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

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.