Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 856495, 14 pages
doi:10.1155/2010/856495
Research Article
Almost Sure Convergence for the Maximum and
the Sum of Nonstationary Guassian Sequences
Shengli Zhao,1 Zuoxiang Peng,2 and Songlin Wu1
1
2
Department of Fundament Studies, Logistical Engineering University, Chongqing 401131, China
School of Mathematics and Statistics, Southwest University, Chongqing 400715, China
Correspondence should be addressed to Shengli Zhao, zhaoshengli83@yahoo.com.cn
Received 18 December 2009; Accepted 5 April 2010
Academic Editor: Jewgeni Dshalalow
Copyright q 2010 Shengli Zhao et al. 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 nonstationary Gaussian sequence. Let Mn max{Xk , 1 ≤ k ≤ n}
denote the partial maximum and Sn nk−1 Xk for the partial sum with σn Var Sn 1/2 . In this
paper, the almost sure convergence of Mn , Sn /σn is derived under some mild conditions.
1. Introduction
There have been more researches on the almost sure convergence of extremes and partial
sums since the pioneer work of Fahrner and Stadtmüller 1 and Cheng et al. 2. For more
related work on almost sure convergence of extremes and partial sums, see Berkes and
Csáki 3, Peng et al. 4, 5, Tan and Peng 6, and references therein. For the almost sure
convergence of extremes for dependent Gaussian sequence, Csáki and Gonchigdanzan 7
and Lin 8 proved
∞
n
√ 1 1 Mk − bk
I
≤x exp −e−x−ρ 2ρz φzdz a.s.
n → ∞ log n
k
ak
−∞
k1
1.1
rn log n − ρlog log n1ε O1,
1.2
lim
provided
where I denotes
an indicator function, Φx is the standard normal distribution function, and
2
φx 1/ 2πe−x /2 Φ x. Mn is the partial maximum of a standard stationary Gaussian
2
Journal of Inequalities and Applications
sequence {Xn , n ≥ 1} with correlation rn EX1 Xn1 , n ≥ 0. The norming constants an and bn
are defined by
an 2 log n−1/2 ,
bn 2 log n1/2 −
log log n log 4π
22 log n1/2
.
1.3
For some extensions of 1.1, see Chen and Lin 9 and Peng and Nadarajah 10.
Sometimes, in practice, one would like to know how partial sums and maxima behave
simultaneously in the limit; see Anderson and Turkman 11 for a discussion of an application
involving extreme wind gusts and average wind speeds. Peng et al. 12 studied the almost
sure limiting behavior for partial sums and maxima of i.i.d. random variables. Dudziński
13, 14 proved the almost sure limit theorems in the joint version for the maxima and the
partial sums of stationary Gaussian sequences, that
is, let X1 , X1 , . . . be stationary Gaussian
sequences and Mk maxi≤k Xi , Sn ni1 Xi , σn VarSn , for all x, y ∈ −∞, ∞
n
1 1 Mk − bk
Sk
I
≤ x,
≤ y exp −e−x Φ y
n → ∞ log n
k
ak
σk
k1
lim
if
a.s.
1.4
1/2
/log log n1ε for some ε > 0,
C1 sups≥n s−1
ts−n |rt | log n
n
C2 t1 n − trt ≥ 0 for all n ≥ 1,
C3 limn → ∞ rn log n 0.
Or
rn Ln
,
nα
n≥1
1.5
for some α > 0. Lx is a positive slowly varying function at infinity. Here a b means
a Ob.
This paper focuses on extending 1.4 to nonstationary Gaussian sequences {Xn , n ≥
1} under some mild conditions similar to C1–C3. The paper is organized as follows: in
Section 2, we give the main results, and related proofs are provided in Section 3.
2. The Main Results
Let rij EXi Xj , i, j ≥ 1, denote the correlations of standard nonstationary Gaussian
sequence {Xn , n ≥ 1}. Mn , Sn , and σn are defined as before. The main results are the
following.
