Volume 2010, Article ID 130915, 10 pagesdoi:10.1155/2010/130915 Research Article Almost Sure Central Limit Theorem for a Nonstationary Gaussian Sequence Qing-pei Zang School of Mathemati
Trang 1Volume 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,zqphunhu@yahoo.com.cn
Received 4 May 2010; Revised 7 July 2010; Accepted 12 August 2010
Academic Editor: Soo Hak Sung
Copyrightq 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{Xn; n ≥ 1} be a standardized non-stationary Gaussian sequence, and let denote S nn
k1 X k,
σ n VarSn Under some additional condition, let the constants {uni; 1≤ i ≤ n, n ≥ 1} satisfy
n
i1 1 − Φuni → τ as n → ∞ for some τ ≥ 0 and min1≤i≤nu ni ≥ clog n1/2, for somec > 0, then,
we have limn → ∞1/ log nn k1 1/kI{∩ k
i1 Xi ≤ uki, Sk /σ k ≤ x} e −τ Φx almost surely for any
x ∈ R, where IA 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 andS n n k1 X k , n ≥ 1, M n max1≤k≤nX kforn ≥ 1 If EX 0, VarX 1, the
so-called almost sure central limit theoremASCLT has the simplest form as follows:
lim
n → ∞
1 logn
n
k1
1
k I
S
k
√
k ≤ x
Φx, 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 Brosamler1 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 ¨uller3 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 logn
n
k1
1
k I
M
k − b k
a k ≤ x
Gx a.s. 1.2
Trang 2for allx ∈ R, where a k > 0 and b k ∈ R satisfy
P
M
k − b k
a k ≤ x
−→ Gx, as k −→ ∞ 1.3
for any continuity pointx of G.
In a related work, Cs´aki and Gonchigdanzan5 investigated the validity of 1.2 for maxima of stationary Gaussian sequences under some mild condition whereas Chen and Lin6 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,
andσ n VarSn Here a 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, where ρ n ≤ 1 for all n ≥ 1 and
sups≥ns−1
is−n ρ i log n1/2 /log log n1 ε, 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 min1≤i≤nu ni ≥ clog n1/2 , for some c > 0, then
lim
n → ∞
1 logn
n
k1
1
k I
k
i1
X i ≤ u ki , S σ k
k ≤ x
e −τ Φx, 2.1
almost surely for any x ∈ R.
Remark 2.2 The condition sup s≥ns−1
is−n ρ i log n1/2 /log log n1 ε, 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 ijexp
−u
2
ni u2
nj
2 1 r ij
log logn1 ε. 3.1 Proof This lemma comes from Chen and Lin6
Trang 3The following lemma isTheorem 2.1and Corollary 2.1 in Li and Shao 9.
Lemma 3.2 (1) Let {ξ n } and {η n } be sequences of standard Gaussian variables with covariance
matrices R1 r1
ij and R0 r0
ij , respectively Put ρ ij max|r1
ij |, |r0
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
r1
ij
− arcsinr0
ij
exp
− u
2
i u2
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
≤ 14
1≤i<j≤n
r ijexp − u2
i u2
j
2 1 r ij
, 3.3
for any real numbers u i , i 1, 2, , n.
Lemma 3.3 Let {Xn } be a sequence of standard Gaussian variables and satisfy the conditions of
Theorem 2.1 , then for 1 ≤ k < n, one has
P
n
ik 1
{X i ≤ u ni }, S σ n
n ≤ y
− P
n
i1
{X i ≤ u ni }, S σ n
n ≤ y
≤ k n C
log logn1 ε 3.4 for any y ∈ R.
Proof By the conditions ofTheorem 2.1, we have
σ nn 2
1≤i<j≤n
r ij ≥√n, 3.5
then, for 1≤ i ≤ n, by sup s≥ns−1
is−n ρ i log n1/2 /log log n1 ε, ε > 0, it follows that
Cov
X i , S n
σ n
≤ √1
n
1
√
n
n
k1
ρ k logn
1/2
√
n log logn1 ε. 3.6
Then, there exist numbersδ, n0, such that, for anyn > n0, we have
sup
1≤i≤nCov
X i , S σ n
n
< δ < 1
Trang 4We can write that
L : P
n
ik 1
{X i ≤ u ni }, S σ n
n ≤ y
− P
n
i1
{X i ≤ u ni }, S σ n
n ≤ y
≤
P
n
ik 1
{X i ≤ u ni }, S σ n
n ≤ y
− P
n
ik 1
{X i ≤ u ni}
P Y n ≤ y
P
n
i1
{X i ≤ u ni }, S σ n
n ≤ y
− P
n
i1
{X i ≤ u ni}
P Y n ≤ y
P
n
ik 1
{X i ≤ u ni}
− P
n
i1
{X i ≤ u ni}
: L1 L2 L3,
3.8
where {Y n } is a random variable, which has the same distribution as {S n /σ n}, but it
is independent of X1, X2, , X n For L1, L2, apply Lemma 3.2 1 with ξ i X i , i
1, , n; ξ n 1 S n /σ n , η j X j , j 1, , n; η n 1 Y n Then r1
ij r0
ij r ijfor 1≤ i < j ≤ n
andr1
ij CovX i , S n /σ n , r0
ij 0 for 1 ≤ i ≤ n, j n 1 Thus, we have for i 1, 2
L i n
i1
Cov
X i , S σ n
n
exp
−21 CovXu2ni y2
i , S n /σ n
. 3.9
Since3.5, 3.7 hold, we obtain
L i logn
1/2
√
n log logn1 ε
n
i1
exp
− u2ni
21 δ
. 3.10
Now defineu nby 1− Φu n 1/n By the well-known fact
1− Φx ∼ φx x , x −→ ∞, 3.11
it is easy to see that
exp
−u2n 2
∼
√
2πu n
2 logn. 3.12
Trang 5Thus, according to the assumption min1≤i≤nu ni ≥ clog n1/2, we haveu ni ≥ cu nfor somec > 0.
