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, zqphunhu@yahoo.com.cn
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.
n
Let {X
n ; n ≥ 1} be a standardized non-stationary Gaussian sequence, and let denote Sn k1 Xk ,
σ VarSn . Under some additional condition, let the constants {uni ; 1 ≤ i ≤ n, n ≥ 1} satisfy
nn
→ ∞ for some τ ≥ 0 and min1≤i≤n uni ≥ clog n1/2 , for some c > 0, then,
i1 1 − Φuni → τ as n we have limn → ∞ 1/ log n nk1 1/kI{∩ki1 Xi ≤ uki , Sk /σk ≤ 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, Xn ; n ≥ 1} is a sequence of independent and identically distributed i.i.d. random
variables and Sn nk1 Xk , n ≥ 1, Mn max1≤k≤n Xk for n ≥ 1. If EX 0, VarX 1, the
so-called almost sure central limit theorem ASCLT has the simplest form as follows:
n
Sk
1 1
I √ ≤ x Φx,
lim
1.1
n → ∞ log n
k
k
k1
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 Stadtmüller 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:
n
1 1 Mk − bk
I
≤ x Gx a.s.
1.2
lim
n → ∞ log n
k
ak
k1
2
Journal of Inequalities and Applications
for all x ∈ R, where ak > 0 and bk ∈ R satisfy
P
Mk − bk
≤x
ak
−→ Gx,
as k −→ ∞
1.3
for any continuity point x of G.
In a related work, Csáki 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, Dudziński 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 {Xn ; n ≥ 1} be a non-stationary standardized normal sequence,
and σn 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 {Xn ; n ≥ 1} be a sequence of non-stationary standardized Gaussian variables
j, where ρn ≤ 1 for all n ≥ 1 and
with covariance matrix rij such that 0 ≤ rij ≤ ρ|i−j| for i /
s−1
1/2
1ε
sups≥n is−n ρi log n /log log n , ε > 0. If the constants {uni ; 1 ≤ i ≤ n, n ≥ 1} satisfy
n
1/2
, for some c > 0,
i1 1 − Φuni → τ as n → ∞ for some τ ≥ 0 and min1≤i≤n uni ≥ clog n
then
n
k
1 1 Sk
I
lim
≤ x e−τ Φx,
Xi ≤ uki ,
n → ∞ log n
k i1
σk
k1
2.1
almost surely for any x ∈ R.
1/2
Remark 2.2. The condition sups≥n s−1
/log log n1ε , ε > 0 is inspired by
is−n ρi log n
a1 in Dudziński 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
u2ni u2nj
rij exp − 2 1 rij
1≤i<j≤n
Proof. This lemma comes from Chen and Lin 6.
≤
C
1ε .
log log n
3.1
Journal of Inequalities and Applications
3
The following lemma is Theorem 2.1 and Corollary 2.1 in Li and Shao 9.
Lemma 3.2. (1) Let {ξn } and {ηn } be sequences of standard Gaussian variables with covariance
matrices R1 rij1 and R0 rij0 , respectively. Put ρij max|rij1 |, |rij0 |. Then one has
⎞
⎛
⎞
n
n
P⎝
ξj ≤ uj ⎠ − P ⎝
η j ≤ uj ⎠
⎛
j1
≤
j1
1
2π 1≤i<j≤n
u2i u2j
1
0
arcsin rij − arcsin rij
exp − ,
2 1 ρij
3.2
for any real numbers ui , i 1, 2, . . . , n.
(2) Let {ξn ; n ≥ 1} be standard Gaussian variables with rij Covξi , ξj . Then
⎛
⎞
n
n
u2i u2j
1 P ⎝
⎠
,
rij exp − −
P ξj ≤ uj ≤
ξj ≤ uj
2 1 rij 4 1≤i<j≤n
j1
j1
3.3
for any real numbers ui , 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
Sn
≤y
{Xi ≤ uni },
σn
ik1
n
C
k
Sn
−P
≤y ≤ {Xi ≤ uni },
1ε
σn
n
log log n
i1
3.4
for any y ∈ R.
