Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2012, Article ID 730962, 13 pages
doi:10.1155/2012/730962
Research Article
Complete Moment Convergence of Weighted Sums
for Arrays of Rowwise ϕ-Mixing Random Variables
Ming Le Guo
School of Mathematics and Computer Science, Anhui Normal University, Wuhu 241003, China
Correspondence should be addressed to Ming Le Guo, mleguo@163.com
Received 5 June 2012; Accepted 20 August 2012
Academic Editor: Mowaffaq Hajja
Copyright q 2012 Ming Le Guo. 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 moment convergence of weighted sums for arrays of rowwise ϕ-mixing random
variables is investigated. By using moment inequality and truncation method, the sufficient
conditions for complete moment convergence of weighted sums for arrays of rowwise ϕ-mixing
random variables are obtained. The results of Ahmed et al. 2002 are complemented. As an
application, the complete moment convergence of moving average processes based on a ϕ-mixing
random sequence is obtained, which improves the result of Kim et al. 2008.
1. Introduction
Hsu and Robbins 1 introduced the concept of complete convergence of {Xn }. A sequence
{Xn , n 1, 2, . . .} is said to converge completely to a constant C if
∞
P |Xn − C| > < ∞,
∀ > 0.
1.1
n1
Moreover, they proved that the sequence of arithmetic means of independent identically
distributed i.i.d. random variables converge completely to the expected value if the variance
of the summands is finite. The converse theorem was proved by Erdös 2. This result has
been generalized and extended in several directions, see Baum and Katz 3, Chow 4, Gut
5, Taylor et al. 6, and Cai and Xu 7. In particular, Ahmed et al. 8 obtained the following
result in Banach space.
2
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Theorem A. Let {Xni ; i ≥ 1, n ≥ 1} be an array of rowwise independent random elements in a
separable real Banach space B, · . Let P Xni > x ≤ CP |X| > x for some random variable X,
constant C and all n, i and x > 0. Suppose that {ani , i ≥ 1, n ≥ 1} is an array of constants such that
sup|ani | O n−r ,
for some r > 0,
i≥1
∞
|ani | Onα ,
1.2
for some α ∈ 0, r.
i1
Let β be such that α β /
− 1 and fix δ > 0 such that 1 α/r < δ ≤ 2. Denote s max1 α β ∞ β
1/r, δ. If E|X|s < ∞ and Sn ∞
i1 ani Xni → 0 in probability, then
n1 n P Sn > < ∞
for all > 0.
Chow 4 established the following refinement which is a complete moment
convergence result for sums of i.i.d. random variables.
Theorem B. Let EX1 0, 1 ≤ p < 2 and r ≥ p. Suppose that E|X1 |r |X1 | log1 |X1 | < ∞.
Then
∞
n1
r/p−2−1/p
n
n
1/p
E Xi − n
< ∞,
i1 1.3
∀ > 0.
The main purpose of this paper is to discuss again the above results for arrays of
rowwise ϕ-mixing random variables. The author takes the inspiration in 8 and discusses
the complete moment convergence of weighted sums for arrays of rowwise ϕ-mixing random
variables by applying truncation methods. The results of Ahmed et al. 8 are extended to ϕmixing case. As an application, the corresponding results of moving average processes based
on a ϕ-mixing random sequence are obtained, which extend and improve the result of Kim
and Ko 9.
For the proof of the main results, we need to restate a few definitions and lemmas for
easy reference. Throughout this paper, C will represent positive constants, the value of which
may change from one place to another. The symbol IA denotes the indicator function of A;
x indicates the maximum integer not larger than x. For a finite set B, the symbol B denotes
the number of elements in the set B.
Definition 1.1. A sequence of random variables {Xi , 1 ≤ i ≤ n} is said to be a sequence of
ϕ-mixing random variables, if
,
P
>
0
−→ 0,
ϕm sup |P B | A − P B| ; A ∈ k1 , B ∈ ∞
A
km
k≥1
as m −→ ∞,
1.4
where kj σ{Xi ; j ≤ i ≤ k}, 1 ≤ j ≤ k ≤ ∞.
Definition 1.2. A sequence {Xn , n ≥ 1} of random variables is said to be stochastically
dominated by a random variable X write {Xi } ≺ X if there exists a constant C, such that
P {|Xn | > x} ≤ CP {|X| > x} for all x ≥ 0 and n ≥ 1.
