ani|q=O

nβ for some q> β

α– , (.)

then,for everyε> ,

∞

n=

n–αE

n

i=

AniXi

–εnα

+

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and

∞

n=

P

n

i=

AniXi

>εnα

<∞. (.)

Thus,by the Borel-Cantelli lemma and(.),the strong law of large numbers is as follows:



nα

n

i=

AniXi→, a.s. as n→ ∞. (.)

Letlogx=ln max(x,e). In addition, for the casep= , we have the following result.

Theorem . Suppose that(A.)and(A.)are fulﬁlled.Letα> and E[|X|log|X|] <∞.

If(.)holds,then,for everyε> ,

∞

n=

n––βE

n

i=

AniXi

–εnα

+

<∞. (.)

So one has

∞

n=

nα––βP

n

i=

AniXi

>εnα

<∞. (.)

Remark . Ifβ=  in (.) and (.), then the randomly weighted conditions are

n

i=

EA_ni=O(n) (.)

and

n

i=

E|Ani|q=O(n) for someq> (αp– )/(α– ), (.)

which are used in Thanhet al.[]. Under the randomly weighted conditions (.), (.) and other conditions, Thanhet al.[] obtained some complete convergence such as

∞

n=

nαp–_P

max ≤k≤n

k

i=

AniXi

>εnα

<∞

for partial sums of randomly weightedρ˜-mixing sequences. Yanget al.[] extended the results of Thanhet al.[] and obtained complete moment convergence such as

∞

n=

nαp––αE

max ≤k≤n

k

i=

aniXi

–εnα

+ <∞

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4 The application to linear-time-invariant systems

As an application of Theorem ., we study the convergence of the state observers of linear-time-invariant systems in this section.

Fort≥, we consider a multi-input-single-output (MISO) linear time invariant system as follows:

⎧ ⎨ ⎩ ˙

x(t) =Ax(t) +Bu(t),

y(t) =Cx(t), (.)

whereA∈Rm×m_,_B∈_Rm×m_,_C∈_R×m_{are known system matrices,}_u₍_t₎∈_Rm _{is the}

control input,x(t)∈Rm_{is the state and}_y₍_t₎∈_R_{is the system output. The initial state}_x_()

is unknown. For some limited observations ony(t), it is interesting to estimatex(t). In the setup, the outputy(t) is only measured at a sequence of sampling time instants{ti}with measured valuesγ(ti) and noiseddisuch that

y(ti) =γ(ti) +di, ≤i≤n.

We would like to estimate the statex(t) from information onu(t),{ti}and{γ(ti)}. LetG denote the transpose ofG. In order to proceed, we need the following assumption.

Assumption . The system (.) is observable,i.e., the observability matrix

W_o=C, (CA), . . . ,CAm–

has full rank.

The solution to system (.) can be checked:

x(t) =eA(t–t)_x₍_t_{) +} t

t

eA(t–τ)Bu(τ)dτ.

From{ti, ≤i≤n}, it follows that

γ(ti) +di=y(ti) =CeA(ti–tn)x(tn) +C

ti

tn

eA(ti–τ)_Bu₍_τ₎_dτ_.

Denote

v(ti,tn) =C

ti

tn

eA(ti–τ)_Bu₍_τ₎_dτ_.

So this leads to the observation

CeA(ti–tn)_x₍_t

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Deﬁne n= ⎡ ⎢ ⎢ ⎢ ⎢ ⎣

CeA(t–tn) .. .

CeA(tn––tn)

C ⎤ ⎥ ⎥ ⎥ ⎥

⎦, n= ⎡ ⎢ ⎢ ⎢ ⎢ ⎣

γ(t) .. .

γ(tn–)

γ(tn)

⎤ ⎥ ⎥ ⎥ ⎥

⎦, Vn= ⎡ ⎢ ⎢ ⎢ ⎢ ⎣

v(t,tn) .. .

v(tn–,tn)  ⎤ ⎥ ⎥ ⎥ ⎥

⎦, Dn= ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ d .. .

dn–

dn ⎤ ⎥ ⎥ ⎥ ⎥ ⎦.

Then we rewrite (.) as follows:

nx(tn) = n–Vn+Dn. (.)

