Complete convergence for negatively orthant dependent random variables

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R E S E A R C H

Open Access

Complete convergence for negatively orthant

dependent random variables

Dehua Qiu

1*

_{, Qunying Wu}

2

_{and Pingyan Chen}

3

*_{Correspondence:}

[email protected] 1_{School of Mathematics and}

Statistics, Guangdong University of Finance and Economics, Guangzhou, 510320, P.R. China Full list of author information is available at the end of the article

Abstract

In this paper, necessary and suﬃcient conditions of the complete convergence are obtained for the maximum partial sums of negatively orthant dependent (NOD) random variables. The results extend and improve those in Kuczmaszewska (Acta Math. Hung. 128(1-2):116-130, 2010) for negatively associated (NA) random variables.

MSC: 60F15; 60G50

Keywords: NOD; complete convergence

1 Introduction

The concept of complete convergence for a sequence of random variables was introduced by Hsu and Robbins [] as follows. A sequence{Un,n≥}of random variablesconverges

completelyto the constantθ if

∞

n=

P|Un–θ|>ε

<∞ for allε> .

Moreover, they proved that the sequence of arithmetic means of independent identically distribution (i.i.d.) random variables converges completely to the expected value if the variance of the summands is ﬁnite. This result has been generalized and extended in sev-eral directions by many authors. One can refer to [–], and so forth. Kuczmaszewska [] proved the following result.

Theorem A Let{Xn,n≥}be a sequence of negatively associated(NA)random variables and X be a random variables possibly deﬁned on a diﬀerent space satisfying the condition



n

i=

P|Xi|>x=DP|X|>x

for all x> ,all n≥and some positive constant D.Letαp> andα> /.Moreover,

additionally assume that EXn= for all n≥if p≥.Then the following statements are

equivalent:

(i) E|X|p_<_∞_,

(ii) ∞_n_=nαp–_P₍_max ≤j≤n|

j

i=Xi| ≥εnα) <∞,∀ε> .

The aim of this paper is to extend and improve Theorem A to negatively orthant de-pendent (NOD) random variables. The tool in the proof of Theorem A is the Rosenthal

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maximal inequality for NA sequence (cf.[]), but no one established the kind of maximal inequality for NOD sequence. So the truncated method is diﬀerent and the proofs of our main results are more complicated and diﬃcult.

The concept of negatively associated (NA) and negatively orthant dependent (NOD) was introduced by Joag-Dev and Proschan [] in the following way.

Deﬁnition . A ﬁnite family of random variables{Xi, ≤i≤n}is said to be negatively associated (NA) if for every pair of disjoint nonempty subsetA,Aof{, , . . . ,n},

Covf(Xi,i∈A),f(Xj,j∈A)≤,

wherefandfare coordinatewise nondecreasing such that the covariance exists. An in-ﬁnite sequence of{Xn,n≥}is NA if every ﬁnite subfamily is NA.

Deﬁnition . A ﬁnite family of random variables{Xi, ≤i≤n}is said to be

(a) negatively upper orthant dependent (NUOD) if

P(Xi>xi,i= , , . . . ,n)≤ n

i=

P(Xi>xi)

for∀x,x, . . . ,xn∈R,

(b) negatively lower orthant dependent (NLOD) if

P(Xi≤xi,i= , , . . . ,n)≤ n

i=

P(Xi≤xi)

for∀x,x, . . . ,xn∈R,

(c) negatively orthant dependent (NOD) if they are both NUOD and NLOD.

A sequence of random variables{Xn,n≥}is said to be NOD if for eachn,X,X, . . . ,Xn are NOD.

Obviously, every sequence of independent random variables is NOD. Joag-Dev and Proschan [] pointed out that NA implies NOD, neither being NUOD nor being NLOD implies being NA. They gave an example that possesses NOD, but does not possess NA, which shows that NOD is strictly wider than NA. For more details of NOD random vari-ables, one can refer to [, , , , –], and so forth.

In order to prove our main results, we need the following lemmas.

Lemma .(Bozorgniaet al.[]) Let X,X, . . . ,Xnbe NOD random variables.

