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Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 130915, 10 pages
doi:10.1155/2010/130915
Research Article
Almost Sure Central Limit Theorem for a
Nonstationary Gaussian Sequence
Qing-pei Zang
School of Mathematical Science, Huaiyin Normal University, Huaian 223300, China
Correspondence should be addressed to Qing-pei Zang,
Received 4 May 2010; Revised 7 July 2010; Accepted 12 August 2010
Academic Editor: Soo Hak Sung
Copyright q 2010 Qing-pei Zang. 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 {X
n
; n ≥ 1} be a standardized non-stationary Gaussian sequence, and let denote S
n
n
k1
X
k
,
σ
n
VarS
n
. Under some additional condition, let the constants {u
ni
;1≤ i ≤ n, n ≥ 1} satisfy
n
i1
1 −Φu
ni
→ τ as n →∞for some τ ≥ 0andmin
1≤i≤n
u
ni
≥ clog n
1/2
,forsomec>0, then,
we have lim
n →∞
1/ log n
n
k1
1/kI{∩
k
i1
X
i
≤ u
ki
,S
k
/σ
k
≤ x} e
−τ
Φx almost surely for any
x ∈ R,whereIA is the indicator function of the event A and Φx stands for the standard normal
distribution function.
1. Introduction
When {X, X
n
; n ≥ 1} is a sequence of independent and identically distributed i.i.d. random
variables and S
n
n
k1
X
k
,n ≥ 1,M
n
max
1≤k≤n
X
k
for n ≥ 1. If EX0, VarX1, the
so-called almost sure central limit theorem ASCLT has the simplest form as follows:
lim
n →∞
1
log n
n
k1
1
k
I
S
k
√
k
≤ x
Φ
x
,
1.1
almost surely for all x ∈ R, where IA is the indicator function of the event A and
Φx stands for the standard normal distribution function. This result was first proved
independently by Brosamler 1 and Schatte 2 under a stronger moment condition; since
then, this type of almost sure version was extended to different directions. For example,
Fahrner and Stadtm
¨
uller 3 and Cheng et al. 4 extended this almost sure convergence for
partial sums to the case of maxima of i.i.d. random variables. Under some natural conditions,
they proved as follows:
lim
n →∞
1
log n
n
k1
1
k
I
M
k
− b
k
a
k
≤ x
G
x
a.s. 1.2
2 Journal of Inequalities and Applications
for all x ∈ R, where a
k
> 0andb
k
∈ R satisfy
P
M
k
− b
k
a
k
≤ x
−→ G
x
, as k −→ ∞ 1.3
for any continuity point x of G.
In a related work, Cs
´
aki and Gonchigdanzan 5 investigated the validity of 1.2 for
maxima of stationary Gaussian sequences under some mild condition whereas Chen and
Lin 6 extended it to non-stationary Gaussian sequences. Recently, Dudzi
´
nski 7 obtained
two-dimensional version for a standardized stationary Gaussian sequence. In this paper,
inspired by the above results, we further study ASCLT in the joint version for a non-stationary
Gaussian sequence.
2. Main Result
Throughout this paper, let {X
n
; n ≥ 1} be a non-stationary standardized normal sequence,
and σ
n
VarS
n
.Herea b and a ∼ b stand for a Ob and a/b → 1, respectively.
Φx is the standard normal distribution function, and φx is its density function; C will
denote a positive constant although its value may change from one appearance to the next.
Now, we state our main result as follows.
Theorem 2.1. Let {X
n
; n ≥ 1} be a sequence of non-stationary standardized Gaussian variables
with covariance matrix r
ij
such that 0 ≤ r
ij
≤ ρ
|i−j|
for i
/
j,whereρ
n
≤ 1 for all n ≥ 1 and
sup
s≥n
s−1
is−n
ρ
i
log n
1/2
/log log n
1ε
,ε>0. If the constants {u
ni
;1 ≤ i ≤ n,n ≥ 1} satisfy
n
i1
1 − Φu
ni
→ τ as n →∞for some τ ≥ 0 and min
1≤i≤n
u
ni
≥ clog n
1/2
,forsomec>0,
then
lim
n →∞
1
log n
n
k1
1
k
I
k
i1
X
i
≤ u
ki
,
S
k
σ
k
≤ x
e
−τ
Φ
x
, 2.1
almost surely for any x ∈ R.