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3
The following lemma is a well-known result.
Lemma 1.3. Let the sequence {Xn , n ≥ 1} of random variables be stochastically dominated by a
random variable X. Then for any p > 0, x > 0
E|Xn |p I|Xn | ≤ x ≤ C E|X|p I|X| ≤ x xp P {|X| > x} ,
1.5
E|Xn |p I|Xn | > x ≤ CE|X|p I|X| > x.
1.6
Definition 1.4. A real-valued function lx, positive and measurable on A, ∞ for some A > 0,
is said to be slowly varying if limx → ∞ lxλ/lx 1 for each λ > 0.
By the properties of slowly varying function, we can easily prove the following lemma.
Here we omit the details of the proof.
Lemma 1.5. Let lx > 0 be a slowly varying function as x → ∞, then there exists C (depends only
on r) such that
i Ckr1 lk ≤
ii Ckr1 lk ≤
k
n1
∞
nk
nr ln ≤ Ckr1 lk for any r > −1 and positive integer k,
nr ln ≤ Ckr1 lk for any r < −1 and positive integer k.
The following lemma will play an important role in the proof of our main results. The
proof is due to Shao 10.
Lemma 1.6. Let {Xi , 1 ≤ i ≤ n} be a sequence of ϕ-mixing random variables with mean zero. Suppose
2
that there exists a sequence {Cn } of positive numbers such that E km
ik1 Xi ≤ Cn for any k ≥ 0, n ≥
1, m ≤ n. Then for any q ≥ 2, there exists C Cq, ϕ· such that
q
kj
q/2
q
Xi ≤ C Cn E max |Xi | .
E max
1≤j≤n
k1≤i≤kn
ik1
Lemma 1.7. Let {Xi , 1 ≤ i ≤ n} be a sequence of ϕ-mixing random variables with
then there exists C such that for any k ≥ 0 and n ≥ 1
E
kn
ik1
2
Xi
≤C
kn
ik1
EXi2 .
1.7
∞
i1
ϕ1/2 i < ∞,
1.8
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Proof. By Lemma 5.4.4 in 11 and Hölder’s inequality, we have
E
kn
2
kn
Xi
ik1
ik1
kn
≤
≤
1/2 1/2
EXj2
ϕ1/2 j − i EXi2
EXi2 4
k1≤i<j≤kn
EXi2 2
kn
kn−1
ϕ1/2 j − i
EXi2 EXj2
1.9
ik1 ji1
ik1
≤
EXi Xj
k1≤i<j≤kn
ik1
kn
EXi2 2
14
n
1/2
ϕ
i
i1
kn
EXi2 .
ik1
Therefore, 1.8 holds.
2. Main Results
Now we state our main results. The proofs will be given in Section 3.
Theorem 2.1. Let {Xni , i ≥ 1, n ≥ 1} be an array of rowwise ϕ-mixing random variables with EXni 1/2
m < ∞. Let lx > 0 be a slowing varying function, and {ani , i ≥ 1, n ≥
0, {Xni } ≺ X and ∞
m1 ϕ
1} be an array of constants such that
sup|ani | O n−r ,
for some r > 0,
i≥1
∞
2.1
|ani | Onα ,
for some α ∈ 0, r.
i1
a If α β 1 > 0 and there exists some δ > 0 such that α/r 1 < δ ≤ 2, and s max1 α β 1/r, δ, then E|X|s l|X|1/r < ∞ implies
k
n lnE sup ani Xni − < ∞,
k≥1 i1
n1
∞
β
∀ > 0.
2.2
b If β −1, α > 0, then E|X|1a/r 1 l|X|1/r < ∞ implies
k
n lnE sup ani Xni − < ∞,
k≥1 i1
n1
∞
−1
∀ > 0.
2.3
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5
Remark 2.2. If α β 1 < 0, then E|X| < ∞ implies that 2.2 holds. In fact,
k
∞
∞
∞
n lnE sup ani Xni − ≤
nβ ln |ani |E|Xni | nβ ln
k≥1 i1
n1
n1
i1
n1
∞
β
≤C
∞
βα
n
lnE|X| n1
∞
2.4
n ln < ∞.