(i) Iff,f, . . . ,fnare Borel functions all of which are monotone increasing(or all

monotone decreasing),thenf(X),f(X), . . . ,fn(Xn)are NOD random variables. (ii) En_i_=X_i+≤n_i_=EX+_i,∀n≥.

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every n≥,then for all n≥,

E

n

i= Xi

q ≤C(q)

n

i=

E|Xi|q+

_n

i= EX_i

q/

.

Lemma . For any q≥,there is a positive constant C(q)depending only on q such that if{Xn,n≥}is a sequence of NOD random variables with EXn= for every n≥,then for

all n≥,

Emax

≤j≤n

j

i= Xi

q

≤C(q)log(n)q n

i=

E|Xi|q+

_n

i= EX_i

q/

.

Proof By Lemma ., the proof is similar to that of Theorem .. in Stout [], so it is

omitted here.

Lemma .(Kuczmaszewska []) Letβ,γ be positive constants.Suppose that{Xn,n≥} is a sequence of random variables and X is a random variable.There exists constant D> 

such that

n

i=

P|Xi|>x≤DnP|X|>x, ∀x> ,∀n≥; (.)

(i) ifE|X|β_<_∞_,_then 

n

j=E|Xj|

β_≤_CE_|_X_|β_;

(ii) _nn_j_=E|Xj|β_I_(|_X_{j| ≤}_γ₎_≤_C_{_E_|_X_|β_I_(|_X_{| ≤}_γ_{) +}_γβ_P_(|_X_|_>_γ_)}_; (iii) _nn_j_=E|Xj|β_I_(|_X_j|_>_γ₎_≤_CE_|_X_|β_I_(|_X_|_>_γ₎_.

Recall that a functionh(x) is said to be slowly varying at inﬁnity if it is real valued, posi-tive, and measurable on [,∞), and if for eachλ> 

lim x→∞

h(λx)

h(x) = .

We refer to Seneta [] for other equivalent deﬁnitions and for a detailed and comprehen-sive study of properties of slowly varying functions.

We frequently use the following properties of slowly varying functions (cf.Seneta []).

Lemma . If h(x)is a function slowly varying at inﬁnity,then for any s> 

Cn–sh(n)≤ ∞

i=n

i––sh(i)≤Cn–sh(n)

and

Cnsh(n)≤ n

i=

i–+sh(i)≤Cnsh(n),

where C,C,C,C> depend only on s.

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2 Main results and proofs

Theorem . Letα> /,p> ,αp> and h(x)be a slowly varying function at inﬁnity.Let

{Xn,n≥}be a sequence of NOD random variables and X be a random variables possibly deﬁned on a diﬀerent space satisfying the condition(.).Moreover,additionally assume that forα≤,EXn= for all n≥.If

E|X|ph|X|/α<∞, (.)

then the following statements hold:

(i) ∞

n=

nαp–h(n)P

max

≤j≤n|Sj| ≥εn α

<∞, ∀ε> ; (.)

(ii) ∞

n=

nαp–_h₍_n₎_P_max ≤k≤nS

(k)

n ≥εnα

<∞, ∀ε> ; (.)

(iii) ∞

n=

nαp–h(n)P

max

≤j≤n|Xj| ≥εn

α

<∞, ∀ε> ; (.)

(iv) ∞

n=

nαp–h(n)P

sup j≥nj

–α_|_S_{j| ≥}_ε_<_∞, _∀_ε_{> ;} _(.)

(v) ∞

n=

nαp–_h₍_n₎_P_sup

j≥nj

–α_|_X_{j| ≥}_ε_<_∞_, _∀_ε_{> .} _(.)

Here Sn=

n

i=Xi,S (k)

n =Sn–Xk,k= , , . . . ,n.