Remark 2.2. The condition sup
s≥n
s−1
is−n
ρ
i
log n
1/2
/log log n
1ε
,ε > 0 is inspired by
a1 in Dudzi
´
nski 8, which is much more weaker.
3. Proof
First, we introduce the following lemmas which will be used to prove our main result.
Lemma 3.1. Under the assumptions of Theorem 2.1, one has
1≤i<j≤n
r
ij
exp
−
u
2
ni
u
2
nj
2
1 r
ij
≤
C
log log n
1ε
.
3.1
Proof. This lemma comes from Chen and Lin 6.
Journal of Inequalities and Applications 3
The following lemma is Theorem 2.1 and C orollary 2.1inLiandShao9.
Lemma 3.2. (1) Let {ξ
n
} and {η
n
} be sequences of standard Gaussian variables with covariance
matrices R
1
r
1
ij
and R
0
r
0
ij
, respectively. Put ρ
ij
max|r
1
ij
|, |r
0
ij
|. Then one has
P
⎛
⎝
n
j1
ξ
j
≤ u
j
⎞
⎠
− P
⎛
⎝
n
j1
η
j
≤ u
j
⎞
⎠
≤
1
2π
1≤i<j≤n
arcsin
r
1
ij
− arcsin
r
0
ij
exp
−
u
2
i
u
2
j
2
1 ρ
ij
,
3.2
for any real numbers u
i
, i 1, 2, ,n.
(2) Let {ξ
n
; n ≥ 1} be standard Gaussian variables with r
ij
Covξ
i
,ξ
j
.Then
P
⎛
⎝
n
j1
ξ
j
≤ u
j
⎞
⎠
−
n
j1
P
ξ
j
≤ u
j
≤
1
4
1≤i<j≤n
r
ij
exp
−
u
2
i
u
2
j
2
1
r
ij
, 3.3
for any real numbers u
i
, i 1, 2, ,n.
Lemma 3.3. Let {X
n
} be a sequence of standard Gaussian variables and satisfy the conditions of
Theorem 2.1, then for 1 ≤ k<n, one has
P
n
ik1
{
X
i
≤ u
ni
}
,
S
n
σ
n
≤ y
− P
n
i1
{
X
i
≤ u
ni
}
,
S
n
σ
n
≤ y
≤
k
n
C
log log n
1ε
3.4
for any y ∈ R.
Proof. By the conditions of Theorem 2.1, we have
σ
n
n 2
1≤i<j≤n
r
ij
≥
√
n,
3.5
then, for 1 ≤ i ≤ n,bysup
s≥n
s−1
is−n
ρ
i
log n
1/2
/log log n
1ε
,ε>0, it follows that
Cov
X
i
,
S
n
σ
n
≤
1
√
n
1
√
n
n
k1
ρ
k
log n
1/2
√
n
log log n
1ε
. 3.6
Then, there exist numbers δ, n
0
, such that, for any n>n
0
, we have
sup
1≤i≤n
Cov
X
i
,
S
n
σ
n
<δ<
1
2
. 3.7
4 Journal of Inequalities and Applications
We can write that
L : P
n
ik1
{
X
i
≤ u
ni
}
,
S
n
σ
n
≤ y
− P
n
i1
{
X
i
≤ u
ni
}
,
S
n
σ
n
≤ y
≤
P
n
ik1
{
X
i
≤ u
ni
}
,
S
n
σ
n
≤ y
− P
n
ik1
{
X
i
≤ u
ni
}
P
Y
n
≤ y
P
n
i1
{
X
i
≤ u
ni
}
,
S
n
σ
n
≤ y
− P
n
i1
{
X
i
≤ u
ni
}
P
Y
n
≤ y
P
n
ik1
{
X
i
≤ u
ni
}
− P
n
i1
{
X
i
≤ u
ni
}
: L
1
L
2
L
3
,
3.8
where {Y
n
} is a random variable, which has the same distribution as {S
n
/σ
n
},butit
is independent of X
1
,X
2
, ,X
n
. For L
