Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2011, Article ID 320932, 17 pages
doi:10.1155/2011/320932
Research Article
Precise Asymptotics in the Law of
Iterated Logarithm for Moving Average
Process under Dependence
Jie Li1, 2
1
2
Wang Yanan Institute for Studies in Economics, Xiamen University, Xiamen 361005, China
School of Mathematics and Statistics, Zhejiang University of Finance and Economics,
Hangzhou 310018, China
Correspondence should be addressed to Jie Li, lijiezufe@gmail.com
Received 10 November 2010; Revised 2 February 2011; Accepted 3 March 2011
Academic Editor: Ulrich Abel
Copyright q 2011 Jie Li. 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.
Let {ξi , −∞ < i < ∞} be a doubly infinite sequence of identically distributed and φ-mixing random
In this
variables, and let {ai , −∞ < i < ∞} be an absolutely summable sequence of real numbers.
paper, we get precise asymptotics in the law of the logarithm for linear process {Xk ∞
i−∞ aik ξi ,
k ≥ 1}, which extend Liu and Lin’s 2006 result to moving average process under dependence
assumption.
1. Introduction and Main Results
Let {ξi , −∞ < i < ∞} be a doubly infinite sequence of identically distributed random
variables with zero means and finite variances, and let {ai , −∞ < i < ∞} be an absolutely
summable sequence of real numbers. Let
Xk ∞
aik ξi ,
k ≥ 1,
1.1
i−∞
be the moving average process based on {ξi , −∞ < i < ∞}. As usual, we denote Sn nk1 Xk ,
n ≥ 1 as the sequence of partial sums.
Under the assumption that {ξi , −∞ < i < ∞} is a sequence of independent identically
distributed random variables, many limiting results have been obtained. Ibragimov 1
established the central limit theorem; Burton and Dehling 2 obtained a large deviation
principle; Yang 3 established the central limit theorem and the law of the iterated logarithm;
2
Journal of Inequalities and Applications
Li et al. 4 obtained the complete convergence result for {Xk , k ≥ 1}. As we know, Xk k ≥ 1
are dependent even if {ξi , −∞ < i < ∞} is a sequence of i.i.d. random variables. Therefore,
we introduce the definition of φ-mixing,
k
∞
−→ 0,
, P A /
0, B ∈ Fkm
φm : sup |P B | A − P B|, A ∈ F−∞
m −→ ∞,
1.2
k≥1
where Fab σξi , a ≤ i ≤ b. Many limiting results of moving average for φ-mixing have been
obtained. For example, Zhang 5 got complete convergence.
Theorem A. Suppose that {ξi , −∞ < i < ∞} is a sequence of identically distributed and φ-mixing
1/2
m < ∞, and {Xk , k ≥ 1} is defined as 1.1. Let hx > 0 x >
random variables with ∞
m1 φ
0 be a slowly varying function and 1 ≤ t < 2, r ≥ 1, then Eξ1 0 and E|ξ1 |rt h|Ytt | < ∞ imply
∞
nr−2 hnP |Sn | ≥ n1/t < ∞,
∀ > 0.
1.3
n1
Li and Zhang 6 achieved precise asymptotics in the law of the iterated logarithm.
Theorem B. Suppose that {ξi , −∞ < i < ∞} is a sequence of identically distributed and φ-mixing
1/2
m < ∞, and 0 < σ 2 Eξ12 random variables with mean zeros and finite variances, ∞
m1 φ
n
δ−1
< ∞, for δ > 0. Suppose that {X, Xk , k ≥ 1} is defined as in
2 k2 Eξ1 ξk < ∞, Eξ12 log |ξ1 |
1.1, where {ai , −∞ < i < ∞} is a sequence of real number with ∞
i−∞ |ai | < ∞, then one has
lim 0
where τ : σ
∞
i−∞
2δ2
∞ log n δ τ 2δ2
P |Sn | ≥ n log nτ E|N|2δ2 ,
n
δ
1
n2
1.4
ai , N is a standard normal random variable.
On the other hand, since Hsu and Robbins 7 introduced the concept of the complete
convergence, there have been extensions in some directions.
