R E S E A R C H
Open Access
Further study of complete convergence
for weighted sums of PNQD random variables
Qunying Wu
**Correspondence: [email protected]
College of Science, Guilin University of Technology, Guilin, 541004, P.R. China
Abstract
Based on Wu (J. Appl. Math. 2012:104390, 2012), the complete convergence for weighed sums of pairwise negative quadrant dependent (PNQD) random variables is further studied under weaker weighted condition. Sufficient and necessary
conditions of complete convergence for weighted sums of PNQD random variables are obtained. Our results generalize and improve those on complete convergence theorems previously obtained by Baum and Katz (Trans. Am. Math. Soc. 120:108-123, 1965), Wu (J. Appl. Math. 2012:104390, 2012) and Zhang (J. Inequal. Appl. 2014:353, 2014).
MSC: 62F12
Keywords: pairwise negative quadrant dependent random variable; complete convergence; sufficient and necessary condition
1 Introduction and lemmas
Random variablesXandYare said to be negative quadrant dependent (NQD) if
P(X≤x,Y≤y)≤P(X≤x)P(Y≤y) (.)
for allx,y∈R. It is important to note that (.) and
P(X>x,Y>y)≤P(X>x)P(Y>y) (.) for allx,y∈R are equivalent. Obviously, iff andgare Borel functions both of which are monotone increasing (or both are monotone decreasing), thenf(X) andg(Y) are NQD. A sequence of random variables{Xn;n≥}is said to be pairwise negative quadrant
de-pendent (PNQD) if every pair of random variables in the sequence is NQD. This definition was introduced by Lehmann []. Obviously, PNQD sequence includes many negatively associated sequences, and NA and pairwise independent random sequence are the most common special cases.
In many mathematical and mechanical models, a PNQD assumption among the random variables in the models is more reasonable than an independence assumption. PNQD se-ries have received more and more attention recently because of their wide applications in mathematical and mechanical models, percolation theory and reliability theory. Many statisticians have investigated PNQD series with interest and have established a series of
useful results. For example, Matula [], Li and Yang [] and Wu and Jiang [] obtained the strong law of large numbers; Wanget al.[] obtained Marcinkiewicz’s weak law of large numbers; Wu [] obtained the strong convergence properties of Jamison weighted sums, the three series theorem and complete convergence theorem; and Li and Wang [] ob-tained the central limit theorem. It is interesting for us to extend the limit theorems to the case of PNQD series. However, so far there has not been the general moment inequality for PNQD sequence, and therefore the study of the limit theory for PNQD sequence is very difficult and challenging. In the above-mentioned conclusions, only the Kolmogorov-type strong law of large numbers obtained by Matula [], Theorem , and Baum and Katz-type complete convergence theorem obtained by Wu [], Theorem , achieve the correspond-ing conclusions of independent cases, and the rest did not achieve the optimal results of independent cases.
Complete convergence is one of the most important problems in probability theory. Re-cent results of the complete convergence can be found in Wu [], Chen and Wang [] and Liet al.[]. In this paper, based on Wu [], we establish the complete convergence theo-rem for weighted sums of PNQD sequence, which extend and improve the corresponding results of Baum and Katz [], Wu [] and Zhang [].
2 Main results and the proof
In the following, the symbolcstands for a generic positive constant which may differ from one place to another. Letanbn(anbn) denote that there exists a constantc> such
thatan≤cbn(an≥cbn) for all sufficiently largen, and letXi≺X(XiX) denote that
there exists a constantc> such thatP(|Xi|>x)≤cP(|X|>x) (P(|Xi|>x)≥cP(|X|>x))
for alli≥ andx> .
Theorem . Let{Xn;n≥}be a sequence of PNQD random variables with Xi≺X.Let
forαp> , <p< ,α> ,and EXi= forα≤.Let{ank;k≤n,n≥}be a sequence of
real numbers such that
n
k=
|ank|p n–αp for some p >p. (.)