Suppose thatnis full rank, which will be established later. Then the least-squares es-timator ofx(tn) is given by

ˆ x(tn) =

_nn

–

_n( n–Vn). (.)

Combining (.) with (.), the estimation error forx(tn) attnis presented as

e(tn) =xˆ(tn) –x(tn) =

_nn

–

_nDn=



nr

nn

– 

nr

nDn for some  <r< .

In order to obtain the convergence, one must consider a typical entry in _nr_nDn. By the Cayley-Hamilton theorem [], the matrix exponential can be expressed by a polynomial function ofAof order at mostm– :

eAt=α(t)I+· · ·+αm(t)A

m–_,

where the time functions αi(t) can be derived by the Lagrange-Hermite interpolation method []. So one has

CeA(ti–tn)₌_α_(_t

i–tn), . . . ,αm(ti–tn) ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ C CA .. .

CAm– ⎤ ⎥ ⎥ ⎥ ⎥ ⎦=ϕ

₍_t

i–tn)Wo,

whereϕ(ti–tn) = [α(ti–tn), . . . ,αm(ti–tn)] andWois the observability matrix.

Denote n= ⎡ ⎢ ⎢ ⎣

ϕ(t–tn) .. .

ϕ()

⎤ ⎥ ⎥ ⎦.

Then

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which reduces to



nr

nn=Wo 

nr

nnWo,



nr

nDn= 

nrW

onDn.

As a result, one has for anyr> 

e(tn) =



nr

nn

– 

nr

nDn=Wo–



nr

nn

– 

nr

nDn. (.)

By Assumption ., it can be found thatW–

 exists. The convergence analysis will be established by the suﬃciently conditions: _nrnDn→, a.s., and _nrnn≥λI, a.s., for someλ> . So we need the following PE condition which was used in Thanhet al.[] and Wanget al.[].

Assumption . For some / <r< ,

λ=inf

n≥σmin



nr

nn

> , a.s., (.)

whereσmin(H) is the small eigenvalue ofHfor a suitable symmetricH.

We focus on the convergence of partial sums of randomly weighted END random vari-ables such that



nr n

i=

Anidi (.)

for some / <r< . Since a typical entry of_nr_nDnis



nr n

i=

αj(ti–tn)di, (.)

the convergence analysis for e(tn) is a special case of (.). Note that when the sam-pling time sequence is a random process, so areαj(ti–tn) in (.), rendering a randomly weighted noise driven by END random variables. As an application of Theorem ., we obtain the following theorem.

Theorem . Let β ≥, / <r<  and Assumptions.and . hold. Suppose that {dn,n≥}is an END sequences stochastically dominated by a random variable d with

E|d|(+β)/r_<_∞_._{Suppose that}_,_{for some q}_> β

r–,one has

n

i=

Eαj(ti–tn) q

=Onβ_, _(.)

where≤j≤m.Then



nr

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Consequently,

e(tn)→, a.s. (.)

Remark . If{ϕ(ti–tn)}is uniformly bounded, a.s., then the condition (.) holds with

β =  for anyq. Wanget al.[] obtained (.) for constant weightedρ˜-mixing errors (see Theorem  of Wanget al.[]). Thanhet al.[] extended the result of Wanget al.

[] to randomly weightedρ˜-mixing errors (see Theorem . of Thanhet al.[]). Yanget al.[] obtained the result (.) for the case of constant weighted martingale diﬀerences (see Theorem  of Yanget al.[]). Generally, our Theorem . generalize the results of Thanhet al.[], Wanget al.[] and Yanget al.[] to the case of randomly weighted END errors.

5 Conclusions

In this paper, we investigate the complete convergence of partial sums of randomly weighted END random variables. Some results of complete moment convergence, com-plete convergence and strong law of large numbers for this dependent structure are pre-sented (see our Theorems .-.). As an application of Theorem ., we study the con-vergence of the state observers of linear-time-invariant systems and obtain the result of the strong law of large numbers for the systems (see our Theorem .). Therefore, we ex-tend some results of Thanhet al.[], Wanget al.[] and Yanget al.[] to the case of randomly weighted END sequences. Furthermore, END random variables contain NA random variables, NOD random variables and NSD random variables, so the results ob-tained in this paper hold true for these negatively dependent random variables.

