Functional central limit theorem for topological functionals of Gaussian critical points

TL;DR

This paper establishes a functional central limit theorem for Betti numbers of Gaussian fields' excursions, advancing understanding in topological data analysis by analyzing their asymptotic behavior as the observation window expands.

Contribution

It introduces a novel functional CLT for topological functionals of Gaussian fields, combining geometric stabilization techniques with asymptotic analysis.

Findings

Proves a functional CLT for Betti numbers with growing observation window.

Establishes fixed-level CLTs using martingale methods.

Identifies non-degenerate limiting variance for the topological functionals.

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.