Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 104390, 10 pages
doi:10.1155/2012/104390
Research Article
Sufficient and Necessary Conditions
of Complete Convergence for Weighted Sums
of PNQD Random Variables
Qunying Wu1, 2
1
2
College of Science, Guilin University of Technology, Guilin 541004, China
Guangxi Key Laboratory of Spatial Information and Geomatics, Guilin University of Technology,
Guilin 541004, China
Correspondence should be addressed to Qunying Wu, wqy666@glite.edu.cn
Received 7 February 2012; Accepted 19 April 2012
Academic Editor: Martin Weiser
Copyright q 2012 Qunying Wu. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
The complete convergence for pairwise negative quadrant dependent PNQD random variables is
studied. 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. We establish a
collection that contains relationship to overcome the difficulties that there is no general moment
inequality. 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 1965 and Wu 2002.
1. Introduction and Lemmas
Random variables X and Y are said to be negative quadrant dependent NQD if
P X ≤ x, Y ≤ y ≤ P X ≤ xP Y ≤ y ,
1.1
for all x, y ∈ R. A sequence of random variables {Xn ; n ≥ 1} is said to be pairwise negative
quadrant dependent PNQD if every pair of random variables in the sequence is NQD.
This definition was introduced by Lehmann 1. Obviously, PNQD sequence includes many
negatively associated sequences, and pairwise independent random sequence is the most
special case.
In many mathematics and mechanic models, a PNQD assumption among the random
variables in the models is more reasonable than an independence assumption. PNQD
series have received more and more attention recently because of their wide applications
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Journal of Applied Mathematics
in mathematics and mechanic 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 2, Li and Yang 3, and Wu and Jiang 4 obtained the
strong law of large numbers, Wang et al. 5 obtained the Marcinkiewicz’s weak law of large
numbers, Wu 6 obtained the strong convergence properties of Jamison weighted sums, the
three-series theorem, and complete convergence theorem, and Li and Wang 7 obtained
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 2, Theorem 1 and Baum and Katz-type complete
convergence theorem obtained by Wu 6, Theorem 4 achieve the corresponding 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.
Recent results of the complete convergence can be found in Wu 6, Chen and Wang 8, and
Li et al. 9. In this paper, we establish a collection that contains relationship to overcome
the difficulties that there is no the general moment inequality and obtain the complete
convergence theorem for weighted sums of PNQD sequence, which extend and improve the
corresponding results of Baum and Katz 10 and Wu 6.
Lemma 1.1 see 1. Let X and Y be NQD random variables. Then
i covX, Y ≤ 0,
ii P X > x, Y > y ≤ P X > xP Y > y, for all x, y ∈ R,
iii if f and g are Borel functions, both of which being monotone increasing (or both are
monotone decreasing), then fX and gY are NQD.
Lemma 1.2 see 6, Lemma 2. Let {Xn ; n ≥ 1} be a sequence of PNQD random variables with
jk
EXn 0, EXn2 < ∞, Tj k
ij1 Xi , j ≥ 0. Then
jk
2 EXi2 ,
E Tj k ≤
ij1
Emax Tj k
2
1≤k≤n
≤
jn
4 log2 n log2 2
1.2
EXi2 .
ij1
∞, then P An ; i.o. 0.
Lemma 1.3 see 2, Lemma 1. i If ∞
n1 P An <
m, and ∞
ii if P Ak Am ≤ P Ak P Am , k /
n1 P An ∞, then P An ; i.o. 1.
Lemma 1.4. Let {Xn ; n ≥ 1} be a sequence of PNQD random variables. Then for any x ≥ 0, there
exists a positive constant c such that for all n ≥ 1,
2 n
1 − P max |Xk | > x
P |Xk | > x ≤ cP max |Xk | > x .
1≤k≤n
k1
1≤k≤n
Proof. We can prove the Lemma by Lemma A.6 of Zhang and Wen 11.
1.3
Journal of Applied Mathematics
3
2. Main Results and the Proof
In the following, the symbol c stands for a generic positive constant which may differ from
one place to another. Let an bn an bn denote that there exists a constant c > 0 such that
an ≤ cbn an ≥ cbn for all sufficiently large n, and let Xi ≺ X Xi X denote that there exists
a constant c > 0 such that P |Xi | > x ≤ cP |X| > x P |Xi | > x ≥ cP |X| > x for all i ≥ 1
and x > 0.