For the case of i.i.d. random
variables, Davis 8 proved ∞
log
n/nP
|S
|
≥
n
log
n
< ∞, for > 0 if and
n
n1
only if EX1 0, EX12 < ∞. Gut and Spătaru 9 gave the precise asymptotics of
∞
δ
n log n. We know that complete convergence can be derived
n2 log n /nP |Sn | ≥
from complete moment convergence. Liu
and Lin 10 introduced a new kind of convergence
∞
δ−1
2
2
of n2 log n /n E|Sn | I{|Sn | ≥
n log n}. In this note, we show that the precise
asymptotics for the moment convergence hold for moving-average process when {ξi , −∞ <
i < ∞} is a strictly stationary φ-mixing sequences. Now, we state the main results.
Theorem 1.1. Suppose that {X, Xk , k ≥ 1} is defined as in 1.1, where {ai , −∞ < i < ∞} is a
sequence of real number with ∞
i−∞ |ai | < ∞, and {ξi , −∞ < i < ∞} is a sequence of identically
Journal of Inequalities and Applications
3
1/2
m < ∞ and
distributed φ-mixing random variables with mean zeros and finite variances, ∞
m1 φ
n
δ
2
2
2
0 < σ Eξ1 2 k2 Eξ1 ξk < ∞, Eξ1 log |ξ1 | < ∞, for 0 < δ ≤ 1, then one has
lim 0
where τ : σ
∞
i−∞
2δ
∞
log nδ−1
n2
n2
2
E|Sn | I |Sn | ≥
n log nτ
τ 2δ2
E|N|2δ2 ,
δ
1.5
ai .
Theorem 1.2. Under the conditions in Theorem 1.1, one has
∞ log log n δ−1
τ 2δ2 E|N|2δ1
2
lim .
ES
I
n
log
log
nτ
≥
|S
|
n
n
0
δ
n2 log n
n3
2δ
1.6
Remark 1.3. In this paper, we generate the results of Liu and Lin 10 to linear process under
dependence based on Theorem B by using the technique of dealing with the innovation
process in Zhang 5.
We first proceed with some useful lemmas.
Lemma 1.4. Let {X, Xk , k ≥ 1} be defined as in 1.1, and let {ξi , −∞ < i < ∞} be a sequence
of identically distributed φ-mixing random variables with Eξ1 0, Eξ12 < ∞, 0 < σ 2 Eξ12 ∞
1/2 m
2 < ∞, then
2 ∞
m1 φ
k2 Eξ1 ξk < ∞,
Sn D
√ −→ N0, 1.
τ n
The proof is similar to Theorem 1 in 11. Set Δn supx |P |Sn | ≥
From Lemma 1.4, one can get Δn → 0 as n → ∞.
Lemma 1.5 see 2. Let
∞
i−∞ ai and k ≥ 1, then
∞
i−∞
1.7
√
nx − P |N| ≥ x|.
ai be an absolutely convergent series of real numbers with a k
∞ in
1 lim
aj |a|k .
n→∞n
i−∞ji1 1.8
Lemma 1.6 see 12. Let {Xi , i ≥ 1} be a sequence of φ-mixing random variables with zero means
and finite second moments. Let Sn ni1 Xi . If exists Cn such that max1≤i≤n ES2n ≤ Cn , then for all
q ≥ 2, there exists C Cq, φ· such that
q/2
q
E max |Si | ≤ C Cn E max |Xi | .
q
1≤i≤n
1≤i≤n
1.9
4
Journal of Inequalities and Applications
2. Proofs
Proof of Theorem 1.1. Without loss of generality, we assume that τ 1. We have
∞ log n δ−1
2
ESn I |Sn | ≥ n log n
n2
n2
∞ log n δ−1 ∞
∞ log n δ P |Sn | ≥ n log n 2xP |Sn | ≥ xdx.
√
n
n2
n log n
n2
n2
2
2.1
Set d expM−2 , where M > 1. By Theorem B, we need to show
lim 2δ
0
∞ log n δ−1 ∞
1
E|N|2δ2 .