Theorem 2.1. Let {Xn , n ≥ 1} be a standardized nonstationary Gaussian sequence. Suppose that
there exists numerical sequence {uni , 1 ≤ i ≤ n, n ≥ 1} such that ni1 1 − Φuni → τ for some
0 < τ < ∞ and n1 − Φλn is bounded, where λn min1≤i≤n uni . If
suprij < δ < 1,
i/
j
2.1
Journal of Inequalities and Applications
3
j−1
n rij on,
2.2
j2 i1
1/2
n log n
sup rij 1ε
i≥1 j1
log log n
2.3
for some ε > 0,
then
n
k
1 1 Sk
lim
I
≤ y e−τ Φ y
Xi ≤ uki ,
n → ∞ log n
k
σ
k
i1
k1
2.4
a.s.
for all y ∈ −∞, ∞.
Theorem 2.2. For the nonstationary Gaussian sequence {Xn , n ≥ 1}, under the conditions 2.1–
2.3, we have
n
1 1
Sk
I Mk ≤ ak x bk ,
≤ y exp −e−x Φ y
n → ∞ log n
k
σ
k
k1
lim
a.s.
2.5
for all x, y ∈ −∞, ∞, where an and bn are defined as in 1.3.
3. Proof of the Main Results
To prove the main results, we need some auxiliary lemmas.
Lemma 3.1. Suppose that the standardized nonstationary Gaussian sequences {Xn , n ≥ 1} satisfy
the conditions 2.1–2.3. Assume that n1 − Φλn is bounded. Then for < l,
l
l
1
k
Sl
Sl
EI
≤y −I
≤y .
Xi ≤ uli ,
Xi ≤ uli ,
1ε
i1
σl
σ
l
l
log
log
l
ik1
3.1
Proof. We will start with the following observations. For all 1 ≤ i ≤ l,
l Cov Xi , Sl 1 | CovXi , Sl | ≤ 1
rij .
σl σl
σl j1
3.2
Clearly,
⎛
⎞1/2
j−1
l rij ⎠ .
σl ⎝l 2
j2 i1
3.3
4
Journal of Inequalities and Applications
By 2.2, for large l there exists c1 > 0 such that
σl ≥ c1 l1/2 .
3.4
log l1/2
Sl sup Cov Xi ,
σl l1/2 log log l1ε
1≤i≤l
3.5
By 2.3 and 3.4, we have
for large l. Obviously,
log l1/2
lim
l → ∞ l1/2 log
log l1ε
3.6
0,
which implies that there exist μ > 0 and l0 such that
Sl <μ<1
sup Cov Xi ,
σl 1≤i≤l
∀l > l0 .
3.7
Notice,
l
l
Sl
Sl
EI
≤y −I
≤y Xi ≤ uli ,
Xi ≤ uli ,
i1
σl
σ
l
ik1
l
Sl
Sl
P
≤y −P
≤y
Xi ≤ uli ,
Xi ≤ uli ,
σl
σl
i1
ik1
l
l
Sl
Sl
≤ P
≤y −P
≤y Xi ≤ uli ,
Xi ≤ uli P
σ
σ
l
l
i1
i1
l
l
Sl
Sl
P
≤y −P
≤y Xi ≤ uli ,
Xi ≤ uli P
σ
σ
l
l
ik1
ik1
l
Sl
P
≤y
σl
3.8
l
l
P
Xi ≤ uli − P
Xi ≤ uli i1
ik1
: A1 l A2 l A3 l.
By the Normal Comparison Lemma 13, Theorem 4.2.1 , we get
l u2li y2
S
l
Cov Xi ,
exp −
A1 l σ 21 | CovX , S /σ |
i1
l
i
l λ2
Cov Xi , Sl exp − l ≤
σl 2 1μ
i1
.
l
l
3.9
Journal of Inequalities and Applications
5
Since n1 − Φλn is bounded, for large n and some absolute positive constant C,
λ2
exp − n
2
∼C
log1/2 n
.
n
3.10
So,
A1 l l1/2 log l1/2 log l1/21μ
1/2
l1/1μ
log log l
log l1/21/21μ
l1/1μ−1/2 log log l
1ε
1
log log l1ε
.
3.11
Similarly,
A2 l 1
log log l1ε
.
3.12
It remains to estimate A3 l. It is easy to check that
A3 l ≤ P
l
Xi ≤ uli −P
l
Xi ≤ uli i1
ik1
l
l
l
l−k
≤ P
Xi ≤ uli − Φ λl P
Xi ≤ uli − Φ λl Φl−k λl − Φl λl i1
ik1
: B1 l B2 l B3 l.
