# On the excursion area of perturbed Gaussian fields

## TL;DR

This paper studies how small random perturbations affect the geometric properties of excursion sets of Gaussian fields, deriving formulas and limit theorems that enable statistical inference on the perturbation.

## Contribution

It provides a new asymptotic analysis of perturbed Gaussian fields' excursion sets, including formulas, limit theorems, and an estimator for the perturbation parameter.

## Key findings

- Derived an expansion formula for mean curvatures under small perturbations.
- Established a non-Gaussian limit theorem in Wasserstein distance for the perturbed excursion area.
- Proposed an asymptotically normal and unbiased estimator for the perturbation.

## Abstract

We investigate Lipschitz-Killing curvatures for excursion sets of random fields on $\mathbb R^2$ under small spatial-invariant random perturbations. An expansion formula for mean curvatures is derived when the magnitude of the perturbation vanishes, which recovers the Gaussian Kinematic Formula at the limit by contiguity of the model. We develop an asymptotic study of the perturbed excursion area behaviour that leads to a quantitative non-Gaussian limit theorem, in Wasserstein distance, for fixed small perturbations and growing domain. When letting both the perturbation vanish and the domain grow, a standard Central Limit Theorem follows. Taking advantage of these results, we propose an estimator for the perturbation which turns out to be asymptotically normal and unbiased, allowing to make inference through sparse information on the field.