Functional central limit theorem for topological functionals of Gaussian critical points
Probability
2025-12-16 v3
Abstract
We consider Betti numbers of the excursion of a smooth Euclidean Gaussian field restricted to a rectangular window, in the asymptotics where the window grows to R^d . With motivations coming from Topological Data Analysis, we derive a functional Central Limit Theorem where the varying argument is the thresholding parameter, under assumptions of regularity and covariance decay for the field and its derivatives. We also show fixed-level CLTs coming from martingale based techniques inspired from the theory of geometric stabilisation, and limiting non-degenerate variance.
Cite
@article{arxiv.2411.11429,
title = {Functional central limit theorem for topological functionals of Gaussian critical points},
author = {Christian Hirsch and Raphaël Lachièze-Rey},
journal= {arXiv preprint arXiv:2411.11429},
year = {2025}
}