Proof First, we prove (.). Chooseqsuch that /αp<q< . LetX_i(n,)= –nαq_I₍_X

i< –nαq) +

XiI(|Xi| ≤nαq) +nαqI(Xi>nαq), Xi(n,)= (Xi–nαq)I(Xi>nαq), Xi(n,) = –(Xi+nαq)I(Xi< –nαq_),_∀_n_≥_{, }_≤_i_≤_n_{. Note that}

Xi=Xi(n,)+X

(n,)

i –X

(n,)

i

and

∞

n=

nαp–h(n)P

max

≤j≤n|Sj|>εn α

≤ ∞

n=

nαp–_h₍_n₎_P

max

≤j≤n

j

i= X_i(n,)

>εnα/

+ ∞

n=

nαp–h(n)P

_n

i=

X_i(n,)>εnα/

+ ∞

n=

nαp–h(n)P

_n

i=

X_i(n,)>εnα/

def

=I+I+I. (.)

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η)q>  andp–η–  >  ifp> . In order to proveI<∞, we ﬁrst prove that

lim n→∞n

–α_max ≤j≤n

j

i= EX_i(n,)

= . (.)

This holds whenα≤. Sinceαp> ,p> . ByEXi= ,i≥, and Lemma ., we have

n–αmax

≤j≤n

j

i= EX_i(n,)

≤ n–αmax≤j≤n

j

i=

E|Xi|I|Xi|>nαq+nαqP|Xi|>nαq

≤ n–α

n

i=

E|Xi|I|Xi|>nαq≤Cn–αE|X|I|X|>nαq

≤ Cn–{α(p–η)q–}–α(–q)E|X|p–η

→, n→ ∞.

Whenα> ,p> ,

n–αmax

≤j≤n

j

i= EX_i(n,)

≤ n–αmax≤j≤n j

i=

E|Xi|I|Xi| ≤nαq+nαqP|Xi|>nαq

≤ n–α

n

i=

E|Xi| ≤Cn–αE|X|

→, n→ ∞.

Whenα> ,p≤,

n–αmax

≤j≤n

j

i= EX_i(n,)

≤ n–αmax≤j≤n

j

i=

≤ n–α

n

i=

≤ n–α

n

i=

nα(–p+η)qE|Xi|p–η

≤ Cn–{α(p–η)q–}–α(–q)E|X|p–η

→, n→ ∞.

Therefore, (.) holds. So, in order to proveI<∞, it is enough to prove that

I_∗:= ∞

n=

nαp–h(n)P

max

≤j≤n

j

i=

X_i(n,)–EX_i(n,) >εnα/

<∞. (.)

By Lemma . for∀n≥,{X_i(n,)–EX(_in,), ≤i≤n}is a sequence of NOD random vari-ables. When  <p≤, byα(p–η) >  and  <q< , we have α– 

 –α( –

p–η  )q>

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 –α( –

p–η  )q),

p–(p–η)q

–q }, we get by the Markov inequality, theCrinequality, the Hölder inequality, and Lemma .,

I_∗≤C

∞

n=

nαp–αv–_h₍_n₎_log_(_n₎vn i=

EX_i(n,)v

+C

∞

n=

nαp–αv–h(n)log(n)v

_n

i=

EX(_in,) v/

def

= I_∗ +I_∗.

By theCrinequality, Lemma ., and Lemma ., we have

I_∗ ≤C

∞

n=

nαp–αv–h(n)log(n)v n

i=

E|Xi|vI|Xi| ≤nαq+nαqvP|Xi|>nαq

≤C

∞

n=

nαp–αv–_h₍_n₎_log_(_n₎v

E|X|vI|X| ≤nαq₊_nαqv_P_|_X_|_>_nαq

≤C

∞

n=

nα{–(–q)v+p–q(p–η)}–h(n)log(n)vE|X|p–η<∞.

By theCrinequality and Lemma .,

I_∗ ≤C

∞

n=

nαp–αv–_h₍_n₎_log_(_n₎v n i=

E|Xi|I|Xi| ≤nαq₊_nαq_P_|_X_i|_>_nαq

v/

≤C

∞

n=

nαp––(α–/)vh(n)log(n)vE|X|I|X| ≤nαq+nαqP|X|>nαqv/.

Whenp> ,

I_∗ ≤C

∞

n=

nαp––(α–/)v_h₍_n₎_log_(_n₎v

EXv/<∞.

When  <p≤,

I_∗ ≤C

∞

n=

nαp––(α–/)vh(n)log(n)vE|X|p–ηv/nαq{–(p–η)}v/

≤C

∞

n=

nαp––{α–_–α(–p–_η)q}v_h₍_n₎_log_(_n₎v <∞.