2xP |Sn | ≥ xdx √
2
δδ
1
n
n log n
n2
2.2
By Proposition 5.1 in 10, we have
∞ log n δ−1 ∞
1
x
lim
E|N|2δ2 .
dx 2xP
≥
|N|
√
√
2
0
δδ 1
n
n
n log n
n2
2.3
Hence, Theorem 1.1 will be proved if we show the following two propositions.
Proposition 2.1. One has
lim 2δ
0
d
δ−1 ∞
∞
log n
x
2xP |Sn | ≥ xdx − √
2xP |N| ≥ √ dx 0.
√
n log n
n2
n
n log n
n2
2.4
Proof. Write
d
n2
δ−1 ∞
∞
log n
x
2xP |Sn | ≥ xdx − √
2xP |N| ≥ √ dx
√
n log n
n2
n
n log n
d
δ−1
log n
≤
log n
n
n2
∞
×
2x P |Sn | ≥ n log nx − P |N| ≥ log nx dx
0
≤
d
log n δ−1
n2
n
Δn1 Δn2 Δn3 ,
2.5
Journal of Inequalities and Applications
5
where
Δn1 log n
1/√log nΔ1/4
n
0
Δn2 log n
∞
1/
Δn3 log n
√
∞
1/
√
log nΔ1/4
n
log nΔ1/4
n
2x P |Sn | ≥ n log nx − P |N| ≥ log nx dx,
2x P |Sn | ≥ n log nx dx,
2x P |N| ≥ log nx dx.
2.6
Since n ≤ d implies
√
log n ≤ M, we have
⎛
⎞2
⎜
Δn1 ≤ C log nΔn ⎝ 1
log nΔ1/4
n
2
⎟
⎠ ≤ C Δ1/4
MΔn .
n 2.7
For Δn3 , by Markov’s inequality, we get
Δn3 ≤ C log n
∞
1/
√
log nΔ1/4
n
1
dx ≤ CΔ1/4
3/2
n .
2
log n
x 2.8
From 2.7 and 2.8, we can get
lim 2δ
0
d
n2
δ−1
log n
Δn1 Δn3 0.
n
2.9
n
∞
n
Note that nk1 Xk ∞
i−∞
i−∞ ani ξi , where ani k1 aki ξi k1 aki . By Lemma 1.5,
we can assume that
∞
|ani |t ≤ n,
i−∞
Set S
n ∞
i−∞
ani ξi I{|ani ξi | ≤
t ≥ 1,
a ∞
|ai | ≤ 1.
i−∞
n log nx }. As Eξ1 0, by 2.10, we have
∞
ESn ≤ CE
a ξ I |ani ξi | > n log nx i−∞ ni i
≤C
∞
|ani |E|ξ1 |I |ani ξ1 | > n log nx i−∞
≤ CnE|ξ1 |I a|ξ1 | > n log nx 2.10
6
Journal of Inequalities and Applications
≤ CnE|ξ1 |I |ξ1 | > n log nx 1/2
1/2 ≤ Cn Eξ12
P |ξ1 | > n log nx ≤ C
√
nEξ12
log nx .
2.11
So, when x ∈ 1/ log nΔ1/4
n , ∞,
|ES
n |
n log nx ≤C
Eξ12
2 < ,
log n 1/ log nΔ1/4
n
for n large enough.
2.12
By 2.12, we have
d
n2
δ−1
log n
Δn2
n
≤
d
∞
n2
1/
√
⎡
log nΔ1/4
n
⎢
× ⎣P
δ
log n x n
⎛
⎞⎤
n
log
nx
⎜
⎟⎥
sup|ani ξi | > n log nx P ⎝S
n − ES
n ≥
⎠⎦dx
2
i
: H1 H2 .
