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
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 →∞
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
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
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
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
j
≤ u
j
⎞
⎠
≤
1
2π
1≤i<j≤n
arcsin
r
1
ij
− arcsin
r
0
ij
exp
−
u
2
P
⎛
⎝
n
j1
ξ
j
≤ u
j
⎞
⎠
−
n
j1
P
ξ
j
≤ u
j
, 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 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
, 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
P
n
ik1
{
X
i
≤ u
ni
}
,
S
n
σ
n
≤ y
− P
n
ik1
{
X
i
≤ u
}
,
S
n
σ
n
≤ y
− P
n
i1
{
X
i
≤ u
ni
}
P
Y
n
≤ y
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
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
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
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 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
}
−
n
ik1
Φ
u
ni
P
n
i1
{
X
u
ni
−
n
i1
Φ
u
ni
: L
31
L
32
L
33
.
3.14
For L
33
, it follows that
L
33
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
k
σ
k
≤ y
,I
n
ik1
{
X
i
≤ u
ni
}
,
S
n
σ
n
≤ y
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
,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
k
σ
k
≤ y
,I
n
ik1
{
X
i
≤ u
ni
}
,
S
n
σ
n
≤ y
σ
k
≤ y,
S
n
σ
n
≤ y
−P
k
i1
{
X
i
≤ u
ki
}
,
S
k
σ
k
≤ y
P
n
−
u
2
ki
u
2
nj
2
1 r
ij
1
4
k
i1
Cov
X
i
,
S
n
σ
n
,
S
k
σ
k
exp
−
u
2
ni
y
2
2
1 Cov
X
i
,S
k
/σ
k
1
4
Cov
1 r
ij
1
4
k
i1
Cov
X
i
,
S
n
σ
n
exp
−
u
2
ki
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
−
u
2
n
2
1 δ
n
ik1
Cov
X
i
,
S
k
σ
k
1
n
1/1δ
n
ik1
1/1δ
1
√
k
k
j1
n
ik1
Cov
X
i
,X
j
1
n
1/1δ
1
√
k
k
j1
n
i1
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
,
S
n
σ
n
≤ y
− P
n
i1
X
i
≤ u
ni
P
−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
log n
n
k1
1
k
I
k
i1
X
i
≤ u
ki
,
S
k
σ
k
≤ x
1
log log n
1ε
1
log n
n
k1
1
k
I
k
i1
X
i
≤ u
ki
,
S
k
σ
k
≤ x
E
1
log n
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
k
i1
{
X
i
≤ u
ki
}
,
S
k
σ
k
≤ x
,I
l
i1
{
X
i
≤ u
li
}
,
S
l
k
σ
k
≤ x
,I
l
i1
{
X
i
≤ u
li
}
,
S
l
σ
l
≤ x
−I
l
ik1
{
X
X
i
≤ u
ki
}
,
S
k
σ
k
≤ x
,I
l
ik1
{
X
i
≤ u
li
}
,
S
l
σ
l
≤ x
l
ik1
{
X
i
≤ u
li
}
,
S
l
σ
l
≤ x
Cov
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
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
⎞
⎠
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
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
aki and K. Gonchigdanzan, “Almost sure limit theorems for the maximum of stationary
Gaussian sequences,” Statistics & Probability Letters, vol. 58, no. 2, pp. 195–203, 2002.
6 S. Chen and Z. Lin, “Almost sure max-limits for nonstationary Gaussian sequence,” Statistics &
Probability Letters, vol. 76, no. 11, pp. 1175–1184, 2006.
7 M. Dudzi
´
nski, “The almost sure central limit theorems in the joint version for the maxima and sums
of certain stationary Gaussian sequences,” Statistics & Probability Letters, vol. 78, no. 4, pp. 347–357,
2008.
8 M. Dudzi
´
nski, “An almost sure limit theorem for the maxima and sums of stationary Gaussian
sequences,” Probability and Mathematical Statistics, vol. 23, no. 1, pp. 139–152, 2003.
9 W. V. Li and Q. Shao, “A normal comparison inequality and its applications,” Probability Theory and
Related Fields, vol. 122, no. 4, pp. 494–508, 2002.
10 M. R. Leadbetter, G. Lindgren, and H. Rootz
´
en, Extremes and Related Properties of Random Sequences and
Processes, Springer Series in Statistics, Springer, New York, NY, USA, 1983.
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