Therefore, (.) holds forI. DeﬁneY_i(n,)= (Xi–nαq)I(nαq<Xi≤nα+nαq) +nαI(Xi>

nα₊_nαq_{), }_≤_i_≤_n_,_n_≥_{, since}_X(n,)

i =Y

(n,)

i + (Xi–nαq–nα)I(Xi>nα+nαq), we have

I≤

∞

n=

nαp–h(n)P

_n

i=

Y_i(n,)>εnα/

+ ∞

n=

nαp–h(n)P

_n

i=

Xi–nαq–nα

IXi>nα+nαq

>εnα/

def

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By Lemma ., (.), and a standard computation, we have

I≤

∞

n=

nαp–h(n) n

i=

PXi>nα+nαq

≤ ∞

n=

nαp–h(n) n

i=

P|Xi|>nα

≤C

∞

n=

nαp–_h₍_n₎_P_|_X_|_>_nα_≤_C₊_CE_|_X_|p_h_|_X_|/α_<_∞_. _(.)

Now we proveI<∞. By (.) and Lemma ., we have

≤n–α

n

i= EY_i(n,)

≤

⎧ ⎨ ⎩

n–αn

i=EXiI(Xi>nαq), ifp> ,

n–αn

i={E|Xi|I(|Xi| ≤n

α_{) +}_nα_P₍_|_X_i|_{> }_nαq₎_}_, _{if  <}_p_≤_

≤

⎧ ⎨ ⎩

Cn–{α(p–η)q–}–α(–q)_E_|_X_|p–η_, _if_p_{> ,}

Cn–α(p–η)qE|X|p–η, if  <p≤→

, n→ ∞.

Therefore, in order to proveI<∞, it is enough to prove that

I_∗ ≤

∞

n=

nαp–h(n)P

_n

i=

Y_i(n,)–EY_i(n,)>εnα/

<∞. (.)

Takingvsuch thatv>max{,_α–/αp–,_α((α_p_–η)–p–)}, we get by Lemma ., the Markov inequality, theCrinequality, the Hölder inequality, and Lemma .,

I_∗ ≤ C

∞

n=

nαp–αv–h(n)E n i=

Y_i(n,)–EY_i(n,) v ≤ C ∞ n=

nαp–αv–h(n) n

i=

EY_i(n,)v+C

∞

n=

nαp–αv–h(n)

_n

i=

EY_i(n,) v/

def

= I_∗ +I_∗ .

By theCrinequality, Lemma ., Lemma ., (.), and a standard computation, we have

I_∗ =C

∞

n=

i=

EY_i(n,)v

≤C

∞

n=

nαp–αv–_h₍_n₎

n

i=

EX_ivInαq_<_X

i≤nαq+nα

+nαv_P_X

i>nαq+nα

≤C

∞

n=

i=

E|Xi|vI|Xi| ≤nα+nαvP|Xi|>nα

≤C

∞

n=

nαp–αv–_h₍_n₎_E_|_X_|v_I_|_X_{| ≤}__nα₊_nαv_P_|_X_|_>_nα

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and

I_∗ ≤C

∞

n=

i=

EX_iInαq<Xi≤nαq+nα

+nαPXi>nαq+nα

v/

≤C

∞

n=

nαp–αv+v/–_h₍_n₎_EX_I_|_X_{| ≤}__nα₊_nα_P_|_X_|_>_nαv/

≤

⎧ ⎨ ⎩

C∞_n_=nαp–(α–/)v–_h₍_n₎₍_EX₎v/_, _if_p_{> ,}

C∞_n_=nαp––{α(p–η)–}v/_h₍_n₎₍_E_|_X_|p–η₎v/_, _if_p_≤_

≤

⎧ ⎨ ⎩

C∞_n_=nαp–(α–/)v–_h₍_n_), _if_p_{> ,}

C∞_n_=nαp––{α(p–η)–}v/_h₍_n_), _if_p_≤_

<∞.

Therefore, (.) holds. By (.)-(.) we getI<∞. In a similar way ofI<∞we can obtainI<∞. Thus, (.) holds.