1
,L
2
, apply Lemma 3.2 1 with ξ
i
X
i
,i
1, ,n; ξ
n1
S
n
/σ
n
, η
j
X
j
,j 1, ,n; η
n1
Y
n
. Then r
1
ij
r
0
ij
r
ij
for 1 ≤ i<j≤ n
and r
1
ij
CovX
i
,S
n
/σ
n
,r
0
ij
0for1≤ i ≤ n, j n 1. Thus, we have for i 1, 2
L
i
n
i1
Cov
X
i
,
S
n
σ
n
exp
−
u
2
ni
y
2
2
1 Cov
X
i
,S
n
/σ
n
. 3.9
Since 3.5, 3.7 hold, we obtain
L
i
log n
1/2
√
n
log log n
1ε
n
i1
exp
−
u
2
ni
2
1 δ
. 3.10
Now define u
n
by 1 − Φu
n
1/n. By the well-known fact
1 − Φ
x
∼
φ
x
x
,x−→ ∞, 3.11
it is easy to see that
exp
−
u
2
n
2
∼
√
2πu
n
n
,u
n
∼
2logn. 3.12
Journal of Inequalities and Applications 5
Thus, according to the assumption min
1≤i≤n
u
ni
≥ clog n
1/2
, we have u
ni
≥ cu
n
for some c>0.
Hence
L
i
≤
log n
1/2
√
n
log log n
1ε
1≤i≤n
exp
−
u
2
ni
2
1 δ
≤
√
n
log n
1/2
log log n
1ε
exp
−
u
2
n
2
1 δ
√
n
2logn
2δ/1δ
n
1/1δ
log log n
1ε
log n
2δ/1δ
n
1/1δ−1/2
1
n
δ
,δ
> 0.
3.13
Now, we are in a position to estimate L
3
. Observe that
L
3
P
n
ik1
{
X
i
≤ u
ni
}
− P
n
i1
{
X
i
≤ u
ni
}
≤
P
n
ik1
{
X
i
≤ u
ni
}
−
n
ik1
Φ
u
ni
P
n
i1
{
X
i
≤ u
ni
}
−
n
i1
Φ
u
ni
n
ik1
Φ
u
ni
−
n
i1
Φ
u
ni
: L
31
L
32
L
33
.
3.14
For L
33
, it follows that
L
33
n
ik1
Φ
u
ni
1 −
k
i1
Φ
u
ni
1 − Φ
k
u
n
1 −
1 −
1
n
k
≤
k
n
.
3.15
By Lemma 3.2 2, we have
L
3i
≤
1
4
1≤i<j≤n
r
ij
exp
−
u
2
ni
u
2
nj
2
1 r
ij
,i 1, 2. 3.16
Thus by Lemma 3.1 we obtain the desired result.
6 Journal of Inequalities and Applications
Lemma 3.4. Let {X
n
} be a sequence of standard Gaussian variables satisfying the c onditions of
Theorem 2.1, then for 1 ≤ k<n, any y ∈ R, one has
Cov
I
k
i1
{
X
i
≤ u
ki
}
,
S
k
σ
k
≤ y
,I
n
ik1
{
X
i
≤ u
ni
}
,
S
n
σ
n
≤ y
k
n
log n
1/2
log log n
1ε
1
log log n
1ε
.
3.17
Proof. Apply Lemma 3.2 1 with ξ
i
X
i
, 1 ≤ i ≤ k, ξ
k1
S
k
/σ
k
,ξ
i1
X
i
,k 1 ≤ i ≤
n, ξ
n2
S
n
/σ
n
, η
j
ξ
j
, 1 ≤ j ≤ k 1,η
j
ξ
j
,k 2 ≤ j ≤ n 2, where ξ
k2
, ,ξ
n2
has
the same distribution as ξ
k2
, ,ξ
n2
, but it is independent of ξ
k2
, ,ξ
n2
. Then,
r
1
ij
r
0
ij
for 1 ≤ i<j≤ k 1ork 2 ≤ i<j≤ n 2;
r
1
ij
r
ij−1
,r
0
ij
0for1≤ i ≤ k, k 2 ≤ j ≤ n 1;
r
1
ij
Cov
X
i
,
S
n
σ
n
,r
0
ij
0for1≤ i ≤ k, j n 2;
r
1
ij
Cov
X
i
,
S
k
σ
k
,r
0
ij
0fork 1 ≤ i ≤ n, j k 1;
r
1
ij
Cov
S
k
σ
k
,
S
n
σ
n
,r
0
ij
0fori k 1,j n 2.
