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Almost Sure Convergence for the Maximum and the Sum of Nonstationary Guassian Sequences
Journal of Inequalities and Applications volume 2010, Article number: 856495 (2010)
Abstract
Let (,
) be a standardized nonstationary Gaussian sequence. Let
max
denote the partial maximum and
for the partial sum with
(Var
. In this paper, the almost sure convergence of (
,
) 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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F856495/MediaObjects/13660_2009_Article_2279_Equ1_HTML.gif)
provided
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F856495/MediaObjects/13660_2009_Article_2279_Equ2_HTML.gif)
where denotes an indicator function,
is the standard normal distribution function, and
.
is the partial maximum of a standard stationary Gaussian sequence
with correlation
. The norming constants
and
are defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F856495/MediaObjects/13660_2009_Article_2279_Equ3_HTML.gif)
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 be stationary Gaussian sequences and
,
,
, for all
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F856495/MediaObjects/13660_2009_Article_2279_Equ4_HTML.gif)
if
(C1)
(C2)
(C3)
Or
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F856495/MediaObjects/13660_2009_Article_2279_Equ5_HTML.gif)
for some .
is a positive slowly varying function at infinity. Here
means
.
This paper focuses on extending (1.4) to nonstationary Gaussian sequences 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 , denote the correlations of standard nonstationary Gaussian sequence
.
, and
are defined as before. The main results are the following.
Theorem 2.1.
Let be a standardized nonstationary Gaussian sequence. Suppose that there exists numerical sequence
such that
for some
and
is bounded, where
. If
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F856495/MediaObjects/13660_2009_Article_2279_Equ6_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F856495/MediaObjects/13660_2009_Article_2279_Equ7_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F856495/MediaObjects/13660_2009_Article_2279_Equ8_HTML.gif)
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F856495/MediaObjects/13660_2009_Article_2279_Equ9_HTML.gif)
for all .
Theorem 2.2.
For the nonstationary Gaussian sequence , under the conditions (2.1)–(2.3), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F856495/MediaObjects/13660_2009_Article_2279_Equ10_HTML.gif)
for all where
and
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 satisfy the conditions (2.1)–(2.3). Assume that
is bounded. Then for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F856495/MediaObjects/13660_2009_Article_2279_Equ11_HTML.gif)
Proof.
We will start with the following observations. For all ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F856495/MediaObjects/13660_2009_Article_2279_Equ12_HTML.gif)
Clearly,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F856495/MediaObjects/13660_2009_Article_2279_Equ13_HTML.gif)
By (2.2), for large there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F856495/MediaObjects/13660_2009_Article_2279_Equ14_HTML.gif)
By (2.3) and (3.4), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F856495/MediaObjects/13660_2009_Article_2279_Equ15_HTML.gif)
for large . Obviously,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F856495/MediaObjects/13660_2009_Article_2279_Equ16_HTML.gif)
which implies that there exist and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F856495/MediaObjects/13660_2009_Article_2279_Equ17_HTML.gif)
Notice,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F856495/MediaObjects/13660_2009_Article_2279_Equ18_HTML.gif)
By the Normal Comparison Lemma [13, Theorem ], we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F856495/MediaObjects/13660_2009_Article_2279_Equ19_HTML.gif)
Since is bounded, for large
and some absolute positive constant
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F856495/MediaObjects/13660_2009_Article_2279_Equ20_HTML.gif)
So,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F856495/MediaObjects/13660_2009_Article_2279_Equ21_HTML.gif)
Similarly,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F856495/MediaObjects/13660_2009_Article_2279_Equ22_HTML.gif)
It remains to estimate . It is easy to check that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F856495/MediaObjects/13660_2009_Article_2279_Equ23_HTML.gif)
By the arguments similar to that of Lemma in Csáki and Gonchigdanzan [7], we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F856495/MediaObjects/13660_2009_Article_2279_Equ24_HTML.gif)
By the Normal Comparison Lemma and (3.4), we derive that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F856495/MediaObjects/13660_2009_Article_2279_Equ25_HTML.gif)
Combining with above analysis, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F856495/MediaObjects/13660_2009_Article_2279_Equ26_HTML.gif)
The proof is complete.