6 The proofs of main results

In the proofs,C,C,C, . . . denote some positive constants not depending onn.

Proof of Theorem. SinceAniXi=Ani+Xi–A–niXi, without loss of generality, we assume

Ani≥ in the proof. Forn≥ and ≤i≤n, let

Xni= –nαI

Xi< –nα

+XiI

|Xi| ≤nα

+nαIXi>nα

,

˜ Xni=nαI

Xi< –nα

+XiI

|Xi|>nα

–nαIXi>nα

.

It can be found that

AniXi=

AniXni–E(AniXni)

+E(AniXni) +AniX˜ni, ≤i≤n.

Therefore, by Lemma . witha=nα_and_q_{= , we obtain}

∞

n=

n

i=

AniXi

–εnα

+

≤C

∞

n=

nαp––β–αE

n

i=

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+

∞

n=

nαp––β–αE n i=

AniX˜ni

+ ∞ n=

nαp––β–α n i=

E(AniXni)

:=H+H+H. (.)

Combining (.) with Hölder’s inequality, one has

n

i=

E|Ani| ≤

_n

i=

EA_ni / _n

i= 

/

≤Cn β+

 ≤_C__nβ_, _(.)

by using the factβ≥.

Since, for everyn≥,{Ani, ≤i≤n}is independent of the sequence{Xn,n≥}, one has by Markov’s inequality, Lemma ., (.) andE|X|p<∞(p> )

H≤

∞

n=

nαp––β–α

n

i=

E|Ani|E|Xi|I

|Xi|>nα

≤C

∞

n=

nαp––αE|X||I|X|>nα

=C

∞

n=

nαp––α

∞

m=n

E|X||Im<|X|/α_≤_m_{+ }

=C

∞

m=

E|X||Im<|X|/α≤m+  m

n=

nα(p–)–

≤C

∞

m=

mαp–αE|X|Im<|X|/α≤m+ ]≤CE|X|p<∞. (.)

In addition, it can be seen thatE(AniXi) =EAniEXi= , ≤i≤n,n≥. So, by the proof of (.), we have

H =

∞

n=

nαp––β–α n i=

–nαEAniI

Xi< –nα

–EAniXiI

|Xi|>nα

· · ·

+· · ·nα_EA

niXiI

|Xi|>nα

≤C

∞

n=

nαp––β–α

n

i=

E|Ani|E

|Xi|I

|Xi|>nα

≤CE|X|p<∞. (.)

In view of Lemma ., one sees that{Xni, ≤i≤n}are END random variables. Combining the assumption of{Ani}with Remark ., we establish that{[AniXni–E(AniXni)], ≤i≤

n} are mean zero END random variables with the same dominating coeﬃcient. So, by Markov’s inequality, (.), Lemma . withp=  and Lemma ., we get

H =C

∞

n=

nαp––β–α_E

n

i=



≤C

∞

n=

nαp––β–α

n

i=

(11)

≤C

∞

n=

nαp––αEXI|X| ≤nα+C

∞

n=

nαp–EI|X|>nα

:=CH+CH. (.)

Sincep<  andEXp_<_∞_{, it can be checked that}

H=

∞

n=

nαp––α

n

i=

EXI(i– )α<|X| ≤iα

=

∞

i=

EXI(i– )α<|X| ≤iα

∞

n=i

nαp––α

≤C

∞

i=

E|X|pX–pI(i– )α_<_|_X_{| ≤}_iα_]_iαp–α_≤_C

E|X|p<∞. (.)

By the proof of (.), it follows that

H≤

∞

n=

nαp––αE|X|I|X|>nα≤CE|X|p<∞. (.)

Combining (.) with (.)-(.), we can get (.) immediately. Moreover, by (.) and Remark . of Sung [], for everyε> , it can be argued that

∞>

∞

n=

n

i=

AniXi

–εnα

+

≥

∞

n=

nαp––β εnα



P

n

i=

AniXi

–εnα>t

dt

≥ε

∞

n=

nαp––βP

n

i=

AniXi

> εnα

, (.)

which implies (.).