β
n1
Remark 2.3. Note that
∞ k
∞
k
β
∞>
n lnE sup ani Xni − n ln
P sup ani Xni − > x dx
0
k≥1 i1
k≥1 i1
n1
n1
∞
β
2.5
k
nβ lnP sup ani Xni > x dx.
k≥1
n1
i1
∞ ∞
0
Therefore, from 2.5, we obtain that the complete moment convergence implies the complete
convergence, that is, under the conditions of Theorem 2.1, result 2.2 implies
k
β
n lnP sup ani Xni > < ∞,
k≥1 i1
n1
2.6
k
n lnP sup ani Xni > < ∞.
k≥1 i1
n1
2.7
∞
and 2.3 implies
∞
−1
Corollary 2.4. Under the conditions of Theorem 2.1,
1 if α β 1 > 0 and there exists some δ > 0 such that α/r 1 < δ ≤ 2, and s max1 α β 1/r, δ, then E|X|s l|X|1/r < ∞ implies
∞
n lnE ani Xni − < ∞,
i1
n1
∞
β
∀ > 0,
2.8
2 if β −1, α > 0, then E|X|1α/r 1 l|X|1/r < ∞ implies
∞
n lnE ani Xni − < ∞,
i1
n1
∞
−1
∀ > 0.
2.9
Corollary 2.5. Let {Xni , i ≥ 1, n ≥ 1} be an array of rowwise ϕ-mixing random variables with
1/2
m < ∞. Suppose that lx > 0 is a slowly varying function.
EXni 0,{Xni } ≺ X and ∞
m1 ϕ
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International Journal of Mathematics and Mathematical Sciences
1 Let p > 1 and 1 ≤ t < 2. If E|X|pt l|X|t < ∞, then
k
1/t
lnE max Xni − n
< ∞,
1≤k≤n
i1
∞
p−2−1/t
n
n1
∀ > 0.
2.10
∀ > 0.
2.11
2 Let 1 < t < 2. If E|X|t 1 l|X|t < ∞, then
∞
k
1/t
lnE max Xni − n
< ∞,
1≤k≤n
i1
−1−1/t
n
n1
Corollary 2.6. Suppose that Xn ∞
i−∞ ain Yi , n ≥ 1, where {ai , −∞ < i < ∞} is a sequence of
∞
real numbers with −∞ |ai | < ∞, and {Yi , −∞ < i < ∞} is a sequence of ϕ-mixing random variables
1/2
m < ∞. Let lx be a slowly varying function.
with EYi 0, {Yi } ≺ Y and ∞
m1 ϕ
1 Let 1 ≤ t < 2, r ≥ 1 t/2. If E|Y |r l|Y |t < ∞, then
∞
r/t−2−1/t
n
n1
n
1/t
lnE Xi − n
< ∞,
i1 ∀ > 0.
2.12
2 Let 1 < t < 2. If E|Y |t 1 l|Y |t < ∞, then
∞
n1
−1−1/t
n
n
1/t
lnE Xi − n
< ∞,
i1 ∀ > 0.
2.13
Remark 2.7. Corollary 2.6 obtains the result about the complete moment convergence of
moving average processes based on a ϕ-mixing random sequence with different distributions.
We extend the results of Chen et al. 12 from the complete convergence to the complete
moment convergence. The result of Kim and Ko 9 is a special case of Corollary 2.6 1.
Moreover, our result covers the case of r t, which was not considered by Kim and Ko.
3. Proofs of the Main Results
Proof of Theorem 2.1. Without loss of generality, we can assume
sup|ani | ≤ n−r ,
i≥1
∞
i1
|ani | ≤ nα .
3.1
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7
Let Snk x ki1 ani Xni I|ani Xni | ≤ n−r x for any k ≥ 1, n ≥ 1, and x ≥ 0. First note that
E|X|s l|X|1/r < ∞ implies E|X|t < ∞ for any 0 < t < s. Therefore, for x > nr ,
k
−r
x n supE|Snk x| x n supE ani Xni I |ani Xni | > n x i1
k≥1
k≥1
−1 r
−1 r
≤
∞
∞
E|ani Xni |I |ani Xni | > n−r x ≤
E|ani X|I |ani X| > n−r x
i1
≤
EXni 0
i1
3.2
∞
|ani |E|X|I|X| > x ≤ nα E|X|I|X| > x
i1
≤ xα/r E|X|I|X| > x ≤ E|X|1α/r I|X| > nr −→ 0 as n −→ ∞.