Theorem 2.1. Let {Xn ; n ≥ 1} be a sequence of PNQD random variables with Xi ≺ X. Let {ank ; k ≤
n, n ≥ 1} be a sequence of real numbers such that
|ank | n−α ,
∀k ≤ n, n ≥ 1.
2.1
Let for αp > 1, 0 < p < 2, α > 0, and EXi 0, for α ≤ 1. If
E|X|p < ∞,
2.2
then
∞
nαp−2 P max |Snk | > ε < ∞,
n1
where Snk k
i1
1≤k≤n
∀ε > 0,
2.3
ani Xi .
Theorem 2.2. Let {Xn ; n ≥ 1} be a sequence of PNQD random variables with Xi X. Let {ank ; k ≤
n, n ≥ 1} be a sequence of real numbers such that |ank | n−α , for all k ≤ n, n ≥ 1. Let for
α > 0, αp > 1, 0 < p < 2. If 2.3 holds, then 2.2 holds.
Remark 2.3. Taking ani n−α , for all i ≤ n, n ≥ 1 in Theorem 2.1, then
∞
n1
αp−2
n
P max |Snk | > ε
1≤k≤n
∞
nαp−2 P
n1
∞
nαp−2 P
n1
k
max n Xi > ε
i1 1≤k≤n
−α k
α
max Xi > εn .
1≤k≤n
i1
2.4
Hence, Theorem 4 in Wu 6 is a particular case of our Theorem 2.1.
Remark 2.4. When {Xn ; n ≥ 1} is i.i.d. and ani n−α , for all i ≤ n, n ≥ 1, then Theorems 2.1
and 2.2 become Baum and Katz 10 complete convergence theorem. Hence, our Theorems
2.1 and 2.2 improve and extend the well-known Baum and Katz theorem.
4
Journal of Applied Mathematics
Proof of Theorem 2.1. Without loss of generality, assume that ank > 0 for k ≤ n, n ≥ 1. Let q > 0
such that 1 1/αp/2 < q < 1. For all i ≤ n, let
αq−1
I ani Xi < −nαq−1 Xi I ani |Xi | ≤ nαq−1
Yni − a−1
ni n
αq−1
a−1
I ani Xi > nαq−1 ,
ni n
Unk 2.5
k
ani Yni .
i1
Write
An n anj Xj ≥ ε ,
j1
Bn αq−1
ani Xi > n
αq−1
, anj Xj > n
αq−1
ani Xi < −n
αq−1
, anj Xj < −n
2.6
.
1≤i<j≤n
Firstly, we prove that
max |Snk | < 6ε
1≤k≤n
⊇
Acn
n anj Xj < ε
max |Unk | < 2ε
max |Unk | < 2ε
1≤k≤n
Bnc
1≤k≤n
j1
ani Xi ≤ nαq−1
anj Xj ≤ nαq−1
2.7
1≤i<j≤n
ani Xi ≥ −nαq−1
anj Xj ≥ −nαq−1
Dn .
For any ω ∈ Dn , we have
anj Xj < ε,
anj Ynj ≤ anj Xj < ε,
∀1 ≤ j ≤ n,
max |Unk | < 2ε,
1≤k≤n
2.8
and for any 1 ≤ i < j ≤ n,
ani Xi ≤ nαq−1 ,
or
anj Xj ≤ nαq−1 ,
ani Xi ≥ −nαq−1 ,
or
anj Xj ≥ −nαq−1 .
2.9
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5
Hence
a
i; 1 ≤ i ≤ n, ani Xi ω > nαq−1 ≤ 1,
b
i; 1 ≤ i ≤ n, ani Xi ω < −nαq−1 ≤ 1,
2.10
where the symbol A denotes the number of elements in the set A.
When a b 0, then |ani Xi ω| ≤ nαq−1 for any 1 ≤ i ≤ n; thus, Yni ω Xi ω, and
therefore by 2.8,
max |Snk | max |Unk | < 2ε < 6ε.