Proof of Theorem. We use the same notation in the proof of Theorem .. Obviously, byp≥, it is easy to see thatq> (αp– )/(α– )≥. Consequently, for any ≤r≤, by Hölder’s inequality and condition (.), one has

n

i=

E|Ani|r≤

_n

i=

E|Ani|q

r/q_n

i= 

–r/q

≤Cnβ r

q+–qr_. _(.)

(12)

follows that

H=C

∞

n=

nαp––β–qαE

n

i=

q

≤C

∞

n=

nαp––β–qα _n

i=

EAniXni–E(AniXni)



q/

+C

∞

n=

nαp––β–qα

n

i=

EAniXni–E(AniXni) q

:=CH+CH. (.)

Obviously, for ≤i≤n, by Lemma ., it follows that

EAniXni–E(AniXni)



≤CEA_niEX_ni

≤CEA_niEXI|X| ≤nα₊_nα_P_|_X_|_>_nα

≤CEA_niEXI|X| ≤nα+EXI|X|>nα=CEA_niEX. (.)

Byp≥ andE|X|p_<_∞_{, one concludes that}_EX_<_∞_{. Thus, we take (.) with}_r_{=  and} (.), and establish

H≤C

∞

n=

nαp––β–qα _n

i=

EA_niEX q/

≤C

∞

n=

nαp––β–qα+(β/q+–/q)q/

=C

∞

n=

nαp–qα+q/–<∞, (.)

by the factq> (αp– )/(α– ). In addition, by theCrinequality, Lemma . and (.),

H≤C

∞

n=

nαp––β–qα

n

i=

E|Ani|qE|Xni|q

≤C

∞

n=

nαp––qαE|X|qI|X| ≤nα+C

∞

n=

nαp–P|X|>nα

:=CH∗ +CH∗∗. (.)

Byp≥ andα> /, one has (αp– )/(α– ) –p≥, which impliesq>p. So, byE|X|p<

∞, it can be argued that

H_∗ =

∞

n=

nαp––qα

n

i=

E|X|q_I₍_i_{– )}α_<_|_X_{| ≤}_iα

=

∞

i=

E|X|qI(i– )α<|X| ≤iα

∞

n=i

(13)

≤C

∞

i=

E|X|p|X|q–pI(i– )α<|X| ≤iαiαp–qα

≤CE|X|p<∞. (.)

By the proof of (.), one has

H_∗∗≤

∞

n=

nαp––α_E_|_X_|_I_|_X_|_>_nα_≤_CE_|_X_|p_<_∞_. _(.)

Consequently, by (.) and (.)-(.), we obtainH<∞. So, we obtain the result (.).

Finally, by the proof of (.), (.) also holds true.

Proof of Theorem. For someβ≥ andα> ( +β)/, we takep= ( +β)/α and have

αp=  +β. Applying Theorem ., we obtain (.) and (.) immediately. Combining (.) with the Borel-Cantelli lemma, we establish the result of (.).

Proof of Theorem. Similar to the proof of Theorem ., by Lemma ., we can check that

∞

n=

n––β_E

n

i=

AniXi

–εnα

+

≤C

∞

n=

n––β–αE

n

i=



+

∞

n=

n––β_E

n

i=

AniX˜ni

+

∞

n=

n––β

n

i=

E(AniXni)

:=Q+Q+Q. (.)

By the proof of (.), it follows that

Q≤

∞

n=

n–E|X|I|X|>nα= 

∞

n=

n–

∞

m=n

E|X|Im<|X|/α≤m+ 

= 

∞

m=

E|X|Im<|X|/α_≤_m_{+ }

m

n=

n–

≤C

∞

m=

logmE|X|Im<|X|/α≤m+ ]≤CE

|X|log|X|<∞. (.)

Similarly, by the proof of (.), one has

Q≤C

∞

n=

n–E|X|I|X|>nα_≤_C

E

(14)

Moreover, by the proof of (.), we obtain

Q≤C

∞

n=

n––β–α

n

i=

E(AniXni)=C

∞

n=

n––β–α

n

i=

EA_niEX_ni

≤C

∞

n=

n––αEXI|X| ≤nα+C

∞

n=

nα–P|X|>nα

≤C

∞

i=

EXI(i– )α_<_{|X| ≤} iα

∞

n=i

n––α+CE

|X|log|X|

≤C

∞

i=

EXI(i– )α<|X| ≤iαi–α+CE

|X|log|X|

≤CE|X|+CE

|X|log|X|<∞. (.)