We also need the following auxiliary result.
Lemma 3.2.
Suppose that the standardized nonstationary Gaussian sequences satisfy the conditions (2.1)–(2.3). Assume that
is bounded; then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F856495/MediaObjects/13660_2009_Article_2279_Equ27_HTML.gif)
for , where
.
Proof.
By (2.2) and (2.3), for , we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F856495/MediaObjects/13660_2009_Article_2279_Equ28_HTML.gif)
Clearly,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F856495/MediaObjects/13660_2009_Article_2279_Equ29_HTML.gif)
which implies that there exist and
such that for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F856495/MediaObjects/13660_2009_Article_2279_Equ30_HTML.gif)
For , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F856495/MediaObjects/13660_2009_Article_2279_Equ31_HTML.gif)
Condition (2.2) implies that there exist positive numbers and
such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F856495/MediaObjects/13660_2009_Article_2279_Equ32_HTML.gif)
So there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F856495/MediaObjects/13660_2009_Article_2279_Equ33_HTML.gif)
for .By applying the inequalities above and the Normal Comparison Lemma, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F856495/MediaObjects/13660_2009_Article_2279_Equ34_HTML.gif)
By (3.10), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F856495/MediaObjects/13660_2009_Article_2279_Equ35_HTML.gif)
Similarly,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F856495/MediaObjects/13660_2009_Article_2279_Equ36_HTML.gif)
While (3.22) implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F856495/MediaObjects/13660_2009_Article_2279_Equ37_HTML.gif)
the proof is complete.
We also need the following auxiliary result.
Lemma 3.3.
Let be a standardized nonstationary Gaussian sequences satisfying assumptions (2.1)–(2.3). Assume that
for some
and
is bounded. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F856495/MediaObjects/13660_2009_Article_2279_Equ38_HTML.gif)
for all .
Proof.
By the Normal Comparison Lemma and the proof of Lemma 3.1, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F856495/MediaObjects/13660_2009_Article_2279_Equ39_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F856495/MediaObjects/13660_2009_Article_2279_Equ40_HTML.gif)
which implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F856495/MediaObjects/13660_2009_Article_2279_Equ41_HTML.gif)
By Theorem of Leadbetter et al. [15], we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F856495/MediaObjects/13660_2009_Article_2279_Equ42_HTML.gif)
Since follows the standard normal distribution, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F856495/MediaObjects/13660_2009_Article_2279_Equ43_HTML.gif)
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 in Csáki and Gonchigdanzan [7]. In order to prove Theorem 2.1, it is enough to show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F856495/MediaObjects/13660_2009_Article_2279_Equ44_HTML.gif)
for all fixed .
Let , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F856495/MediaObjects/13660_2009_Article_2279_Equ45_HTML.gif)
Since are bounded,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F856495/MediaObjects/13660_2009_Article_2279_Equ46_HTML.gif)
The remainder is to estimate . Notice
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F856495/MediaObjects/13660_2009_Article_2279_Equ47_HTML.gif)
By Lemmas 3.1 and 3.2, we infer that if and
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F856495/MediaObjects/13660_2009_Article_2279_Equ48_HTML.gif)
for some . By the arguments similar to that of Theorem
in Dudziński [13], we can get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F856495/MediaObjects/13660_2009_Article_2279_Equ49_HTML.gif)
So by Lemma of Csáki and Gonchigdanzan [7] and Lemma 3.3,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F856495/MediaObjects/13660_2009_Article_2279_Equ50_HTML.gif)
which completes the proof.