Hence, for n large enough we have supk≥1 E|Snk x| < /2n−r x. Then
k
n lnE sup ani Xni − k≥1 i1
n1
∞
β
∞ k
n ln
P sup ani Xni ≥ x dx
k≥1 i1
n1
∞
∞
β
β−r
n
n1
≤C
∞
∞ k
−r
ln
P sup ani Xni ≥ n x dx
nr
k≥1 i1
β−r
n
∞ −r
ln
P sup|ani Xni | > n x dx
nr
n1
i
∞ ∞
β−r
−r C
dx : I1 I2 .
n ln
P sup|Snk x − ESnk x| ≥ n x
2
nr
k≥1
n1
Noting that α β > −1, by Lemma 1.5, Markov inequality, 1.6, and 3.1, we have
I1 ≤ C
∞
nβ−r ln
nr
n1
≤C
∞
nβ−r ln
∞
nβα ln
≤C
n1
i1
nr x−1
∞
∞
E|ani Xni |I |ani Xni | > n−r x dx
i1
x−1 E|X|I|X| > xdx
nr
n1
∞
∞
P |ani Xni | > n−r x dx
nr
n1
≤C
∞ ∞
βα
n
ln
∞ kr1
kn
kr
x−1 E|X|I|X| > xdx
3.3
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International Journal of Mathematics and Mathematical Sciences
≤C
∞
nβα ln
n1
≤C
∞
∞
k−1 E|X|I|X| > kr ≤ C
kn
∞
k−1 E|X|I|X| > kr k
nβα ln
n1
k1
kβα lkE|X|I|X| > kr ≤ CE|X|11αβ/r l |X|1/r < ∞.
k1
3.4
Now we estimate I2 , noting that
sup E
1≤m<∞
m
∞
m1
ϕ1/2 m < ∞, by Lemma 1.7, we have
−r
ani Xni I |ani Xni | ≤ n x − E
i1
≤C
∞
m
2
−r
ani Xni I |ani Xni | ≤ n x
i1
2
Ea2ni Xni
I
−r
3.5
|ani Xni | ≤ n x .
i1
By Lemma 1.6, Markov inequality, Cr inequality, and 1.5, for any q ≥ 2, we have
≤ Cnrq x−q Esup|Snk x − ESnk x|q
P sup|Snk x − ESnk x| ≥ n x
2
k≥1
k≥1
⎤
⎡
q/2
∞
∞
2
≤ Cnrq x−q ⎣
Ea2ni Xni
I |ani Xni | ≤ n−r x
E|ani Xni |q I |ani Xni | ≤ n−r x ⎦
−r
i1
rq −q
≤ Cn x
∞
i1
Ea2ni X 2 I
q/2
−r
|ani X| ≤ n x
Cnrq x−q
i1
∞
3.6
E|ani X|q I |ani X| ≤ n−r x
i1
q/2
∞
∞
−r
C
P |ani X| > n x
C
P |ani X| > n−r x
i1
i1
: J1 J2 J3 J4 .
So,
I2 ≤
∞
n1
nβ−r ln
∞
nr
J1 J2 J3 J4 dx.
∞
β−r
From 3.4, we have ∞
ln nr J4 dx < ∞.
n1 n
For J1 , we consider the following two cases.
3.7
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9
If s > 2, then EX 2 < ∞. Taking q ≥ 2 such that β qα − r/2 < −1, we have
∞
β−r
n
∞
ln
J1 dx
nr
n1
≤C
∞
β−rrq
n
∞
ln
x
∞
∞
nr
n1
≤C
−q
q/2
a2ni
3.8
dx
i1
nβ−rrq lnnqα−r/2 nr−q1 ≤ C
n1
∞
nβqα−r/2 ln < ∞.
n1
If s ≤ 2, we choose s such that 1 α/r < s < s. Taking q ≥ 2 such that β qr/21 α/r −
s < −1, we have
∞
β−r
n
∞
ln
J1 dx
nr
n1
≤C
∞
β−rrq
n
∞
ln
x
nr
n1
≤C
∞
−q
∞
|ani ||ani |
i1
nβ−rrq lnnqα/2 n−qr/2s −1
∞
∞
x
E|ani X|
−q
n−r x
2−s
s
q/2
−r
|X| I |ani X| ≤ n x
q/22−s dx
3.9
dx
nr
n1
≤C
s −1
nβqr/21α/r−s ln < ∞.
n1
∞
nβ−r ln nr J1 dx < ∞.