1≤k≤n
1≤k≤n
2.11
When a 1, b 0 or a 0, b 1, then there exists only an i0 : 1 ≤ i0 ≤ n such that
ani0 Xi0 ω > nαq−1 or ani0 Xi0 ω < −nαq−1 , the remaining j, |anj Xnj ω| ≤ nαq−1 ; thus,
Xj ω Ynj ω. If 1 ≤ k ≤ i0 − 1, then Snk ω Unk ω. If i0 ≤ k ≤ n, then by 2.8,
max |Snk ω| max ani Xi ω ani0 Xi0 ω
1≤k≤n
1≤k≤n
1≤i≤k,i /
i0
k
max ani Yni ω − an0 Yni0 ω ani0 Xi0 ω
1≤k≤n
i1
2.12
k
≤ max ani Yni ω
|ani0 Yni0 ω| |ani0 Xi0 ω|
1≤k≤n
i1
< 2ε ε ε < 6ε.
When a b 1, then there exist 1 ≤ i1 , i2 ≤ n such that ani1 Xi1 ω >
, ani2 Xi2 ω < −nαq−1 , the remaining j, |anj Xj ω| ≤ nαq−1 ; thus, Xj ω Ynj ω.
n
Without loss of generality, assume that i1 ≤ i2 . If 1 ≤ k ≤ i1 − 1, then Snk ω Unk ω; if
i1 ≤ k < i2 , then by 2.8,
αq−1
max |Snk ω| ≤ max |Unk ω| |ani1 Yni1 ω| |ani1 Xi1 ω|
1≤k≤n
1≤k≤n
< 2ε ε ε < 6ε.
If k ≥ i2 , then by 2.8,
ani Xi ω ani1 Xi1 ω ani2 Xi2 ω
max |Snk ω| max 1≤k≤n
1≤k≤n
1≤i≤k,i / i1 ,i2
≤ max |Unk ω| |ani1 Yni1 ω| |ani2 Yni2 ω|
1≤k≤n
2.13
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Journal of Applied Mathematics
|ani1 Xi1 ω| |ani2 Xi2 ω|
< 6ε.
2.14
Hence, 2.7 holds, that is:
max |Snk | ≥ 6ε
⊆ An
1≤k≤n
max |Unk | ≥ 2ε
1≤k≤n
Bn .
2.15
Therefore, in order to prove 2.3, we only need to prove that
∞
nαp−2 P An < ∞,
2.16
∞
nαp−2 P Bn < ∞,
2.17
n1
n1
∞
nαp−2 P max |Unk | ≥ 2ε < ∞,
1≤k≤n
n1
∀ε > 0.
2.18
By 2.1, 2.2, Xi ≺ X, and αp > 1,
∞
nαp−2 P An ≤
n1
∞
n
nαp−2 P anj Xj ≥ ε
n1
≤
j1
∞
n
α
nαp−2 P Xj ≥ εa−1
nj ≥ εcn
n1
j1
∞
nαp−1 P |X| ≥ εcnα n1
∞
∞
α P εcj α ≤ |X| < εc j 1
n1
jn
nαp−1
j
∞ α nαp−1 P εcj α ≤ |X| < εc j 1
j1 n1
≤
∞
α j αp P εcj α ≤ |X| < εc j 1
j1
E|X|p < ∞.
That is, 2.16 holds.
2.19
Journal of Applied Mathematics
7
By Lemma 1.1ii, Xi ≺ X, and the definition of q, αp1 − 2q < −1,
∞
∞
P ani Xi > nαq−1 P anj Xj > nαq−1
nαp−2 P Bn ≤
nαp−2
n1
n1
1≤i<j≤n
P ani Xi < −nαq−1 P anj Xj < −nαq−1
2.20
∞
∞
2
nαp P 2 |X| > cnαq ≤
nαp n−2αpq E|X|p
n1
n1
∞
nαp1−2q < ∞.
n1
That is, 2.17 holds.
In order to prove 2.18, firstly, we prove that
k
max |EUnk | max E ani Yni −→ 0,
1≤k≤n
1≤k≤n
i1
n −→ ∞.