4. Numerical Analysis
The aim of this section is to calculate the actual convergence rate of
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F856495/MediaObjects/13660_2009_Article_2279_Equ51_HTML.gif)
for finite; that is, calculate
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F856495/MediaObjects/13660_2009_Article_2279_Equ52_HTML.gif)
where and
Firstly, we will construct a standardized triangular Gaussian array with equal correlation
in
th array for
. Meanwhile, the sequence
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
to a convex sequence,
is a standardized i.i.d Gaussian sequence, and
is also a standardized normal random variable which is independent of
. For each
, let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F856495/MediaObjects/13660_2009_Article_2279_Equ53_HTML.gif)
where . Obviously,
is a zero mean normal sequence with equal correlation. By this way, we get the Gaussian array needed.
Figures 1 to 3 give the actual error, , for
and
. In each figure, the actual error shocks tend to zero as
increases. The overall performance of the actual error becomes better as
.
Figures 4 to 6 give the actual error, , for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F856495/MediaObjects/13660_2009_Article_2279_Equ54_HTML.gif)
In each figure, the actual error shocks also tend to zero as increases. Also the overall performance of the actual error becomes better as
.
References
Fahrner I, Stadtmüller U: On almost sure max-limit theorems. Statistics and Probability Letters 1998, 37(3):229–236. 10.1016/S0167-7152(97)00121-1
Cheng S, Peng L, Qi Y: Almost sure convergence in extreme value theory. Mathematische Nachrichten 1998, 190: 43–50. 10.1002/mana.19981900104
Berkes I, Csáki E: A universal result in almost sure central limit theory. Stochastic Processes and Their Applications 2001, 94(1):105–134. 10.1016/S0304-4149(01)00078-3
Peng Z, Li J, Nadarajah S: Almost sure convergence of extreme order statistics. Electronic Journal of Statistics 2009, 3: 546–556. 10.1214/08-EJS303
Peng Z, Wang L, Nadarajah S: Almost sure central limit theorem for partial sums and maxima. Mathematische Nachrichten 2009, 282(4):632–636. 10.1002/mana.200610760
Tan Z, Peng Z: Almost sure convergence for non-stationary random sequences. Statistics and Probability Letters 2009, 79(7):857–863. 10.1016/j.spl.2008.11.005
Csáki E, Gonchigdanzan K: Almost sure limit theorems for the maximum of stationary Gaussian sequences. Statistics and Probability Letters 2002, 58(2):195–203. 10.1016/S0167-7152(02)00128-1
Lin F: Almost sure limit theorem for the maxima of strongly dependent gaussian sequences. Electronic Communications in Probability 2009, 14: 224–231.
Chen S, Lin Z: Almost sure max-limits for nonstationary Gaussian sequence. Statistics and Probability Letters 2006, 76(11):1175–1184. 10.1016/j.spl.2005.12.018
Peng Z, Nadarajah S: Almost sure limit theorems for Gaussian sequences. Theory Probability and its Applidations. In press
Anderson CW, Turkman KF: Limiting joint distributions of sums and maxima in a statistical context. Theory Probability and its Applidations 1993, 37: 314–316. 10.1137/1137063
Peng Z, Wang L, Nadarajah S: Almost sure central limit theorem for partial sums and maxima. Mathematische Nachrichten 2009, 282(4):632–636. 10.1002/mana.200610760
Dudziński M: An almost sure limit theorem for the maxima and sums of stationary Gaussian sequences. Statistics and Probability Letters 2003, 23: 139–152.
Dudziński M: The almost sure central limit theorems in the joint version for the maxima and sums of certain stationary Gaussian sequences. Statistics and Probability Letters 2008, 78(4):347–357. 10.1016/j.spl.2007.07.007
Leadbetter MR, Lindgren G, Rootzén H: Extremes and Related Properties of Random Sequences and Processes. Springer, New York, NY, USA; 1993.
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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Zhao, S., Peng, Z. & Wu, S. Almost Sure Convergence for the Maximum and the Sum of Nonstationary Guassian Sequences. J Inequal Appl 2010, 856495 (2010). https://doi.org/10.1155/2010/856495
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DOI: https://doi.org/10.1155/2010/856495