Now, we estimate J2 . Set Inj {i ≥ 1 | nj 1−r < |ani | ≤ nj−r }, j 1, 2, . . .. Then
∪j≥1 Inj N, where N is the set of positive integers. Note also that for all k ≥ 1, n ≥ 1,
So,
∞
n1
nα ≥
∞
|ani | i1
≥
∞ |ani |
j1 i∈Inj
∞
∞
−r
−rq
≥ n−r
Inj n j 1
Inj j 1
k 1rq−r .
j1
3.10
jk
Hence, we have
∞
Inj j −rq ≤ Cnαr kr−rq .
jk
3.11
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International Journal of Mathematics and Mathematical Sciences
Note that
∞
β−r
n
∞
ln
J2 dx
nr
n1
C
∞
β−rrq
n
∞
C
β−rrq
n
ln
n1
≤C
∞
C
nβ−rrq ln
∞
nβ−r ln
∞
Inj
nj
∞
−rq ∞
Inj
∞
nβ−r ln
∞
β−r
n
ln
n1
kr
nj
∞
−rq kr−q1−1
r kr−q1−1 E|X|q I |X| ≤ k 1r j 1
∞
3.12
Inj j −rq
k1j1−1
kr−q1−1
∞
2k1−1
E|X|q I ir < |X| ≤ i 1r
Inj j −rq
i0
j1
k
E|X|q I ir < |X| ≤ i 1r
i0
j1
∞
r x−q E|X|q I |X| ≤ x j 1 dx
kn
kn
∞
k1r
kn
kn
n1
C
j1 i∈Inj
j1
n1
≤C
E|ani X|q I |ani X| ≤ n−r x dx
j1
n1
∞
∞ nr
n1
∞
x−q
ln
r−q1−1
∞
Inj j
−rq
j1
kn
k1j1
E|X|q I ir < |X| ≤ i 1r
i2k1
: J2 J2 .
Taking q ≥ 2 large enough such that β α − rq r < −1, for J2 , by Lemma 1.6 and 3.11, we
get
J2 ≤ C
∞
nβ−r ln
n1
C
∞
∞
kr−q1−1 nαr
kr−q1−1
2k1−1
k
E|X|q I ir < |X| ≤ i 1r
nβα ln
n1
2k1−1
E|X|q I ir < |X| ≤ i 1r
i0
k1
∞
i3
≤CC
E|X|q I ir < |X| ≤ i 1r
i0
kβα−rqr lk
≤CC
2k1−1
i0
kn
k1
≤C
∞
∞
i3
∞
E|X|q I ir < |X| ≤ i 1r
kβα−rqr lk
ki/2
iβα−rqr1 liE|X|q I ir < |X| ≤ i 1r ≤ C CE|X|1βα1/r l |X|1/r < ∞.
3.13
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11
For J2 , we obtain
J2 ≤ C
∞
nβ−r ln
n1
≤C
∞
∞
nβ−r ln
C
nβ−r ln
∞
∞
∞
k−1
∞
i2k1
∞
∞
kβα lk
jik1−1 −1
i2k1
kr−q1−1
∞
E|X|q I ir < |X| ≤ i 1r
Inj j −rq
nrα ir1−q k−r1−q E|X|q I ir < |X| ≤ i 1r
i2k1
3.14
ir1−q E|X|q I ir < |X| ≤ i 1
k
r
nβα ln
n1
i2k1
k1
≤C
kr−q1−1
kn
k1
≤C
∞
j1k1
E|X|q I ir < |X| ≤ i 1r
Inj j −rq
∞
j1
kn
n1
∞
kr−q1−1
kn
n1
≤C
∞
∞
ir1−q E|X|q I ir < |X| ≤ i 1r
i2k1
iβα1r−rq E|X|q I ir < |X| ≤ i 1r ≤ CE|X|1βα1/r l |X|1/r < ∞.
