On the extreme order statistics for stationary Gaussian sequences subject to random missing observations

TL;DR

This paper investigates the asymptotic behavior of extreme values in stationary Gaussian sequences with missing observations, deriving joint limit distributions for maxima and exceedance processes.

Contribution

It introduces a novel analysis of extreme order statistics in Gaussian sequences with random missing data, providing joint limit distributions for maxima and exceedance points.

Findings

Derived joint limit distribution of extreme order statistics.

Established the limiting behavior of exceedance point processes.

Obtained joint distributions of maxima locations and heights.

Abstract

Let $\mathbf{X}=\{X_{n}\}_{n\geq 1}$ be a sequence of stationary Gaussian variables and suppose that only some of the random variables from $\mathbf{X}$ can be observed. In this paper, by studying the limiting properties of multidimensional exceedance point processes for $\mathbf{X}$, we derived the joint limit distribution of extreme order statistics for the Gaussian sequence $\mathbf{X}$ and its observed ones. The joint limit distribution of the locations and heights of the maxima for the Gaussian sequence $\mathbf{X}$ and its observed ones are also obtained.