Proof. By the conditions of Theorem 2.1, we have
σn √
n2
rij ≥ n,
1≤i<j≤n
then, for 1 ≤ i ≤ n, by sups≥n
s−1
is−n
3.5
ρi log n1/2 /log log n1ε , ε > 0, it follows that
1/2
n
log n
1 Sn
1
ρk √ ≤√ √
Cov Xi ,
1ε .
σn
n
n k1
n log log n
3.6
Then, there exist numbers δ, n0 , such that, for any n > n0 , we have
Sn
1
sup Cov Xi ,
<δ< .
σn
2
1≤i≤n
3.7
4
Journal of Inequalities and Applications
We can write that
L : P
n
Sn
≤y
{Xi ≤ uni },
σn
ik1
−P
n
Sn
≤y
{Xi ≤ uni },
σn
i1
n
n
Sn
≤ P
≤y −P
{Xi ≤ uni },
{Xi ≤ uni } P Yn ≤ y σn
ik1
ik1
n
n
Sn
P
≤y −P
{Xi ≤ uni },
{Xi ≤ uni } P Yn ≤ y σn
i1
i1
P
n
{Xi ≤ uni }
−P
3.8
n
{Xi ≤ uni }
i1
ik1
: L1 L2 L3 ,
where {Yn } is a random variable, which has the same distribution as {Sn /σn }, but it
is independent of X1 , X2 , . . . , Xn . For L1 , L2 , apply Lemma 3.2 1 with ξi Xi , i 1, . . . , n; ξn1 Sn /σn , ηj Xj , j 1, . . . , n; ηn1 Yn . Then rij1 rij0 rij for 1 ≤ i < j ≤ n
and rij1 CovXi , Sn /σn , rij0 0 for 1 ≤ i ≤ n, j n 1. Thus, we have for i 1, 2
u2ni y2
Sn
.
Cov Xi ,
exp −
Li σn
21 CovXi , Sn /σn i1
n
3.9
Since 3.5, 3.7 hold, we obtain
1/2
log n
Li √ 1ε
n log log n
n
u2ni
exp −
21 δ
i1
.
3.10
Now define un by 1 − Φun 1/n. By the well-known fact
1 − Φx ∼
φx
,
x
x −→ ∞,
3.11
it is easy to see that
u2
exp − n
2
√
∼
2πun
,
n
un ∼
2 log n.
3.12
Journal of Inequalities and Applications
5
Thus, according to the assumption min1≤i≤n uni ≥ clog n1/2 , we have uni ≥ cun for some c > 0.
Hence
1/2
log n
u2ni
exp
−
Li ≤ √ 1ε
21 δ
n log log n
1≤i≤n
1/2
√ n log n
u2n
≤
1ε exp − 21 δ
log log n
2δ/1δ
√ n
2 log n
1ε
n1/1δ log log n
3.13
2δ/1δ
log n
n1/1δ−1/2
1
,
nδ
δ > 0.
Now, we are in a position to estimate L3 . Observe that
n
n
L3 P
{Xi ≤ uni } − P
{Xi ≤ uni }
i1
ik1
n
n
n
n
Φuni P
Φuni ≤ P
{Xi ≤ uni } −
{Xi ≤ uni } −
i1
i1
ik1
ik1
n
n
Φuni −
Φuni ik1
i1
3.14
: L31 L32 L33 .
For L33 , it follows that
L33 n
Φuni 1 −
ik1
k
Φuni i1
3.15
1 − Φk un 1 k k
1− 1−
≤ .
n
n
By Lemma 3.2 2, we have
2
uni u2nj
1 rij exp − L3i ≤
,
4 1≤i<j≤n
2 1 rij
Thus by Lemma 3.1 we obtain the desired result.
i 1, 2.