2.13
Set Inj {j ∈ L, 1/j 1 < |anj | ≤ 1/j, j 1, 2, . . .}, then
can get
k
#Inj ≤ nk 1.
j1
Then,
"
P sup|ani ξi | >
#
n log nx i
≤
∞
P |ani ξi | > n log nx i−∞
!
j≥1 Inj
L referred by 4. We
2.14
Journal of Inequalities and Applications
≤
7
∞ ∞
P |ξ1 | ≥ n log njx ≤
#Inj P |ξ1 | ≥ n log njx j1 i∈Inj
≤
j1
n log nkx ≤ |ξ1 | < n log nk 1x #Inj P
∞ j1 k≥j
≤
k
∞ #Inj
n log nkx ≤ |ξ1 | < n log nk 1x P
k1 j1
≤
∞
nk 1P
n log nkx ≤ |ξ1 | < n log nk 1x k1
√
nE|ξ1 |I |ξ1 | ≥ n log nx ≤
.
log nx 2.15
So, we get
H1 ≤ CE|ξ1 |
d
δ−1/2
log n
∞
1/
√
×
n1/2
log nΔ1/4
n
n2
∞ I
k log kx < |ξ1 | ≤ k 1 logk 1x dx
kn
≤C
∞ ∞
E|ξ1 |I
k log kx < |ξ1 | ≤
0 k2
≤ CEξ12
≤ CEξ12
∞
0
∞
x −1
∞
log k
δ−1
I
k log n δ−1/2
k 1 logk 1x dx
n1/2
n2
k log kx < |ξ1 | ≤ k 1 logk 1x dx
k2
log |ξ1 | − logx δ−1 x −1 I{|ξ1 | > x }dx
0
δ
δ
δ
≤ CEξ12 log |ξ1 | − log ≤ CEξ12 log |ξ1 | CEξ12 − log .
2.16
Therefore,
lim 2δ H1 0.
0
2.17
8
Journal of Inequalities and Applications
By Lemma 1.6, noting that
H2 ≤ C
d
∞
m1
δ−q/2
log n
×
φ1/2 m < ∞, for q > 2,
n1q/2
0
n2
∞
⎧
∞
⎨ ⎩
x 1−q
q/2
Eani ξ1 I |ani ξ1 | ≤ n log nx 2
i−∞
E|ani ξ1 |q I |ani ξ1 | ≤ n log n
∞
2.18
i−∞
⎫
⎬
dx
E|ani ξ1 |q I
n log n < |ani ξ1 | ≤ n log nx ⎭
i−∞
∞
: H21 H22 H23 .
For H21 , we have
H21 ≤
δ−q/2
q/2
log n
1−q
2
dx
Eξ1 I |ani ξ1 | ≤ n log nx x n
d
∞
0
n2
2.19
≤ C−2δ Mδ1−q/2 .
Then, for 0 < δ ≤ 1, q > 2, we have
lim lim sup 2δ H21 0.
M→∞
2.20
0
For H22 , we decompose it into two parts,
H22 ≤
d
∞
n2
≤
2−q
δ−q/2
log n
n1q/2
0
x 1−q
∞ |ani |q E|ξ1 |q I |ani ξ1 | ≤ n log n dx
j1 i∈Inj
d
δ−q/2
log n
n1q/2
⎧
∞
2n
−q ⎨
k log k ≤ |ξ1 | < k 1 logk 1
×
E|ξ1 |q I
#Inj j
⎩ k0
j1
n2
⎫
⎬
E|ξ1 |q I
k log k ≤ |ξ1 | < k 1 logk 1
⎭
k2n1
j1n
: H221 H222 .
2.21
Journal of Inequalities and Applications
9
It is easy to see that
∞
#Inj
∞
∞
∞ −q
−1 j 1 m 1q−1 ≤
#Inj j 1
≤
|ani | |ani | ≤ n.
jm
i−∞
j1
2.22
j1 i∈Inj
So,
∞
#Inj j −q ≤ Cnm−q−1 .