(.)⇒(.). Note that|S(nk)|=|Sn–Xk| ≤ |Sn|+|Xk|=|Sn|+|Sk–Sk–| ≤ |Sn|+|Sk|+

|Sk–| ≤max≤j≤n|Sj|, we have (max≤k≤n|Sn(k)| ≥εnα)⊆(max≤j≤n|Sj| ≥εnα/), hence,

from (.), (.) holds.

(.)⇒(.). Since _|Sn| ≤n–_n |Sn|=|_nn_k_=Sn(k)| ≤max≤k≤n|Sn(k)|,∀n≥, and|Xk|= |Sn–S(nk)| ≤ |Sn|+|S(nk)|, we have (max≤k≤n|Xk| ≥εnα)⊆(|Sn| ≥εnα/)∪(max≤k≤n|S(nk)| ≥ εnα_/)_⊆₍_max

≤k≤n|S(nk)| ≥εnα/),∀n≥, hence, from (.), (.) holds. (.)⇒(.). By Lemma . and (.), we have

∞

n=

nαp–h(n)P

sup j≥nj

–α_|_S_{j| ≥}_ε

= ∞

i=

i–_≤n_<i

nαp–_h₍_n₎_P_sup

j≥nj

–α_|_S_{j| ≥}_ε

≤C

∞

i=

i(αp–)hiP

sup j≥i–

j–α|Sj| ≥ε

≤C

∞

i=

i(αp–)hi ∞

k=i

P

max

k–_≤j_<k|Sj| ≥ε

α(k–)

≤C

∞

k= P

max

k–_≤j_<k|Sj| ≥ε

α(k–)

k

i=

i(αp–)hi

≤C

∞

k=

k(αp–)hkP

max

≤j<k|Sj| ≥ε

α(k–)_<_∞.

(.)⇒(.). The proof of (.)⇒(.) is similar to that of (.)⇒(.), so it is

omit-ted.

Theorem . Letα> /,p> ,αp> and h(x)be a slowly varying function at inﬁnity.Let

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deﬁned on a diﬀerent space.Moreover,additionally assume that forα≤,EXn= for all

n≥.If there exist constant D> and D> such that

D n

n–

i=n

P|Xi|>x≤P|X|>x≤D n

n–

i=n

P|Xi|>x, ∀x> ,n≥,

then(.)-(.)are equivalent.

Proof From the proof of Theorem ., in order to prove Theorem ., it is enough to show that (.)⇒(.) and (.)⇒(.). The proof of (.)⇒(.) is similar to that of (.)⇒(.). Now, we prove (.)⇒(.). Firstly we prove that

lim n→∞P

sup j≥nj

–α_|_X_{j| ≥}_ε_{= ,} _∀_ε_{> .} _(.)

Otherwise, there areε> ,δ> , and a sequence of positive integers{nk,k≥},nk↑ ∞ such thatP(sup_j≥nkj–α_|_X_{j| ≥}_ε

)≥δ,∀k≥. Without loss of generality, we can assume

thatnk+≥nk,∀k≥. Therefore, we have

P

sup j≥nk

j–α|Xj| ≥ε

≥δ, ∀k≥.

Byαp>  we have

∞

n=

nαp–h(n)P

sup j≥nj

–α_|_X_{j| ≥}_ε 

≥ ∞

k= nk

n=nk+

nαp–h(n)P

sup j≥nj

–α_|_X_{j| ≥}_ε 

≥C

∞

k=

nα_kp–h(nk)P

sup j≥nk

j–α|Xj| ≥ε

=∞,

which is in contradiction with (.), thus, (.) holds. By Lemma ., we get

P

sup j≥nj

–α_|_X_{j| ≥}_ε_≥_P_max

n≤j<nj

–α_|_X_{j| ≥}_ε

≥P

max

n≤j<n|Xj| ≥(n)

α_ε

≥ –Pmax

n≤j<nXj< (n)

α_ε_{=  –}_E __n_–

j=n

IXj< (n)αε

≥ –

n–

j=n

PXj< (n)αε

=  –

n–

j=n

 –PXj≥(n)αε

≥ –exp

–

n–

j=n

PXj≥(n)αε

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By (.), we havelim_n→∞_j₌n_n–P(Xj≥(n)αε) = ,∀ε> . Therefore, whennis large enough, we have

Pmax n≤j<nj

–α_|_X_{j| ≥}_ε_≥_{ –} _{ –} n–

j=n

PXj≥(n)αε

+ 

__n_–

j=n

PXj≥(n)αε



≥C n–

j=n

PXj≥(n)αε

, ∀ε> .

