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.
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DOI : 10.1051/ps/2020002
Mots-clés : LK curvatures, Gaussian fields, perturbed fields, quantitative limit theorems, sojourn times, sparse inference for random fields, spatial-invariant random perturbations
@article{PS_2020__24_1_252_0,
author = {Di Bernardino, Elena and Estrade, Anne and Rossi, Maurizia},
title = {On the excursion area of perturbed {Gaussian} fields},
journal = {ESAIM: Probability and Statistics},
pages = {252--274},
publisher = {EDP-Sciences},
volume = {24},
year = {2020},
doi = {10.1051/ps/2020002},
mrnumber = {4090340},
zbl = {1447.60078},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps/2020002/}
}
TY - JOUR AU - Di Bernardino, Elena AU - Estrade, Anne AU - Rossi, Maurizia TI - On the excursion area of perturbed Gaussian fields JO - ESAIM: Probability and Statistics PY - 2020 SP - 252 EP - 274 VL - 24 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps/2020002/ DO - 10.1051/ps/2020002 LA - en ID - PS_2020__24_1_252_0 ER -
%0 Journal Article %A Di Bernardino, Elena %A Estrade, Anne %A Rossi, Maurizia %T On the excursion area of perturbed Gaussian fields %J ESAIM: Probability and Statistics %D 2020 %P 252-274 %V 24 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps/2020002/ %R 10.1051/ps/2020002 %G en %F PS_2020__24_1_252_0
Di Bernardino, Elena; Estrade, Anne; Rossi, Maurizia. On the excursion area of perturbed Gaussian fields. ESAIM: Probability and Statistics, Tome 24 (2020), pp. 252-274. doi : 10.1051/ps/2020002. http://www.numdam.org/articles/10.1051/ps/2020002/
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