So, by (.)-(.), (.) holds. In addition, by (.) withp= , (.) also holds true under

the conditions of Theorem ..

Proof of Theorem. It is easy to check that



nr

nDn=

⎡ ⎢ ⎢ ⎣

 nr

n

i=α(ti–tn)di ..

. 

nr

n

i=αm(ti–tn)di ⎤ ⎥ ⎥ ⎦.

In order to prove (.), it suﬃces to look at the jth component _nr

n

i=αj(ti–tn)di of 

nr_nDn. For someq> __rβ_–, by (.), we apply Theorem . withα=r,Ani=αj(ti–tn) in (.) andXn=dn, and obtain the result of (.).

Moreover, by Assumption ., W–

 exists. In addition, by (.) in Assumption ., (

nr_nn)–exists and

σmax



nr

nn

–

≤  λ, a.s.,

whereσmax(·) is the largest eigenvalue. Therefore, combining

e(tn) =Wo–



nr

nn

– 

nr

nDn

with (.), we obtain (.) immediately.

Acknowledgements

The authors are deeply grateful to the editors and anonymous referees, whose insightful comments and suggestions have contributed substantially to the improvement of this paper. This work is supported by National Natural Science Foundation of China (11501005, 61403053, 61403115), Natural Science Foundation of Anhui Province (1508085J06, 1608085QA02) and Science Research Project of Anhui Colleges (KJ2017A027, KJ2014A020, KJ2015A065, KJ2016A027).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

(15)

Author details

1_{Automotive Electronics Engineering Research Center, College of Automation, Chongqing University of Posts and}

Telecommunications, Chongqing, 400065, China.2_{School of Mathematical Sciences, Anhui University, Hefei, 230601,}

China.3_{Training Center of Anhui Provincial Electric Power Company, Hefei, 230000, China.}

Publisher’s Note

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

Received: 5 February 2017 Accepted: 19 July 2017

References

1. Liu, L: Precise large deviations for dependent random variables with heavy tails. Stat. Probab. Lett.79, 1290-1298 (2009)

2. Joag-Dev, K, Proschan, F: Negative association of random variables with applications. Ann. Stat.11, 286-295 (1983) 3. Hu, TZ: Negatively superadditive dependence of random variables with applications. Chinese J. Appl. Probab. Statist.

16, 133-144 (2000)

4. Wang, XJ, Shen, AT, Chen, ZY, Hu, SH: Complete convergence for weighted sums of NSD random variables and its application in the EV regression model. Test24, 166-184 (2015)

5. Bulinski, AV, Shaskin, A: Limit Theorems for Associated Random Fields and Related Systems. World Scientiﬁc, Singapore (2007)

6. Prakasa Rao, BLS: Associated Sequences, Demimartingales and Nonparametric Inference. Springer, Basel (2012) 7. Oliveira, PE: Asymptotics for Associated Random Variables. Springer, Berlin (2012)

8. Liu, L: Necessary and suﬃcient conditions for moderate deviations of dependent random variables with heavy tails. Sci. China Ser. A53, 1421-1434 (2010)

9. Chen, YQ, Chen, AY, Ng, KW: The strong law of large numbers for extended negatively dependent random variables. J. Appl. Probab.47, 908-922 (2010)

10. Shen, AT: Probability inequalities for END sequence and their applications. J. Inequal. Appl.2011, 98 (2011) 11. Wang, XJ, Hu, T-C, Volodin, A, Hu, SH: Complete convergence for weighted sums and arrays of rowwise extended

negatively dependent random variables. Commun. Stat., Theory Methods42, 2391-2401 (2013)

12. Hu, T-C, Rosalsky, A, Wang, KL: Complete convergence theorems for extended negatively dependent random variables. Sankhya, Ser. A77, 1-29 (2015)

