Comparison Inequalities for Order Statistics of Gaussian Arrays

TL;DR

This paper extends classical comparison inequalities like Slepian's inequality to order statistics of Gaussian arrays, enabling new insights into the behavior of Gaussian process maxima and their applications.

Contribution

It introduces new comparison inequalities for order statistics of Gaussian arrays, expanding tools for analyzing Gaussian processes.

Findings

Derived inequalities for order statistics of Gaussian arrays.

Analyzed lower tail behavior of order statistics in self-similar Gaussian processes.

Established mixed Gumbel limit theorems for these order statistics.

Abstract

Normal comparison lemma and Slepian's inequality are essential tools in the study of Gaussian processes. In this paper we extend normal comparison lemma and derive various related comparison inequalities including Slepian's inequality for order statistics of two Gaussian arrays.The derived results can be applied in numerous problems related to the study of the supremum of order statistics of Gaussian processes. In order to illustrate the range of possible applications, we analyze the lower tail behaviour of order statistics of self-similar Gaussian processes and derive mixed Gumbel limit theorems.