Functional central limit theorem for topological functionals of Gaussian critical points

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.

💡 Research Summary

The paper addresses a fundamental problem at the intersection of stochastic geometry and topological data analysis (TDA): establishing central limit theorems for topological descriptors of Gaussian excursion sets when the observation window expands to the whole Euclidean space. The authors focus on Betti numbers, which count k‑dimensional holes in the excursion set {x∈ℝⁿ : F(x) ≥ u}, and treat them as additive functionals of the connected components that lie strictly inside a growing rectangular window.

Model and Assumptions
The random field F: ℝᵈ → ℝ is assumed to be stationary, centered, unit‑variance, and sufficiently smooth: its covariance function C belongs to C^{ℓ₀‑2}(ℝᵈ) for some ℓ₀>2 and satisfies a polynomial decay |∂^α C(x)| ≤ c(1+‖x‖)^{‑η} with η>2d. Moreover, C is non‑degenerate (C(x)≠0) so that almost surely the field is a Morse function, guaranteeing isolated non‑degenerate critical points. These regularity conditions ensure that the excursion sets have well‑behaved geometry and that the number of critical points in any bounded region has finite moments of order ℓ₀.

Topological Functionals
For a fixed level u, the excursion set E_u = {F ≥ u} intersected with a window W_n yields a collection of bounded connected components C(E_u∩W_n, B_{W_n}) that do not touch the window boundary. The k‑th Betti number β_k(u;F,W_n) is defined as the number of k‑dimensional homology classes (holes) among these components. The authors embed Betti numbers into a broader class of “topologically additive” functionals: any functional that (i) depends only on the isotopy class of each component and (ii) grows at most linearly with the number of critical points above the level. This abstraction allows the same proof machinery to apply to other topological descriptors.

Fixed‑Level Multivariate CLT (Theorem 5)
Let u₁,…,u_K belong to an interval I that satisfies a percolation‑type decay assumption (Assumption 2). Define the centered, normalized vector

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