Hence
L i≤ logn
1/2
√
n log logn1 ε
1≤i≤n
exp
− u2ni
21 δ
≤
√
n logn1/2
log logn1 ε exp
−21 δu2n
√
n2 logn2 δ/1 δ
n1/1 δ log logn1 ε
logn2 δ/1 δ
n1/1 δ−1/2
1
n δ , δ > 0.
3.13
Now, we are in a position to estimateL3 Observe that
L3 P
n
ik 1
{X i ≤ u ni}
− P
n
i1
{X i ≤ u ni}
≤
P
n
ik 1
{X i ≤ u ni}
− n
ik 1 Φu ni
P
n
i1
{X i ≤ u ni}
−n
i1 Φu ni
n
ik 1
Φu ni −n
i1
Φu ni
: L31 L32 L33.
3.14
ForL33, it follows that
L33 n
ik 1
Φu ni
1−k
i1
Φu ni
1 − Φk u n
1 −
1−n1
k
≤ k n
3.15
ByLemma 3.22, we have
L3i≤ 1 4
1≤i<j≤n
r ijexp
−u
2
ni u2
nj
2 1 r ij
Thus byLemma 3.1we obtain the desired result
Trang 6Lemma 3.4 Let {Xn } be a sequence of standard Gaussian variables satisfying the conditions of
Theorem 2.1 , then for 1 ≤ k < n, any y ∈ R, one has
Cov
I
k
i1
{X i ≤ u ki }, S σ k
k ≤ y
, I
n
ik 1
{X i ≤ u ni }, S σ n
n ≤ y
k n
logn1/2
log logn1 ε
1 log logn1 ε.
3.17
Proof ApplyLemma 3.21 with ξ i X i , 1 ≤ i ≤ k, ξ k 1 S k /σ k , ξ i 1 X i , k 1 ≤ i ≤
n, ξ n 2 S n /σ n , η j ξ j , 1 ≤ j ≤ k 1, η j ξ j , k 2 ≤ j ≤ n 2, where ξ k 2 , , ξ n 2 has the same distribution asξ k 2 , , ξ n 2 , but it is independent of ξ k 2 , , ξ n 2 Then,
r1
ij r0
ij for 1≤ i < j ≤ k 1 or k 2 ≤ i < j ≤ n 2;
r1
ij r ij−1 , r0
ij 0 for 1 ≤ i ≤ k, k 2 ≤ j ≤ n 1;
r1
ij Cov
X i , S σ n
n
, r0
ij 0 for 1 ≤ i ≤ k, j n 2;
r1
ij Cov
X i , S σ k
k
, r0
ij 0 for k 1 ≤ i ≤ n, j k 1;
r1
ij Cov
S
k
σ k , S n
σ n
, r0
ij 0 for i k 1, j n 2.
3.18
Thus, combined with3.5, 3.7, it follows that
Cov
I
k
i1
{X i ≤ u ki }, S k
σ k ≤ y
, I
n
ik 1
{X i ≤ u ni }, S n
σ n ≤ y
P
k
i1
{X i ≤ u ki }, n
ik 1
{X i ≤ u ni }, S σ k
k ≤ y, S σ n
n ≤ y
−P
k
i1
{X i ≤ u ki }, S σ k
k ≤ y
P
n
ik 1
{X i ≤ u ni }, S σ n
n ≤ y
≤ 1
4
1≤i≤k
k 1≤j≤n
r ijexp
−u
2
ki u2
nj
2 1 r ij
1 4
k
i1
Cov
X i , S σ n
n
exp
2
ki y2
21 CovXi , S n /σ n
1
4
n
ik 1
Cov
X i , S σ k
k
exp
−21 CovXu2ni y2
i , S k /σ k
1
4Cov
S
k
σ k ,
S n
σ n
≤ 1
4
1≤i≤k
k 1≤j≤n
r ijexp
−u
2
ki u2
nj
2 1 rij
1 4
k
i1
Cov
X i , S σ n
n
exp
− u
2
ki
21 δ
1
4
n
ik 1
Cov
X i , S σ k
k
exp
−21 δu2ni
1
4Cov
S
k
σ k ,
S n
σ n
: T1 T2 T3 T4.
