Journal of Inequalities and Applications
Volume 2010 (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

1Department of Fundament Studies, Logistical Engineering University, Chongqing 401131, China
2School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

Received 18 December 2009; Accepted 5 April 2010

Academic Editor: Jewgeni Dshalalow

Copyright © 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.

Abstract

Let (Xn, n≥1) be a standardized nonstationary Gaussian sequence. Let Mn= max{Xk,1≤k≤n} denote the partial maximum and Sn=∑k−1nXk 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.