On the excursion area of perturbed Gaussian fields

¹ Conservatoire National des Arts et Métiers, Paris, EA4629, 292 rue Saint-Martin, Paris Cedex 03, France.
² MAP5 UMR CNRS 8145, Université Paris Descartes, 45 rue des Saints-Pères, 75270 Paris Cedex 06, France.
³ Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano-Bicocca Via Roberto Cozzi, 55, 20125 Milano MI, Italy.

^* Corresponding author: elena.dibernardino@lecnam.net

Received: 3 June 2019
Accepted: 13 January 2020

Abstract

We investigate Lipschitz-Killing curvatures for excursion sets of random fields on ℝ² under a very specific perturbation, namely a small spatial-invariant random perturbation with zero mean. 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 variance which turns out to be asymptotically normal and unbiased, allowing to make inference through sparse information on the field.