2.23
jm
Now, we estimate H221 , by 2.23,
H221 ≤ 2−q
d
δ−q/2 2n
log n
E|ξ1 |q I
k log k ≤ |ξ1 | <
k 1 logk 1
nq/2
k0
d
d
log n δ−q/2
q
2−q
≤
E|ξ1 | I
k log k ≤ |ξ1 | < k 1 logk 1
nq/2
k2
nk/2
d
log k δ−q/2
q
2−q
≤
E|ξ
I
k
log
k
≤
1
logk
1
<
k
|ξ
|
|
1
1
kq/2−1
k2
d
δ−1 2 Eξ1 I
k log k ≤ |ξ1 | < k 1 logk 1
≤
log k
n2
k2
≤
2.24
δ 2 −1 log |ξ1 | − log Eξ1 I
k log k ≤ |ξ1 | < k 1 logk 1
log k
d
k2
δ
δ
≤ CEξ12 log |ξ1 | CEξ12 − log .
For H222 , we have
H222 ≤ 2−q
d
δ−q/2
log n
n1q/2
−q
q
k log k ≤ |ξ1 | < k 1 logk 1
#Inj j E|ξ1 | I
n2
∞
×
k2n1 j≥k/n−1
≤
2−q
d
δ−q/2 ∞
log n
n2
≤
d
log k
n1−q/2
δ−1
k
1−q
q
E|ξ1 | I
k log k ≤ |ξ1 | <
k 1 logk 1
k2n1
Eξ12 I
k log k ≤ |ξ1 | < k 1 logk 1
k2
≤
d
log k
−1 log |ξ1 | − log δ Eξ2 I
k
log
k
≤
1
logk
1
<
k
|ξ
|
1
1
k2
δ
≤ CEξ12 log |ξ1 | CEξ12 − log δ .
2.25
10
Journal of Inequalities and Applications
From 2.24 and 2.25, we can get
lim 2δ H22 lim 2δ H221 lim 2δ H222 0.
0
0
0
2.26
Finally, q > 2, and we will get
H23 ≤ C
d
δ−q/2
log n
n1q/2
n2
×
∞
x q
E|ξ1 | I |
aξ1 | >
∞
1−q
#Inj j −q
0
j1
×I
≤C
d
δ−q/2
log n
nq/2
n2
⎧
2n
⎨
⎩ k0
n log n
⎫
j1
n ⎬
k2n1
⎭
k log kx < |ξ1 | ≤ k 1 logk 1x dx
E|ξ1 |q I |
aξ1 | > n log n
d
log n δ−q/2
×
E|ξ1 |q
x I |ξ1 | ≤ 2n log2nx dx C
1−q/2
n
0
n2
∞ ∞
1−q
k
×
I
k
log
kx
<
1
logk
1x
dx
≤
k
|ξ
|
1
q−1
0 k2n1 x ∞
≤C
∞
1−q
Eξ12 I
k2
CEξ12
∞
k log n δ−1
k log k < |ξ1 | ≤ k 1 logk 1
n
n2
x −1
0
×
∞
log k
δ−1
I
k log kx < |ξ1 | ≤ k 1 logk 1x dx
k4
≤C
∞
δ
k log k < |ξ1 | ≤ k 1 logk 1
log k Eξ12 I
k2
CEξ12
∞
log |ξ1 | − logx δ−1 x −1 I{|ξ1 | > x }dx
0
δ
δ
δ
≤ CEξ12 log |ξ1 | − log ≤ CEξ12 log |ξ1 | CEξ12 − log ,
2.27
then
lim 2δ H23 0.
0
Hence, 2.4 can be referred from 2.9, 2.17, 2.20, 2.26, and 2.28.
2.28
Journal of Inequalities and Applications
11
Proposition 2.2. One has
lim 0
δ−1 ∞
∞
log n
x
2xP |Sn | ≥ xdx − √
2xP |N| ≥ √ dx 0.
√
n log n
n2
n
n log n
∞
2δ
nd1
2.29
Proof. Consider the following:
δ−1 ∞
∞
log n
x
2xP |Sn | ≥ xdx − √
2xP |N| ≥ √ dx
√
n log n
n2
n
n log n
∞
nd1
∞
≤
nd1
δ ∞
log n
2x P |N| ≥ log nx n
0
2.30
∞
nd1
δ ∞
log n
2x P |Sn | ≥ n log nx dx
n
0
: G1 G2 .