13. Wang, XJ, Zheng, LL, Xu, C, Hu, SH: Complete consistency for the estimator of nonparametric regression models based on extended negatively dependent errors. Statistics49, 396-407 (2015)

14. Yang, WZ, Xu, HY, Chen, L, Hu, SH: Complete consistency of estimators for regression models based on extended negatively dependent errors. Stat. Pap. (2016, in press). doi:10.1007/s00362-016-0771-x

15. Yang, WZ, Zhao, ZR, Wang, XH, Hu, SH: The large deviation results for the nonlinear regression model with dependent errors. Test26(2), 261-283 (2017)

16. Wang, XJ, Wu, Y, Hu, SH: Exponential probability inequality for m-END random variables and its applications. Metrika 79, 127-147 (2016)

17. Deng, X, Wang, XJ, Xia, FX: Hajek-Renyi-type inequality and strong law of large numbers for END sequences. Commun. Stat., Theory Methods46(2), 672-682 (2017)

18. Wang, XJ, Xu, C, Hu, T-C, Volodin, A: On complete convergence for widely orthant-dependent random variables and its applications in nonparametric regression models. Test23(3), 607-629 (2014)

19. Wu, YF, Hu, T-C: Complete convergence and complete moment convergence for weighted sums of m-NA random variables. J. Inequal. Appl.2015, 200 (2015)

20. Yang, WZ: Complete moment convergence of pairwise NQD random variables. Stochastics87(2), 199-208 (2015) 21. Shen, AT, Zhang, Y: On the rate of convergence in the strong law of large numbers for negatively orthant-dependent

random variables. Commun. Stat., Theory Methods45(21), 6209-6222 (2016)

22. Li, XQ, Liu, X, Yang, WZ: The inverse moment for widely orthant dependent random variables. J. Inequal. Appl.2016, 161 (2016)

23. Thanh, LV, Yin, G: Almost sure and complete convergence of randomly weighted sums of independent random elements in Banach spaces. Taiwan. J. Math.15, 1759-1781 (2011)

24. Thanh, LV, Yin, G, Wang, LY: State observers with random sampling times and convergence analysis of double-indexed and randomly weighted sums of mixing processes. SIAM J. Control Optim.49, 106-124 (2011) 25. Cabrera, MO, Rosalsky, A, Volodin, A: Some theorems on conditional mean convergence and conditional almost sure

convergence for randomly weighted sums of dependent random variables. Test21, 369-385 (2012)

26. Shen, AT, Wu, RC, Chen, Y, Zhou, Y: Conditional convergence for randomly weighted sums of random variables based on conditional residualh-integrability. J. Inequal. Appl.2013, 122 (2013)

27. Yang, WZ, Wang, YW, Wang, XH, Hu, SH: Complete moment convergence for randomly weighted sums of martingale diﬀerences. J. Inequal. Appl.2013, 396 (2013)

28. Yao, M, Lin, L: The moment of maximum normed randomly weighted sums of martingale diﬀerences. J. Inequal. Appl.2015, 264 (2015)

29. Han, J: Complete moment convergence of double-indexed randomly weighted sums of mixing sequences. J. Inequal. Appl.2016, 313 (2016)

30. Li, XQ, Zhao, ZR, Yang, WZ, Hu, SH: The inequalities of randomly weighted sums of pairwise NQD sequences and its application to limit theory. J. Math. Inequal.11, 323-334 (2017)

31. Wang, LY, Li, C, Yin, G, Guo, L, Xu, CZ: State observability and observers of linear-time-invariant systems under irregular sampling and sensor limitations. IEEE Trans. Autom. Control56, 2639-2654 (2011)

32. Yang, WZ, Wang, XH, Li, XQ, Hu, SH: The convergence of double-indexed weighted sums of martingale diﬀerences and its application. Abstr. Appl. Anal.2014, Article ID 893906 (2014)

(16)

34. Adler, A, Rosalsky, A: Some general strong laws for weighted sums of stochastically dominated random variables. Stoch. Anal. Appl.5, 1-16 (1987)

35. Adler, A, Rosalsky, A, Taylor, RL: Strong laws of large numbers for weighted sums of random elements in normed linear spaces. Int. J. Math. Math. Sci.12, 507-530 (1989)