3.18
Thus, combined with 3.5 , 3.7, it follows that
Cov
I
k
i1
{
X
i
≤ u
ki
}
,
S
k
σ
k
≤ y
,I
n
ik1
{
X
i
≤ u
ni
}
,
S
n
σ
n
≤ y
P
k
i1
{
X
i
≤ u
ki
}
,
n
ik1
{
X
i
≤ u
ni
}
,
S
k
σ
k
≤ y,
S
n
σ
n
≤ y
−P
k
i1
{
X
i
≤ u
ki
}
,
S
k
σ
k
≤ y
P
n
ik1
{
X
i
≤ u
ni
}
,
S
n
σ
n
≤ y
≤
1
4
1≤i≤k
k1≤j≤n
r
ij
exp
−
u
2
ki
u
2
nj
2
1 r
ij
1
4
k
i1
Cov
X
i
,
S
n
σ
n
exp
−
u
2
ki
y
2
2
1 Cov
X
i
,S
n
/σ
n
1
4
n
ik1
Cov
X
i
,
S
k
σ
k
exp
−
u
2
ni
y
2
2
1 Cov
X
i
,S
k
/σ
k
1
4
Cov
S
k
σ
k
,
S
n
σ
n
≤
1
4
1≤i≤k
k1≤j≤n
r
ij
exp
−
u
2
ki
u
2
nj
2
1 r
ij
1
4
k
i1
Cov
X
i
,
S
n
σ
n
exp
−
u
2
ki
2
1 δ
1
4
n
ik1
Cov
X
i
,
S
k
σ
k
exp
−
u
2
ni
2
1 δ
1
4
Cov
S
k
σ
k
,
S
n
σ
n
: T
1
T
2
T
3
T
4
.
3.19
Journal of Inequalities and Applications 7
Using Lemma 3.1, we have
T
1
≤
C
log log n
1ε
,ε>0. 3.20
By the similar technique that was applied to prove 3.10,weobtain
T
2
1
n
α
,α>0. 3.21
For T
3
,bysup
s≥n
s−1
is−n
ρ
i
log n
1/2
/log log n
1ε
,ε>0, and 3.12, we have
T
3
exp
−
u
2
n
2
1 δ
n
ik1
Cov
X
i
,
S
k
σ
k
1
n
1/1δ
n
ik1
Cov
X
i
,
S
k
σ
k
1
n
1/1δ
1
√
k
n
ik1
Cov
X
i
,S
k
1
n
1/1δ
1
√
k
k
j1
n
ik1
Cov
X
i
,X
j
1
n
1/1δ
1
√
k
k
j1
n
i1
ρ
i
√
k
n
1/1δ
log n
1/2
log log n
1ε
1
n
β
,β>0.
3.22
As to T
4
,by3.5 and 3.6, we have
T
4
1
σ
k
k
i1
Cov
X
i
,
S
n
σ
n
k
n
log n
1/2
log log n
1ε
. 3.23
Thus the proof of this lemma is completed.