We first estimate G1 , for θ > 2δ, by Markov’s inequality,
G1 ≤
∞
nd1
≤ CM
δ ∞
log n
1
dx
θ2/2
n
0
log n
x θ1
δ−θ/2 −2δ
2.31
.
Hence,
lim lim sup 2δ G1 0.
M→∞
2.32
0
Now, we estimate G2 . Here, n > M−2 , so
|ES
n |
n log nx Eξ12
Eξ12
< ,
<
M
log n x 2
<
for M −→ ∞.
2.33
12
Journal of Inequalities and Applications
We have
G2 ≤
⎡
δ ∞
log n
⎢
x ⎣P sup|ani ξi | > n log nx n
0
i
∞
nd1
⎛
⎞⎤
n
log
nx
⎜
⎟⎥
P ⎝S
n − ES
n ≥
⎠⎦dx
2
2.34
: G21 G22 .
We estimate G21 first. Similar to the proof of 2.16, we have
G21 ≤ CE|ξ1 |
∞
δ−1/2
log n
∞
n1/2
0 nd1
∞ × I
k log kx < |ξ1 | ≤ k 1 logk 1x dx
≤ CE|ξ1 |
kn
∞
∞
I
k log kx < |ξ1 | ≤ k 1 logk 1x dx
0 kd1
×
δ−1/2
log n
k
nd1
≤ CEξ12
∞
n1/2
∞
δ−1
log n
x −1
0 kd1
×I
≤ CEξ12
∞
k log kx < |ξ1 | ≤
k 1 logk 1x dx
log |ξ1 | − logx δ−1 x −1 I{|ξ1 | > x }dx
0
δ
δ
δ
≤ CEξ12 log |ξ1 | − log ≤ CEξ12 log |ξ1 | CEξ12 − log ,
2.35
then
lim lim sup 2δ G21 0.
M→∞
0
2.36
Journal of Inequalities and Applications
13
By Lemma 1.6, for q > 2, we have
G22 ⎫
⎧
δ ∞
⎪
⎪
⎬
⎨
n
log
nx
log n
dx
x P Sn − ESn ≥
⎪
⎪
n
2
⎭
⎩
0
∞
nd1
≤
δ−q/2
log n
∞
n1q/2
⎧
∞
q/2
∞
⎨ 2
1−q
×
Eani ξ1 I |ani ξi | ≤ n log nx x ⎩ i−∞
0
nd1
∞
E|ani ξ1 |q I |ani ξi | ≤ n log n
i−∞
∞
E|ani ξ1 |q I
n log n < |ani ξi | ≤
i−∞
⎫
⎬
n log nx dx
⎭
: G221 G222 G223 .
2.37
For G221 , we have
G221 ≤
∞
∞
nd1
0
q/2
log nδ−q/2
dx
x 1−q Eξ12 I{|ani ξ1 | ≤ nx }
n
≤ C2−q
2.38
∞
δ−q/2
log n
≤ CMδ1−q/2 −2δ .
n
nd1
Next, turning to G222 , it follows that
G222 ≤ δ−q/2
log n
∞
2−q
nd1
×
∞
#Inj j −q
j1
n1q/2
⎧
2n
⎨
⎩ k2
E|ξ1 |q I
j1n
k2n1
: G2221 G2222 ,
E|ξ1 |q I
k log k ≤ |ξ1 | <
k 1 logk 1
k log k ≤ |ξ1 | <
k 1 logk 1
⎫
⎬
⎭
2.39
14
Journal of Inequalities and Applications
then
∞
G2221 ≤ C2−q
δ−q/2 2n
log n
q
E|ξ
I
k
log
k
≤
1
logk
1
<
k
|ξ
|
|
1
1
nq/2
k2
nd1
≤ C
2−q
∞
q
E|ξ1 | I
k2
≤ C
2−q
∞ log k δ−q/2
kq/2−1
k2
≤C
δ−q/2
∞
log n
k log k ≤ |ξ1 | < k 1 logk 1
nq/2
nk/2
∞
log k
δ−1
Eξ12 I
E|ξ1 |q I
k log k ≤ |ξ1 | < k 1 logk 1
k log k ≤ |ξ1 | < k 1 logk 1
k2
≤C
δ 2 −1 log |ξ1 | − log Eξ1 I
k log k ≤ |ξ1 | < k 1 logk 1
log k
∞
k2
δ
δ
≤ CEξ12 log |ξ1 | CEξ12 − log .