i4
∞
∞
β−r
β−r
So ∞
ln nr J2 dx < ∞. Finally, we prove ∞
ln nr J3 dx < ∞. In fact, noting
n1 n
n1 n
1 a/r < s < s and β qr/21 α/r − s < −1, using Markov inequality and 3.1, we
get
∞
n1
β−r
n
∞
ln
J3 dx ≤ C
nr
∞
β−r
n
ln
∞ ∞
nr
n1
≤C
∞
β−r
n
rs −s
n x E|ani X|
i1
qrs /2 −rs −1q/2 αq/2
lnn
n
∞
n1
q/2
dx
∞
n
x−s q/2 dx
nr
n1
≤C
s
nβ−rrq/2αq/2 lnnr−s q/21 ≤ C
∞
nβqr/21α/2−s ln < ∞.
n1
3.15
Thus, we complete the proof in a. Next, we prove b. Note that E|X|1α/r < ∞ implies that
3.2 holds. Therefore, from the proof in a, to complete the proof of b, we only need to
prove
I2 C
∞
n1
−1−r
n
∞ −r dx < ∞.
ln
P sup|Snk x − ESnk x| ≥ n x
2
nr
k≥1
3.16
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International Journal of Mathematics and Mathematical Sciences
In fact, noting β −1, α β 1 > 0, α β − r < −1 and E|X|1α/r l|X|1/r < ∞. By taking q 2
in the proof of 3.12, 3.13, and 3.14, we get
C
∞
−1r
n
∞
ln
x−2
∞
nr
n1
Ea2ni X 2 I |ani X| ≤ n−r x dx ≤ C CE|X|1α/r l |X|1/r < ∞. 3.17
i1
Then, by 3.17, we have
I2 ≤ C
∞
n−1−r ln
nr
n1
≤C
∞
n−1r ln
≤C
∞
n−1r ln
∞
∞
n−1−r ln
n1
≤C
∞
x−2
n−1r ln
x−2
∞
2
Ea2ni Xni
I |ani Xni | ≤ n−r x dx
Ea2ni X 2 I |ani X| ≤ n−r x dx
3.18
i1
∞ ∞
∞
nr i1
x−2
nr
n1
∞
i1
nr
n1
C
n2r x−2 E|Sxn − ESxn |2 dx
nr
n1
∞
∞
P |ani X| > n−r x dx
∞
Ea2ni X 2 I |ani X| ≤ n−r x dx C < ∞.
i1
The proof of Theorem 2.1 is completed.
Proof of Corollary 2.4. Note that
∞
k
ani Xni − ≤ sup ani Xni − .
i1
k≥1 i1
3.19
Therefore, 2.8 and 2.9 hold by Theorem 2.1.
Proof of Corollary 2.5. By applying Theorem 2.1, taking β p − 2, ani n−1/t for 1 ≤ i ≤ n,
and ani 0 for i > n, then we obtain 2.10. Similarly, taking β −1, ani n−1/t for 1 ≤ i ≤ n,
and ani 0 for i > n, we obtain 2.11 by Theorem 2.1.
Proof of Corollary 2.6. Let Xni Yi and ani n−1/t nj1 aij for all n ≥ 1, −∞ < i < ∞.
∞
1−1/t
. By applying
Since −∞ |ai | < ∞, we have supi |ani | On−1/t and ∞
i−∞ |ani | On
Corollary 2.4, taking β r/t − 2, r 1/t, α 1 − 1/t, we obtain
∞
n1
r/t−2−1/t
n
n
∞
∞
1/t
β
lnE Xi − n
n lnE a X − < ∞,
i1 i−∞ ni ni n1
Therefore, 2.12 and 2.13 hold.
∀ > 0.
3.20
International Journal of Mathematics and Mathematical Sciences
13
Acknowledgment
The paper is supported by the National Natural Science Foundation of China no. 11271020
and 11201004, the Key Project of Chinese Ministry of Education no. 211077, the Natural
Science Foundation of Education Department of Anhui Province KJ2012ZD01, and the
Anhui Provincial Natural Science Foundation no. 10040606Q30 and 1208085MA11.
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