Asymptotic behaviour of high Gaussian minima

TL;DR

This paper analyzes the asymptotic probability and shape of smooth Gaussian processes when their sample paths stay above a high level, providing detailed probabilistic and geometric insights.

Contribution

It offers a precise asymptotic analysis of the probability, shape, and location of minima of Gaussian processes conditioned on high-level exceedance.

Findings

Asymptotic probability of high-level exceedance derived

Conditional shape of the process above the high level characterized

Location of the process minimum given high-level exceedance identified

Abstract

We investigate what happens when an entire sample path of a smooth Gaussian process on a compact interval lies above a high level. Specifically, we determine the precise asymptotic probability of such an event, the extent to which the high level is exceeded, the conditional shape of the process above the high level, and the location of the minimum of the process given that the sample path is above a high level.