Almost sure central limit theorems for the maxima of Gaussian functions

Abstract

In this article, we proved the almost sure central limit theorem for the maxima of two types of functions of stationary Gaussian sequence including chi-random sequence and Gaussian order statistics sequence, where the covariance function r_n of the Gaussian sequence satisfies the condition $r_n\hspace{0.17em}\text{log}\hspace{0.17em}n{(\hspace{0.17em}\text{log}\hspace{0.17em}\hspace{0.17em}\text{log}\hspace{0.17em}n)}^{1+\epsilon }=O(1)$ for some $\epsilon >0.$

Keywords:

• Almost sure central limit theorem
• maxima
• stationary Gaussian sequences
• chi-random sequences
• Gaussian order statistic sequences

AMS Classification:

• Primary 62F15
• secondary 60G70

Additional information

Funding

Research was supported by Zhejiang Provincial Natural Science Foundation of China (No. LY18A010020), Xinmiao Talents Project of Zhejiang Province of China (Grant No. 2020R417018) and SRT of Jiaxing University.