- Research Article
- Open Access

# 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

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.

## Keywords

- Standard Normal Distribution
- Average Wind Speed
- Actual Error
- Wind Gust
- Extreme Wind

## 1. Introduction

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

if

(C1)

(C2)

(C3)

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.

for all .

Theorem 2.2.

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.

Proof.

The proof is complete.

We also need the following auxiliary result.

Lemma 3.2.

for , where .

Proof.

the proof is complete.

We also need the following auxiliary result.

Lemma 3.3.

for all .

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.

for all fixed .

which completes the proof.

## 4. Numerical Analysis

where and

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

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## 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.