On the maxima of continuous and discrete time Gaussian order statistics processes

TL;DR

This paper investigates the asymptotic relationship between the maxima of continuous and discretely sampled Gaussian order statistics processes, revealing conditions for their independence or dependence based on process dependence and sampling density.

Contribution

It provides a detailed analysis of the asymptotic dependence structure of maxima in Gaussian order statistics processes under various sampling schemes.

Findings

Maxima are asymptotically independent for weakly dependent processes with sparse sampling.

Maxima are asymptotically dependent in other cases.

Conditions delineating dependence and independence regimes are established.

Abstract

In this paper, we study the asymptotic relation between the maximum of acontinuous order statistics process formed by stationary Gaussian processesand the maximum of this process sampled at discrete time points. It is shown that, these two maxima are asymptotically independent when the Gaussian processes are weakly dependent and the discrete points are sufficient sparse, while for other case, these two maxima are asymptotically dependent.