Abstract
We prove some almost sure central limit theorems for the maxima of strongly dependent nonstationary Gaussian vector sequences under some mild conditions. The results extend the ASCLT to nonstationary Gaussian vector sequences and give substantial improvements for the weight sequence obtained by Lin et al. (Comput. Math. Appl. 62(2):635-640, 2011).
Highlights
The almost sure central limit theorem (ASCLT) has served as a basis for a large group of investigations of fundamental significance both in the theory of probability and in its numerous applications to statistics, natural sciences, engineering, and economics
An influential work is Csáki and Gonchigdanzan [ ], which proved the almost sure limit theorem for the maximum of stationary weakly dependent sequence
Lin [ ] considered the theorem which ASCLT version of the theorem proved by Leadbetter et al [ ]
Summary
The almost sure central limit theorem (ASCLT) has served as a basis for a large group of investigations of fundamental significance both in the theory of probability and in its numerous applications to statistics, natural sciences, engineering, and economics. Lin et al [ ] partially extended [ ] to the case of strongly dependent nonstationary Gaussian sequences and obtained the following theorem. Theorem A Let {ξn : n ≥ } be a sequence of nonstationary standard Gaussian random variables with covariances rij satisfying Ξi(d)) : i ≥ } be a standardized nonstationary Gaussian vector sequence with
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