3.16
6
Journal of Inequalities and Applications
Lemma 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
k
n
Sk
Sn
≤ y ,I
≤y
{Xi ≤ uki },
{Xi ≤ uni },
Cov I
σ
σ
k
n
i1
ik1
1/2
log n
k
1
.
n log log n1ε log log n1ε
3.17
Proof. Apply Lemma 3.2 1 with ξi Xi , 1 ≤ i ≤ k, ξ k1 Sk /σk , ξi1 Xi , k 1 ≤ i ≤
n, ξn2 Sn /σn , ηj ξj , 1 ≤ j ≤ k 1, ηj ξ j , k 2 ≤ j ≤ n 2, where ξk2 , . . . , ξ n2 has
the same distribution as ξk2 , . . . , ξn2 , but it is independent of ξk2 , . . . , ξn2 . Then,
rij1 rij0
for 1 ≤ i < j ≤ k 1
or k 2 ≤ i < j ≤ n 2;
rij1 rij−1 , rij0 0 for 1 ≤ i ≤ k, k 2 ≤ j ≤ n 1;
Sn
, rij0 0 for 1 ≤ i ≤ k, j n 2;
rij1 Cov Xi ,
σn
Sk
rij1 Cov Xi ,
, rij0 0 for k 1 ≤ i ≤ n, j k 1;
σk
Sk Sn
rij1 Cov
,
, rij0 0 for i k 1, j n 2.
σk σn
3.18
Thus, combined with 3.5, 3.7, it follows that
k
n
Sk
Sn
≤ y ,I
≤y
{Xi ≤ uki },
{Xi ≤ uni },
Cov I
σ
σ
k
n
i1
ik1
k
n
Sn
Sk
P
≤ y,
≤y
{Xi ≤ uki },
{Xi ≤ uni },
σ
σn
k
i1
ik1
k
n
Sk
Sn
−P
≤y P
≤y {Xi ≤ uki },
{Xi ≤ uni },
σ
σ
k
n
i1
ik1
2
k
uki u2nj
u2ki y2
1 1
Sn
≤
exp −
rij exp − Cov Xi ,
4 1≤i≤k k1≤j≤n
4 i1
σn
21 CovXi , Sn /σn 2 1 rij
n
u2ni y2
Sk Sn
1
Sk
1 Cov
Cov Xi ,
,
exp −
4 ik1
σk
21 CovXi , Sk /σk 4
σk σn
2
k
uki u2nj
u2ki
1
1 Sn
≤
rij exp − Cov Xi ,
exp −
4 1≤i≤k k1≤j≤n
4 i1
σn
21 δ
2 1 rij
n
u2ni
1
Sk Sn
Sk
1 Cov
exp −
Cov Xi ,
,
4 ik1
σk
21 δ
4
σk σn
: T1 T2 T3 T4 .
3.19
Journal of Inequalities and Applications
7
Using Lemma 3.1, we have
C
T1 ≤ 1ε ,
log log n
ε > 0.
3.20
By the similar technique that was applied to prove 3.10, we obtain
T2 For T3 , by sups≥n
s−1
is−n
1
,
nα
α > 0.
3.21
ρi log n1/2 /log log n1ε , ε > 0, and 3.12, we have
u2n
T3 exp −
21 δ
Sk
Cov Xi ,
σk
ik1
n
Sk
Cov
X
,
i
σk
n1/1δ ik1
n
1 CovXi , Sk √
n1/1δ k ik1
n
k 1 Cov Xi , Xj
√
1/1δ
n
k j1 ik1
n
k 1 ρi
√
n1/1δ k j1 i1
n
1
1
1
3.22
1
1/2
log n
1/1δ 1ε
n
log log n
√
k
1
,
nβ
β > 0.
As to T4 , by 3.5 and 3.6, we have
1/2
k
log n
k
1
Sn
Cov Xi ,
.
T4 σk i1
σn
n log log n1ε
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
n
P
Xi ≤ uni −→ e−τ .
i1
3.24
8
Journal of Inequalities and Applications
Let Yn denote a random variable which has the same distribution as Sn /σn , but it is
independent of X1 , X2 , . . . , Xn , then by 3.10, we derive
n
n
Sn
P
≤y −P
Xi ≤ uni ,
Xi ≤ uni P Yn ≤ y −→ 0,
σn
i1
i1
as n −→ ∞.
3.25
Thus, by the standard normal property of Yn , we have
n
Sn
lim P
≤y
Xi ≤ uni ,
n→∞
σn
i1
e−τ Φ y ,
y ∈ R.
3.26
Hence, to complete the proof, it is sufficient to show
n
1 1
lim
n → ∞ log n
k
k1
k
k
Sk
Sk
I
≤x −P
≤x
0 a.s.