Proof of Theorem 2.1. First, by assumptions and Theorem 6.1.3 in Leadbetter et al. 10,we
have
P
n
i1
X
i
≤ u
ni
−→ e
−τ
. 3.24
8 Journal of Inequalities and Applications
Let Y
n
denote a random variable which has the same distribution as S
n
/σ
n
, but it is
independent of X
1
,X
2
, ,X
n
, then by 3.10, we derive
P
n
i1
X
i
≤ u
ni
,
S
n
σ
n
≤ y
− P
n
i1
X
i
≤ u
ni
P
Y
n
≤ y
−→ 0, as n −→ ∞. 3.25
Thus, by the standard normal property of Y
n
, we have
lim
n →∞
P
n
i1
X
i
≤ u
ni
,
S
n
σ
n
≤ y
e
−τ
Φ
y
,y∈ R. 3.26
Hence, to complete the proof, it is sufficient to show
lim
n →∞
1
log n
n
k1
1
k
I
k
i1
X
i
≤ u
ki
,
S
k
σ
k
≤ x
− P
k
i1
X
i
≤ u
ki
,
S
k
σ
k
≤ x
0a.s. 3.27
In order to show this, by Lemma 3.1 in Cs
´
aki and Gonchigdanzan 5, we only need to prove
Var
1
log n
n
k1
1
k
I
k
i1
X
i
≤ u
ki
,
S
k
σ
k
≤ x
1
log log n
1ε
, 3.28
for ε>0andanyx ∈ R.Letη
k
I{
k
i1
X
i
≤ u
ki
,S
k
/σ
k
≤ x}−P{
k
i1
X
i
≤ u
ki
,S
k
/σ
k
≤ x}.
Then
Var
1
log n
n
k1
1
k
I
k
i1
X
i
≤ u
ki
,
S
k
σ
k
≤ x
E
1
log n
n
k1
1
k
η
k
2
1
log
2
n
n
k1
1
k
2
E
η
k
2
2
log
2
n
1≤k<l≤n
E
η
k
η
l
kl
: S
1
S
2
.
3.29
Since |η
k
|≤2, it follows that
S
1
1
log
2
n
.
3.30
Journal of Inequalities and Applications 9
Now, we turn to estimate S
2
. Observe that f or l>k
E
η
k
η
l
Cov
I
k
i1
{
X
i
≤ u
ki
}
,
S
k
σ
k
≤ x
,I
l
i1
{
X
i
≤ u
li
}
,
S
l
σ
l
≤ x
≤
Cov
I
k
i1
{
X
i
≤ u
ki
}
,
S
k
σ
k
≤ x
,I
l
i1
{
X
i
≤ u
li
}
,
S
l
σ
l
≤ x
−I
l
ik1
{
X
i
≤ u
li
}
,
S
l
σ
l
≤ x
Cov
I
k
i1
{
X
i
≤ u
ki
}
,
S
k
σ
k
≤ x
,I
l
ik1
{
X
i
≤ u
li
}
,
S
l
σ
l
≤ x
≤ E
I
l
i1
{
X
i
≤ u
li
}
,
S
l
σ
l
≤ x
− I
l
ik1
{
X
i
≤ u
li
}
,
S
l
σ
l
≤ x
Cov
I
k
i1
{
X
i
≤ u
ki
}
,
S
k
σ
k
≤ x
,I
l
ik1
{
X
i
≤ u
li
}
,
S
l
σ
l
≤ x
: S
21
S
22
.
3.31
By Lemma 3.3, we have
S
21
≤
k
l
C
log log l
1ε
. 3.32
Using Lemma 3.4, it follows that
S
22
≤
k
l
log l
1/2
log log l
1ε
C
log log l
1ε
.
3.33
Hence for l>k, we have
E
η
k
η
l
≤
k
l
C
log log l
1ε
k
l
log l
1/2
log log l
1ε
. 3.34
10 Journal of Inequalities and Applications
Consequently
S
2
1
log
2
n
⎛
⎝
1≤k<l≤n
1
kl
⎛
⎝
k
l
k
l
log l
1/2
log log l
1ε
⎞
⎠
⎞
⎠
1≤k<l≤n
1
kl
log log l
1ε
1
log
2
n
1≤k<l≤n
1
l
2
1
log
2
n
log n
1/2
log log n
1ε
n
l2
1
l
3/2
l−1
k1
1
√
k
1
log
2
n
n
l3
1
l
log log l
1ε
l−1
k1
1
k
1
log n
1
log n
log log n
1ε
1
log
2
n
n
l3
log l
l
log log l
1ε
1
log n
1
log log n
1ε
.
3.35
Thus, we complete the proof of 3.28 by 3.30 and 3.35. Further, our main result is proved.
Acknowledgments
The author thanks the referees for pointing out some errors in a previous version, as well as
for several comments that have led to improvements in this paper. The authors would like to
thank Professor Zuoxiang Peng of Southwest University in China for his help. The paper has
been supported by the young excellent talent foundation of Huaiyin Normal University.
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