If
E|X|p<∞, (.)
then
∞
n=
nαp–P
max
≤k≤n|Snk|>ε
<∞, ∀ε> , (.)
where Snk=
k
i=aniXi.
Theorem . Let{Xn;n≥}be a sequence of PNQD random variables with XiX.Let
forα> ,αp> , <p< .Let{ank;k≤n,n≥}be a sequence of real numbers such that
n
Remark . Obviously, if|ank| n–α(or|ank| n–α) for allk≤n,n≥, then (.) (or
n
k=|ank|p n–αp) holds. Hence, Theorems . and . in Wu [] are the particular
cases of our Theorems . and ..
Remark . Theorems . and . remain valid if we replace (.) byP(X≤x,Y≤y)≤ MP(X≤x)P(Y ≤y) for someM> and allx,y∈R. Hence, our Theorems . and . improve and extend Theorem . in Zhang [].
Proof of Theorem. Forn≥, letani=ani, if|ani| ≤n–α,ani= otherwise, anda ni=ani,
if|ani|>n–α,a ni= otherwise. Then ∞
n=
nαp–Pmax
≤k≤n|Snk|> ε
≤
∞
n=
nαp–P
max ≤k≤n
k
i=
aniXi
>ε
+
∞
n=
nαp–P
max ≤k≤n
k
i=
a niXi
>ε
:=I+I.
Since|ani| ≤n–α, by Theorem . in Wu [], we haveI
<∞. Hence, in order to prove (.), it suffices to proveI<∞. For convenience, we still use the symbolanisaida ni. Without
loss of generality, assume thatank>n–αfork≤n,n≥, and (.) is n
k=
apnk≤n–αp for somep >p. (.)
Letq> such that ( + /αp)/ <q< . For alli≤n, let Yni= –a–nin
α(q–)I
(aniXi<–nα(q–))+XiI(ani|Xi|≤nα(q–))+a
–
nin
α(q–)I
(aniXi>nα(q–)),
Unk= k
i=
aniYni.
Write
An= n
i=
|aniXi| ≥ε
,
Bn=
≤i<j≤n
aniXi>nα(q–),anjXj>nα(q–)
∪aniXi< –nα(q–),anjXj< –nα(q–)
.
Using (.) in Wu [], in order to proveI<∞, it suffices to prove that
∞
n=
nαp–P(An) <∞, (.)
∞
n=
nαp–P(Bn) <∞, (.)
∞
n=
nαp–Pmax
≤j≤n|Unj| ≥ε
For ≤j≤n– andn≥, let
Dnj=
i; ≤i≤n,n–αp(j+ )–<apni≤n–αpj–.
Then{Dnj; ≤j≤n– }are disjoint, and
n–
j= Dnj={, , . . . ,n}from (.) andani>n–α,
i≤n.
For ≤k≤n– , by (.),
n–αp ≥ n
i=
apni=
n–
j=
i∈Dnj
apni
≥ n– j= i∈Dnj
n–αp(j+ )–≥n–αp
k+
k
j=
Dnj,
where the symbolAdenotes the number of elements in the setA. We have
k
j=
Dnjk for ≤k≤n– . (.)