Xi ≤ uki ,
Xi ≤ uki ,
σk
σk
i1
i1
3.27
In order to show this, by Lemma 3.1 in Csáki and Gonchigdanzan 5, we only need to prove
Var
n
k
1 1 1
Sk
I
≤x
Xi ≤ uki ,
1ε ,
log n k1 k i1
σk
log log n
3.28
for ε > 0 and any x ∈ R. Let ηk I{ ki1 Xi ≤ uki , Sk /σk ≤ x} − P { ki1 Xi ≤ uki , Sk /σk ≤ x}.
Then
2
n
k
n
1 1
1 1 Sk
I
ηk
≤x
E
Var
Xi ≤ uki ,
log n k1 k i1
σk
log n k1 k
n
E ηk ηl 1 2
1 2
E ηk kl
log2 n k1 k2
log2 n 1≤k<l≤n
3.29
: S1 S2 .
Since |ηk | ≤ 2, it follows that
S1 1
log2 n
.
3.30
Journal of Inequalities and Applications
9
Now, we turn to estimate S2 . Observe that for l > k
k
l
S
S
k
l
E ηk ηl Cov I
≤ x ,I
≤x
{Xi ≤ uki },
{Xi ≤ uli },
σ
σ
k
l
i1
i1
k
l
Sk
Sl
≤ Cov I
≤ x ,I
≤x
{Xi ≤ uki },
{Xi ≤ uli },
σk
σl
i1
i1
l
Sl
−I
≤x
{Xi ≤ uli },
σl
ik1
k
l
Sk
Sl
Cov I
≤ x ,I
≤x
{Xi ≤ uki },
{Xi ≤ uli },
σ
σ
k
l
i1
ik1
3.31
l
l
Sl
Sl
≤ EI
≤x −I
≤x {Xi ≤ uli },
{Xi ≤ uli },
i1
σl
σ
l
ik1
k
l
Sk
Sl
Cov I
≤ x ,I
≤x
{Xi ≤ uki },
{Xi ≤ uli },
σ
σ
k
l
i1
ik1
: S21 S22 .
By Lemma 3.3, we have
S21 ≤
C
k
.
l log log l1ε
3.32
Using Lemma 3.4, it follows that
S22 ≤
1/2
log l
k
C
.
l log log l1ε log log l1ε
3.33
Hence for l > k, we have
C
E ηk ηl ≤ k 1ε l
log log l
1/2
log l
k
.
l log log l1ε
3.34
10
Journal of Inequalities and Applications
Consequently
⎛
⎛
1/2
⎞⎞
k
1
⎠⎠ 1ε
1ε
l log log l
log n
1≤k<l≤n kl log log l
1/2
l−1
n
1
log n
1 1
1
1
√
2
2
2
1ε
3/2
log n 1≤k<l≤n l
log n log log n
l2 l
k1 k
1
S2 2
1
⎝
1 ⎝k
kl
l
1≤k<l≤n
n
1
log2 n l3 l log log l
1ε
log l
l−1
1
k1
k
n
log l
1
1
1 2
1ε
log n
log n l3 l log log l 1ε
log n log log n
1
1
.
log n log log n1ε
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.
References
1 G. A. Brosamler, “An almost everywhere central limit theorem,” Mathematical Proceedings of the
Cambridge Philosophical Society, vol. 104, no. 3, pp. 561–574, 1988.
2 P. Schatte, “On strong versions of the central limit theorem,” Mathematische Nachrichten, vol. 137, pp.
249–256, 1988.
3 I. Fahrner and U. Stadtmüller, “On almost sure max-limit theorems,” Statistics & Probability Letters,
vol. 37, no. 3, pp. 229–236, 1998.
4 S. Cheng, L. Peng, and Y. Qi, “Almost sure convergence in extreme value theory,” Mathematische
Nachrichten, vol. 190, pp. 43–50, 1998.
5 E. Csáki 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. Dudziński, “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. Dudziński, “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. Rootzén, Extremes and Related Properties of Random Sequences and
Processes, Springer Series in Statistics, Springer, New York, NY, USA, 1983.