# Almost Sure Convergence for the Maximum and the Sum of Nonstationary Guassian Sequences

- Shengli Zhao
^{1}Email author, - Zuoxiang Peng
^{2}and - Songlin Wu
^{1}

**2010**:856495

https://doi.org/10.1155/2010/856495

© Shengli Zhao et al. 2010

**Received: **18 December 2009

**Accepted: **5 April 2010

**Published: **19 May 2010

## Abstract

## 1. Introduction

For some extensions of (1.1), see Chen and Lin [9] and Peng and Nadarajah [10].

if

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.

Theorem 2.2.

## 3. Proof of the Main Results

To prove the main results, we need some auxiliary lemmas.

Lemma 3.1.

Proof.

The proof is complete.

We also need the following auxiliary result.

Lemma 3.2.

Proof.

the proof is complete.

We also need the following auxiliary result.

Lemma 3.3.

Proof.

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.

which completes the proof.

## 4. Numerical Analysis

where . Obviously, is a zero mean normal sequence with equal correlation. By this way, we get the Gaussian array needed.

## Declarations

### 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).

## Authors’ Affiliations

## 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)00121-1MATHMathSciNetView ArticleGoogle Scholar - Cheng S, Peng L, Qi Y: Almost sure convergence in extreme value theory.
*Mathematische Nachrichten*1998, 190: 43–50. 10.1002/mana.19981900104MATHMathSciNetView ArticleGoogle Scholar - 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)00078-3MATHMathSciNetView ArticleGoogle Scholar - Peng Z, Li J, Nadarajah S: Almost sure convergence of extreme order statistics.
*Electronic Journal of Statistics*2009, 3: 546–556. 10.1214/08-EJS303MATHMathSciNetView ArticleGoogle Scholar - 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.200610760MATHMathSciNetView ArticleGoogle Scholar - 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.2008.11.005MATHMathSciNetView ArticleGoogle Scholar - 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)00128-1MATHMathSciNetView ArticleGoogle Scholar - Lin F: Almost sure limit theorem for the maxima of strongly dependent gaussian sequences.
*Electronic Communications in Probability*2009, 14: 224–231.MATHMathSciNetView ArticleGoogle Scholar - 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.2005.12.018MATHMathSciNetView ArticleGoogle Scholar - Peng Z, Nadarajah S: Almost sure limit theorems for Gaussian sequences. Theory Probability and its Applidations. In pressGoogle Scholar
- Anderson CW, Turkman KF: Limiting joint distributions of sums and maxima in a statistical context.
*Theory Probability and its Applidations*1993, 37: 314–316. 10.1137/1137063MathSciNetView ArticleGoogle Scholar - 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.200610760MATHMathSciNetView ArticleGoogle Scholar - Dudziński M: An almost sure limit theorem for the maxima and sums of stationary Gaussian sequences.
*Statistics and Probability Letters*2003, 23: 139–152.MATHGoogle Scholar - 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.2007.07.007MATHMathSciNetView ArticleGoogle Scholar - Leadbetter MR, Lindgren G, Rootzén H:
*Extremes and Related Properties of Random Sequences and Processes*. Springer, New York, NY, USA; 1993.Google Scholar

## Copyright

This article is published under license to BioMed Central Ltd. 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.