3.13
By the arguments similar to that of Lemma 2.4 in Csáki and Gonchigdanzan 7, we get
B3 l k
.
l
3.14
By the Normal Comparison Lemma and 3.4, we derive that
u2li λ2l
λ2l
≤ l
B1 l rij exp − rij exp −
1δ
2 1 rij 1≤i<j≤l
1≤i≤l
l
B2 l log l1/2
log log l1ε
1
log log l1ε
1
log log l1ε
,
.
log l1/1δ
l2/1δ
3.15
6
Journal of Inequalities and Applications
Combining with above analysis, we have
A3 l 1
k
.
l log log l1ε
3.16
The proof is complete.
We also need the following auxiliary result.
Lemma 3.2. Suppose that the standardized nonstationary Gaussian sequences {Xn , n ≥ 1} satisfy
the conditions 2.1–2.3. Assume that n1 − Φλn is bounded; then
k
l
k1/2 log l1/2
Sl
Sl
≤ y ,I
≤y
Xi ≤ uki ,
Xi ≤ uki ,
Cov I
σl
σl
l1/2 log log l1ε
i1
ik1
3.17
for k < minβ2 llog log 22ε /c22 log l, l, where 0 < β < 1, c2 > 0.
Proof. By 2.2 and 2.3, for i > k 1, we get
Cov Xi , Sl σ l
log l1/2
.
3.18
−→ 0 as l −→ ∞,
3.19
l1/2 log log l1ε
Clearly,
log l1/2
l1/2 log log l1ε
which implies that there exist > 0 and k0 such that for k > k0 ,
sup
i≥k1
Sl
Cov Xi ,
σl
< < 1.
3.20
For k < l, we have
Cov Sk , Sl 1 σ 2 CovSk , Sl σk σl σk σl k
k l σk
1 rij .
≤
σl σk σl i1 ji1
3.21
Journal of Inequalities and Applications
7
Condition 2.2 implies that there exist positive numbers c3 and c4 such that c3 k1/2 ≤ σk ≤
c4 k1/2 and
1/2
k1/2 log l1/2
Cov Sk , Sl k
σk σl
l1/2 l1/2 log log l1ε
k
1/2
1/2
log l
l1/2 log
log l1ε
3.22
.
So there exists 0 < ν < 1 such that
Cov Sk , Sl < ν < 1
σ σ k
l
3.23
for l1 < k < minβ2 llog log l22ε /c22 log l, l.By applying the inequalities above and the
Normal Comparison Lemma, we get
k
k
Sk
Sl
< y ,I
<y
Xi ≤ uki ,
Xi ≤ uki ,
Cov I
σ
σ
k
l
i1
ik1
Sk
Sl
P X1 ≤ uk1 , . . . , Xk ≤ ukk ,
≤ y, Xk1 ≤ ulk1 , . . . , Xll ≤ ull ,
≤y
σk
σl
Sk
Sl
−P X1 ≤ uk1 , . . . , Xk ≤ ukk ,
≤ y P Xk1 ≤ ulk1 , . . . , Xll ≤ ull ,
≤ y σk
σl
⎞
⎛
k l u2ki u2lj
rij exp⎝− ⎠
2 1 rij i1 jk1
k u2ki y2
S
l
exp −
Cov Xi ,
σl 21 |CovXi , Sl /σl |
i1
⎛
⎞
l u2lj y2
S
k
exp⎝− Cov Xj ,
⎠
σk 2 1 Cov Xj , Sk /σk jk1
Sk Sl 1
exp
−
Cov
,
σk σl 1 | CovSk /σk , Sl /σl |
: D1 l D2 l D3 l D4 l.
3.24
8
Journal of Inequalities and Applications
By 3.10, we have
2
l k λk λ2l
rij exp −
D1 l ≤
21 δ
i1 jk1
k
log l1/2
log log l1ε
log k1/21δ log l1/21δ
k1/1δ
l1/1δ
k1/2 log l1/2
,
l1/2 log log l1ε
k u2ki
Cov Xi , Sl D2 l < exp − σl 2 1μ
i1
log k1/21μ
k1/1μ
k
3.25
log l1/2
l1/2 log log l1ε
k1/2 log l1/2
l1/2 log log l1ε
.