2.21
i When α ≤ 1, then p > 1/α ≥ 1; from EXi 0 and the definition of q, we have
q < 1, αpq > αp 1 − αpq 1 αp1 − q > 1 :
k
max E ani Yni 1≤k≤n
i1
≤
n
ani |EYni | i1
≤
n
i1
i1
αq−1 −1 αq−1 αq−1
αq−1
ani E
Xi a−1
n
E
−
a
n
I
I
X
i
ani Xi <−n
ani Xi >n
ni
ni
n
i1
n
i1
n
ani |EXi − Yni |
ani E|Xi |I|ani Xi |>nαq−1 2.22
n
ani |Xi | p−1
≤
ani E|Xi | αq−1
n
i1
p
ani E|Xi |p nα1−qp−1
n−αp1αp−α−αpqαq n−αpq−1−α1−q
−→ 0,
n −→ ∞.
ii When α > 1, and p ≥ 1, then E|X| < ∞ from 2.2, thus,
k
n
ani E|X| n−α1 −→ 0,
max E ani Yni ≤
1≤k≤n
i1
i1
n −→ ∞.
2.23
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Journal of Applied Mathematics
iii When α > 1, and p < 1, by −αp−1−α1−q1−p < 0, and −α1−q−αpq−1 < 0,
we get
k
n
ani E|Xi |Iani |Xi |≤nαq−1 a−1
nαq−1 P |ani Xi | > nαq−1
max E ani Yni ≤
ni
i1
1≤k≤n
i1
≤
n
n
p
p
ani E|Xi |p |ani Xi |1−p Iani |Xi |≤nαq−1 nαq−1 ani n−αpq−1 E|Xi |p
i1
2.24
i1
n−αp−1−α1−p1−q n−α1−q−αpq−1 −→ 0.
Hence, 2.21 holds; that is, for any ε > 0, we have max1≤k≤n |EUnk | < ε for all sufficiently large
n. Thus,
P max
Unj ≥ 2ε
1≤j≤n
≤ P max Unj − EUnj > ε .
1≤j≤n
2.25
Let Yni Yni − EYni . Obviously, Yni is monotonic on Xi . By Lemma 1.1iii, {Yni ; n ≥
1, i ≤ n} is also a sequence of PNQD random variables with EYni 0, by Lemma 1.2 and
−1 − α1 − q2 − p < −1:
∞
αp−2
n
P max
Unj − EUnj > ε
1≤j≤n
n1
∞
n1
∞
n1
≤
n
2
nαp−2 log2 n Ea2nj Ynj
j1
n Ea2nj Xj2 Ianj |Xj |≤nαq−1 n2αq−1 P anj Xj > nαq−1
nαp−2 log2 n
j1
∞
n p
p E
anj Xj nαq−12−p n2αq−1−αpq−1 E
anj Xj nαp−2 log2 n
n1
j1
∞ nαp−1−αpαq−12−p n−1αp−αpq2αq−2α log2 n
n1
∞
2 n−1−α1−q2−p log2 n
n1
< ∞.
This completes the proof of Theorem 2.1.
2.26
Journal of Applied Mathematics
9
Proof of Theorem 2.2. Noting that max1≤k≤n |ank Xk | ≤ 2 max1≤k≤n |Snk | and |ank | n−α , from
2.3,
∞
nαp−2 P max |Xk | > εnα < ∞,
∀ε > 0.
1≤k≤n
n1
2.27
Thus, by αp − 2 > −1, we get
∞ ∞
P max |Xk | > ε2αj1 n−1 P max |Xk | > εnα < ∞,
j1
∀ε > 0.
2.28
max P max
Xj > ε22α nα ≤ P maxm Xj > ε2αm1 −→ 0.
2.29
1≤k≤2j
n1
1≤k≤n
This implies that
2m−1 ≤n≤2m
1≤j≤n
1≤j≤2
Hence, for all sufficiently large n,
1
P max
Xj > ε22α nα < .
1≤j≤n
2
2.30
By Lemma 1.4,
n
P |Xk | > εnα ≤ 4cP max |Xk | > εnα ,
1≤k≤n
k1
∀ε > 0,
2.31
which together with 2.27,
∞
n1
nαp−2
n
P |Xk | > εnα < ∞,
∀ε > 0.
k1
2.32
By Xk X, we obtain
E|X|p ∞
nαp−1 P |X| > εnα < ∞.
n1
2.33
This completes the proof of Theorem 2.2.
Acknowledgments
The author is very grateful to the referees and the editors for their valuable comments and
some helpful suggestions that improved the clarity and readability of the paper. Supported
by the National Natural Science Foundation of China 11061012, and project supported by
10
Journal of Applied Mathematics
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 Key Laboratory of
Spatial Information and Geomatics 1103108-08.
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