# High minima of non-smooth Gaussian processes

## TL;DR

This paper investigates the asymptotic behavior of the minimum values of non-smooth Gaussian processes over compact intervals, linking large deviation estimates with the small-ball problem and analyzing the distribution of the minimum's location.

## Contribution

It establishes new connections between large deviation estimates and the small-ball problem for non-smooth Gaussian processes, and studies the distribution of the minimum's location.

## Key findings

- Asymptotic estimates for the minimum of non-smooth Gaussian processes.
- Relation between large deviation behavior and the small-ball problem.
- Distribution of the minimum's location conditioned on high thresholds.

## Abstract

In this short note we study the asymptotic behaviour of the minima over compact intervals of Gaussian processes, whose paths are not necessarily smooth. We show that, beyond the logarithmic large deviation Gaussian estimates, this problem is closely related to the classical small-ball problem. Under certain conditions we estimate the term describing the correction to the large deviation behaviour. In addition, the asymptotic distribution of the location of the minimum, conditionally on the minimum exceeding a high threshold, is also studied.