Similarly,
⎛
⎞
l Sk ⎠
⎝
D3 l < exp − Cov Xj , σ 2 1
k
jk1
u2lj
⎛
⎞
u2lj
l k 1
rij < exp⎝− ⎠
σk i1 j1
2 1
3.26
log l1/2
log l1/21 k
k1/2 log log l1ε
l1/1
k1/2
l1/2 log log l1ε
.
While 3.22 implies
Sk Sl D4 l < Cov
,
σ σ k
l
the proof is complete.
We also need the following auxiliary result.
k1/2 log l1/2
l1/2 log log l1ε
,
3.27
Journal of Inequalities and Applications
9
Lemma 3.3. Let X1 , X2 , . . . be a standardized nonstationary Gaussian sequences satisfying
assumptions 2.1–2.3. Assume that ni1 1 − Φuni → τ for some 0 < τ < ∞ and n1 − Φλn is bounded. Then
k
Sk
lim P
≤ y e−τ Φ y
Xi ≤ uki ,
k→∞
σk
i1
3.28
for all ∈ −∞, ∞.
Proof. By the Normal Comparison Lemma and the proof of Lemma 3.1, we have
k
k
1
Sk
Sk
≤y −P
≤y ,
Xi ≤ uki ,
Xi ≤ uki P
P
σk
σk
log log k1
i1
i1
3.29
where
1
lim
k → ∞ log
0,
log k1
3.30
which implies
lim P
k→∞
k
i1
Sk
≤y
Xi ≤ uki ,
σk
lim P
k
k→∞
i1
Sk
≤y .
Xi ≤ uki P
σk
3.31
By Theorem 6.1.3 of Leadbetter et al. 15, we have
lim P
k→∞
k
Xi ≤ uki e−τ .
3.32
i1
Since Sk /σk follows the standard normal distribution, we get
lim P
k
k→∞
i1
Sk
≤y
Xi ≤ uki ,
σk
e−τ Φ y ,
3.33
which completes the proof.
We now only give the proof of Theorem 2.1. Theorem 2.2 is a special case of
Theorem 2.1.
Proof of Theorem 2.1. The idea of this proof is similar to that of Theorem 1.1 in Csáki and
Gonchigdanzan 7. In order to prove Theorem 2.1, it is enough to show that
Var
n
1
k1
for all fixed ∈ −∞, ∞.
k
I
k
i1
Sk
≤y
Xi ≤ uki ,
σk
log n2
log log n1
3.34
10
Journal of Inequalities and Applications
Let ξk I
k
≤ uki , Sk /σk ≤ y − P ki1 Xi ≤ uki , Sk /σk ≤ y, we have
n
k
1 Sk
I
Var
≤y
Xi ≤ uki ,
k i1
σk
k1
i1 Xi
n
1
1 2
≤
Eξk 2
|Eξk ξl | : F1 F2 .
2
kl
k
k1
1≤k<l≤n
3.35
Since {ξk } are bounded,
F1 n
1
< ∞.
k2
k1
3.36
The remainder is to estimate F2 . Notice
k
l
Sk
Sk
≤ y ,I
≤y
Xi ≤ uki ,
Xi ≤ uki ,
|Eξk ξl | Cov I
σk
σk
i1
i1
l
Sk
≤y
−I
Xi ≤ uki ,
σk
ik1
k
l
Sk
Sk
Cov I
≤ y ,I
≤y
Xi ≤ uki ,
Xi ≤ uki ,
σ
σ
k
k
i1
i1
3.37
l
l
Sk
Sk
≤ EI
≤y −I
≤y Xi ≤ uki ,
Xi ≤ uki ,
i1
σk
σ
k
ik1
k
l
Sk
Sk
Cov I
≤ y ,I
≤y
Xi ≤ uki ,
Xi ≤ uki ,
.
σ
σ
k
k
i1
i1
By Lemmas 3.1 and 3.2, we infer that if k < β2 llog log l22ε /c22 log l and k < 1,
|Eξk ξl | 1
1
log log n
k1/2 log l1/2
1ε
l1/2 log log l
k
l
3.38
for some > 0. By the arguments similar to that of Theorem 1 in Dudziński 13, we can get
F2 log n2
log log n1
So by Lemma 3.1 of Csáki and Gonchigdanzan 7 and Lemma 3.3,
n
k
1 1 Sk
lim
I
≤ y e−τ Φ y
Xi ≤ uki ,
n → ∞ log n
k i1
σk
k1
which completes the proof.