Letβ–=α– /p, by (.), (.), (.),X
i≺X, it follows that ∞
n=
nαp–P(An)≤ ∞
n=
nαp–
n
i=
P|aniXi| ≥ε
=
∞
n=
nαp–
n
j=
i∈Dnj
P|Xi| ≥εa–ni
≤
∞
n=
nαp–
n
j=
P|X|β≥εnjβ/p D nj
=
∞
n=
nαp–
n
j=
Dnj
k≥njβ/p
Pεk≤ |X|β<ε(k+ )
=
∞
n=
nαp–
∞
k=n
Pεk≤ |X|β<ε(k+ )
j≤min(n,(k/n)p/β) Dnj
∞
n=
nαp–
∞
k=n
Pεk≤ |X|β<ε(k+ )min
n,
k n
p/β
≤
∞
n=
nαp–
n≤k≤n+β/p
Pεk≤ |X|β<ε(k+ )k n
p/β
+
∞
n=
nαp–
k≥n+β/p+
Pεk≤ |X|β<ε(k+ )n
Byαp– –p/β= –α(p –p) – < –,β(p/β–αp(p–p)/(p+β)) =p, andαppβ/(p +β) = p, we get
I
∞
n=
nαp––p/β
n≤k≤n+β/p
Pεk≤ |X|β<ε(k+ )kp/β
≤
∞
k=
Pεk≤ |X|β<ε(k+ )kp/β
n≥kp/(p+β)
nαp––p/β
∞
k=
Pεk≤ |X|β<ε(k+ )kp/β–αp(p–p)/(p+β)
∞
k=
E|X|pIεk≤ |X|β<ε(k+ )
E|X|p<∞, (.)
and
I =
∞
n=
nαp–
k≥n+β/p
Pεk≤ |X|β<ε(k+ )
≤
∞
k=
Pεk≤ |X|β<ε(k+ )
n≤kp/(p+β) nαp–
∞
k=
Pεk≤ |X|β<ε(k+ )kαpp/(p+β)
∞
k=
E|X|pIεk≤ |X|β<ε(k+ ) < ∞.
This together with (.) and (.) implies that (.) holds.
n
i=(
ani n–α)p≤
n
i=(
ani
n–α)p ≤nfrom (.),ani>n–α, andp >p. That is, n
i=
apni≤n–αp. (.)
By (.), (.), (.),Xi≺X, and the definition ofq,αp( – q) < –, ∞
n=
nαp–P(Bn)
≤
∞
n=
nαp–
≤i<j≤n
PaniXi>nα(q–)
PanjXj>nα(q–)
+
∞
n=
nαp–
≤i<j≤n
PaniXi< –nα(q–)
PanjXj< –nα(q–)
∞
n=
nαp–
n
i=
n–α(q–)papniE|X|p
∞
n=
nαp(–q)<∞.
That is, (.) holds.
Finally, we prove (.). Using (.), similarly to the proof of (.) in Wu [], we can provemax≤j≤n|EUnj| →,n→ ∞. Hence, by Lemma in Wu [] and ––α(–q)(–p) <
–,
∞
n=
nαp–P
max
≤j≤n|Unj| ≥ε
≤
∞
n=
nαp–P
max
≤j≤n|Unj–EUnj|>ε
∞
n=
nαp–logn
n
j=
EanjYnj
∞
n=
nαp–logn
n
j=
EanjXjI(anj|Xj|≤nα(q–))+nα(q–)P
anj|Xj|>nα(q–)
≤
∞
n=
nαp–logn
n
j=
E|anjXj|pnα(q–)(–p)+nα(q–)–αp(q–)E|anjXj|p
∞
n=
nαp––αp+α(q–)(–p)+n–+αp–αpq+αq–αlogn
=
∞
n=
n––α(–q)(–p)logn
<∞.
This completes the proof of Theorem ..
Proof of Theorem. Taking|ank| n–αfor alli≤n,n≥, then{ani}satisfies
n
i=a
p ni
n–αp. According to Theorem . in Wu [], (.) holds. This completes the proof of
The-orem ..
Competing interests
The author declares that they have no competing interests.
Author’s contributions
QW conceived of the study, drafted, completed, read and approved the final manuscript.
Author’s information
Qunying Wu, Professor, Doctor, working in the field of probability and statistics.
Acknowledgements
Supported by the National Natural Science Foundation of China (11361019), project supported by Program to Sponsor Teams for Innovation in the Construction of Talent Highlands in Guangxi Institutions of Higher Learning ([2011] 47), and the Support Program of the Guangxi China Science Foundation (2013GXNSFDA019001).
Received: 4 May 2015 Accepted: 3 September 2015 References
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