3.19
Trang 7UsingLemma 3.1, we have
log logn1 ε, ε > 0. 3.20
By the similar technique that was applied to prove3.10, we obtain
ForT3, by sups≥ns−1
is−n ρ i log n1/2 /log log n1 ε, ε > 0, and 3.12, we have
T3 exp
−21 δu2n
n
ik 1
Cov
X i , S σ k
k
1
n1/1 δ
n
ik 1
Cov
X i , S σ k
k
1
n1/1 δ
1
√
k
n
ik 1
CovXi , S k
1
n1/1 δ
1
√
k
k
j1
n
ik 1
Cov X i , X j
1
n1/1 δ
1
√
k
k
j1
n
i1
ρ i
√
k
n1/1 δ
logn1/2
log logn1 ε
1
n β , β > 0.
3.22
As toT4, by3.5 and 3.6, we have
T4 σ1
k
k
i1
Cov
X i , S σ n
n
k n
logn1/2
log logn1 ε. 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
Trang 8Let Y n denote a random variable which has the same distribution as S n /σ n, but it is independent ofX1, X2, , X n , then by 3.10, we derive
P
n
i1
X i ≤ u ni , S σ n
n ≤ y
− P
n
i1
X i ≤ u ni
PY n ≤ y−→ 0, as n −→ ∞. 3.25
Thus, by the standard normal property ofY n, we have
lim
n → ∞ P
n
i1
X i ≤ u ni , S σ n
n ≤ y
e −τΦ y, y ∈ R. 3.26
Hence, to complete the proof, it is sufficient to show
lim
n → ∞
1
logn
n
k1
1
k
I
k
i1
X i ≤ u ki , S σ k
k ≤ x
− P
k
i1
X i ≤ u ki , S σ k
k ≤ x
0 a.s. 3.27
In order to show this, byLemma 3.1in Cs´aki and Gonchigdanzan5, we only need to prove
Var
1 logn
n
k1
1
k I
k
i1
X i ≤ u ki , S σ k
k ≤ x
log logn1 ε, 3.28
forε > 0 and any x ∈ R Let η 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 logn
n
k1
1
k I
k
i1
X i ≤ u ki , S σ k
k ≤ x
E
1 logn
n
k1
1
k η k
2
1 log2n
n
k1
1
k2E η k2 2
log2n
1≤k<l≤n
k η l
kl
: S1 S2.
3.29
Since|η k| ≤ 2, it follows that
S1 1 log2n . 3.30
Trang 9Now, we turn to estimateS2 Observe that forl > k
k η l Cov
I
k
i1
{X i ≤ u ki }, S σ k
k ≤ x
, I
l
i1
{X i ≤ u li }, S σ l
l ≤ x
≤
Cov
I
k
i1
{X i ≤ u ki }, S σ k
k ≤ x
, I
l
i1
{X i ≤ u li }, S σ l
l ≤ x
−I
l
ik 1
{X i ≤ u li }, S σ l
l ≤ x
Cov
I
k
i1
{X i ≤ u ki }, S σ k
k ≤ x
, I
l
ik 1
{X i ≤ u li }, S σ l
l ≤ x
≤ E
I
l
i1
{X i ≤ u li }, S σ l
l ≤ x
− I
l
ik 1
{X i ≤ u li }, S σ l
l ≤ x
Cov
I
k
i1
{X i ≤ u ki }, S k
σ k ≤ x
, I
l
ik 1
{X i ≤ u li }, S l
σ l ≤ x
: S21 S22.
3.31
ByLemma 3.3, we have
S21≤ k l C
log logl1 ε. 3.32
UsingLemma 3.4, it follows that
S22≤
k l
logl1/2
log logl1 ε
C
log logl1 ε. 3.33
Hence forl > k, we have
k η l ≤ k l C
log logl1 ε
k l
logl1/2
log logl1 ε. 3.34
Trang 10S2 1
log2n
⎛
⎝
1≤k<l≤n
1
kl
⎛
⎝k
l
k l
logl1/2
log logl1 ε
⎞
⎠
⎞
⎠
1≤k<l≤n
1
kl log logl1 ε
1
log2n
1≤k<l≤n
1
l2 1 log2n
logn1/2
log logn1 ε
n
l2
1
l3/2
l−1
k1
1
√
k
1
log2n
n
l3
1
l log logl1 ε
l−1
k1
1
k
1
logn
1
logn log logn1 ε
1 log2n
n
l3
logl
l log logl1 ε
1
logn
1 log logn1 ε.
3.35
Thus, we complete the proof of3.28 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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... Trang 6Lemma 3.4 Let {Xn } be a sequence of standard Gaussian variables satisfying the... This lemma comes from Chen and Lin6
Trang 3The following lemma isTheorem 2.1and Corollary 2.1... T3 T4.
3.19
Trang 7UsingLemma 3.1, we have
log