In a similar way, whennis large enough,

P

max n≤j<nj

–α_|_X_{j| ≥}_ε_≥_C n–

j=n

P–Xj≥(n)αε

, ∀ε> .

Thus, whennis large enough, we have

P

max n≤j<nj

–α_|_X_{j| ≥}_ε_≥_C n–

j=n

P|Xj| ≥(n)α_ε_≥

CnP|X| ≥(n)α_ε_, _∀_ε_{> . (.)}

Takingε= –α_{, by (.), (.), Lemma ., and a standard computation, we have}

∞> ∞

n=

nαp–h(n)P

sup j≥nj

–α_|_X_{j| ≥}_–α_≥

∞

n=

nαp–h(n)P

max n≤j<nj

–α_|_X_{j| ≥}_–α

≥C

∞

n=

nαp–h(n)P|X| ≥nα

≥CE|X|ph|X|α.

Thus, (.) holds.

In the following, let{τn,n≥}be a sequence of non-negative, integer valued random variables andτ a positive random variable. All random variables are deﬁned on the same probability space.

Theorem . Letα > /, p> , αp>  and h(x) >  be a slowly varying function as x→+∞.Let{Xn,n≥} be a sequence of NOD random variables and X be a random variables possibly deﬁned on a diﬀerent space satisfying the condition(.)and(.). More-over,additionally assume that forα≤,EXn= for all n≥.If there existsλ> such that

∞

n=nαp–h(n)P( τn

n <λ) <∞,then

∞

n=

nαp–h(n)P|Sτn| ≥ετ_nα<∞, ∀ε> . (.)

Proof Note that

|Sτn| ≥ετ_nα⊆(τn/n<λ)∪|Sτn| ≥ετ

α

n,τn≥λn

⊆(τn/n<λ)∪

sup j≥λn

j–α|Sj| ≥ε

.

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Theorem . Letα> /,p> ,αp> and h(x)be a slowly varying function at infin-ity. Let {Xn,n≥}be a sequence of NOD random variables and X be a random vari-ables possibly defined on a different space satisfying the condition(.)and(.).Moreover,

additionally assume that forα≤,EXn= for all n≥.If there existsθ > such that

∞

n=nαp–h(n)P(| τn

n –τ|>θ) <∞with P(τ≤B) = for some B> ,then

∞

n=

nαp–_h₍_n₎_P_|_Sτ_{n| ≥}_ε_nα_<_∞_, _∀_ε_{> .} _(.)

Proof Note that

|Sτn| ≥εnα_⊆τn n –τ

>θ

∪

|Sτn| ≥εnα_,τn n –τ

≤θ

⊆τn

n –τ >θ

∪|Sτn| ≥εnα,τn≤(τ+θ)n

⊆τn

n –τ >θ

∪|Sτn| ≥εnα_,_τ

n≤(B+θ)n

⊆τn

n –τ >θ

∪ max

≤j≤(B+θ)n|Sj| ≥εn

α

.

Thus, by (.) of Theorem ., we have (.).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the ﬁnal manuscript.

Author details

1_{School of Mathematics and Statistics, Guangdong University of Finance and Economics, Guangzhou, 510320, P.R. China.} 2_{College of Science, Guilin University of Technology, Guilin, 541004, P.R. China.}3_{Department of Mathematics, Jinan}

University, Guangzhou, 510630, P.R. China.

Acknowledgements

The authors would like to thank the referees and the editors for the helpful comments and suggestions. This work was supported by the National Natural Science Foundation of China (Grant No. 11271161).

Received: 14 November 2013 Accepted: 26 March 2014 Published:09 Apr 2014

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