Almost sure limit theorem for the order statistics of stationary Gaussian sequences

Abstract

In this paper, by using a new comparison inequality for order statistics of Gaussian variables,
we proved an almost sure central limit
theorem for extreme order statistics of stationary Gaussian sequences with covariance
$r_{n}$ under the condition $r_{n}\log n(\log\log n)^{1+\varepsilon}=O(1)$.
A similar result on intermediate order statistics is also proved for stationary Gaussian sequences.
The obtained results improve some of the existing results.