3.39
.
a.s.
3.40
Journal of Inequalities and Applications
11
0.013
0.0125
0.012
Actual error
0.0115
0.011
0.0105
0.01
0.0095
0.009
0.0085
0
100
200
300
400
500
600
700
800
n
Figure 1: T he actual error, Δn , for rn 1/nlog n1/2 log log n and x, y −1, −1.
0.04
0.038
Actual error
0.036
0.034
0.032
0.03
0.028
0.026
0
100
200
300
400
500
600
700
800
n
Figure 2: T he actual error, Δn , for rn 1/nlog n1/2 log log n and x, y 0, 0.
4. Numerical Analysis
The aim of this section is to calculate the actual convergence rate of
n
1 1
Sk
I Mk ≤ a−1
x
b
,
≤
y
−→ exp −e−x Φ y
k
k
log n k1 k
σk
for finite; that is, calculate
n
−x 1 1
Sk
Δn x, y I Mk ≤ ak x bk ,
≤ y − exp −e Φ y ,
log n k1 k
σk
where an 2 log n−1/2 and bn 2 log n1/2 − log log n log 4π/22 log n1/2 .
4.1
4.2
12
Journal of Inequalities and Applications
0.13
0.125
0.12
Actual error
0.115
0.11
0.105
0.01
0.095
0.09
0.085
0
100
200
300
400
500
600
700
800
n
Figure 3: T he actual error, Δn , for rn 1/nlog n1/2 log log n and x, y 1, 1.
0.055
0.05
0.045
Actual error
0.04
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
100
200
300
400
500
600
700
800
n
Figure 4: T he actual error, Δn , for rn 1/nlog n1/2 log log nlog log log n and x, y −1, −1.
Firstly, we will construct a standardized triangular Gaussian array {Xn,j , 1 ≤ j ≤
n, n ≥ 1} with equal correlation rn in n th array for ≥ 1. Meanwhile, the sequence rn must
satisfy the conditions 2.1, 2.2, and 2.3. By Leadbetter et al. 15, we can construct the
Gaussian array by i.i.d Gaussian sequence; that is, let rn to a convex sequence, ξ1 , ξ2 , . . . is a
standardized i.i.d Gaussian sequence, and η is also a standardized normal random variable
which is independent of ξk k ≥ 1. For each ≥ 1, let
Xij 1 − ri 1/2 ξj ri1/2 η,
4.3
where 1, 2, . . . , i. Obviously, Xij 1 ≤ j ≤ i is a zero mean normal sequence with equal
correlation. By this way, we get the Gaussian array needed.
Journal of Inequalities and Applications
13
0.11
0.1
0.09
Actual error
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0
100
200
300
400
500
600
700
800
n
Figure 5: T he actual error, Δn , for rn 1/nlog n1/2 log log nlog log log n and x, y 0, 0.
0.25
Actual error
0.2
0.15
0.1
0.05
0
0
100
200
300
400
500
600
700
800
n
Figure 6: T he actual error, Δn , for rn 1/nlog n1/2 log log nlog log log n and x, y 1, 1.
Figures 1 to 3 give the actual error, Δn , for rn 1/nlog n1/2 log log n and x, y −1, −1, 0, 0, 1, 1. In each figure, the actual error shocks tend to zero as n increases. The
overall performance of the actual error becomes better as x, y 0, 0.
Figures 4 to 6 give the actual error, Δn , for
1
rn ,
1/2 log log n log log log n
n log n
4.4
x, y −1, −1, 0, 0, 1, 1.
In each figure, the actual error shocks also tend to zero as n increases. Also the overall
performance of the actual error becomes better as x, y 0, 0.
14
Journal of Inequalities and Applications
Acknowledgments
The authors wish to thank the referees for some useful comments. The research was partially
supported by the National Natural Science Foundation of China Grant no. 70371061, the
National Natural Science Foundation of China Grant no. 10971227, and the Program for ET
of Chongqing Higher Education Institutions Grant no. 120060-20600204.
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