2.40
For G2222 , it follows that
G2222 ≤ C
δ−q/2
log n
∞
2−q
n1q/2
nd1
k log k ≤ |ξ1 | < k 1 logk 1
#Inj j −q E|ξ1 |q I
∞
×
k2n1 j≥k/n−1
≤ C
2−q
∞
k
1−q
q
E|ξ1 | I
kd1
≤C
∞
log k
δ−1
Eξ12 I
k/2
log n δ−q/2
k log k ≤ |ξ1 | < k 1 logk 1
n1−q/2
nd1
k log k ≤ |ξ1 | < k 1 logk 1
kd1
≤C
δ 2 −1 log |ξ1 | − log Eξ1 I
k log k ≤ |ξ1 | < k 1 logk 1
log k
∞
kd1
δ
δ
≤ CEξ12 log |ξ1 | CEξ12 − log .
2.41
Journal of Inequalities and Applications
15
Finally, q > 2, we have
G223 ≤ C
δ−q/2
log n
∞
n1q/2
nd1
×
∞
0
E|ξ1 |q I |
aξ1 | > n log n
⎧
⎫
n ⎬
j1
∞
2n
⎨
#Inj j −q
x 1−q
⎩ k0 k2n1⎭
j1
×I
≤C
δ−q/2
log n
∞
nq/2
nd1
×
∞
1−q
x 0
×
∞ ∞
0
≤
k log kx < |ξ1 | ≤ k 1 logk 1x dx
E|ξ1 | I |
aξ1 | > n log n
q
δ−q/2
∞
log n
I |ξ1 | ≤ 2n log2nx dx C
E|ξ1 |q
1−q/2
n
nd1
k1−q
I
q−1
k2n1 x ∞
Eξ12 I
C
kd1
CEξ12
∞
k log kx < |ξ1 | ≤
k 1 logk 1x dx
δ−1
k
log n
k log k < |ξ1 | ≤ k 1 logk 1
n
nd1
x −1
0
×
∞
log k
δ−1
I
k log kx < |ξ1 | ≤ k 1 logk 1x dx
kd1
≤C
δ
k log k < |ξ1 | ≤ k 1 logk 1
log k Eξ12 I
∞
k2
CEξ12
∞
log |ξ1 | − logx δ−1 x −1 I{|ξ1 | > x }dx
0
δ
δ
δ
≤ CEξ12 log |ξ1 | − log ≤ CEξ12 log |ξ1 | CEξ12 − log .
2.42
From 2.38 to 2.42, we can get
lim lim 2δ G22 0.
M → ∞ 0
2.29 can be derived by 2.32, 2.36, and 2.43.
2.43
16
Journal of Inequalities and Applications
Proof of Theorem 1.2. Without loss of generality, we set τ 1. It is easy to see that
∞ log log n δ−1
2
ESn I |Sn | ≥ n log log n
n2 log n
n3
2
∞ log log n δ P |Sn | ≥ n log log n
n log n
n3
2.44
∞ log log n δ−1 ∞
2xP |Sn | ≥ xdx.
√
n2 log n
n log log n
n3
So, we only prove the following two propositions:
2δ2
∞ log log n δ E|N|2δ1
P |Sn | ≥ n log log n ,
n log n
δ1
n3
∞ log log n δ−1 ∞
E|N|2δ1
.
2xP
≥
xdx
|S
|
n
√
δδ 1
n2 log n
n log log n
n3
2δ
2.45
2.46
The proof of 2.45 can be referred to 6, and the proof of 2.46 is similar to Propositions 2.1
and 2.2.
Acknowledgments
The author would like to thank the referee for many valuable comments. This research was
supported by Humanities and Social Sciences Planning Fund of the Ministry of Education of
PRC. no. 08JA790118 References
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