A central limit theorem for the Euler integral of a Gaussian random field

Abstract

Euler integrals of deterministic functions have recently been shown to have a wide variety of possible applications, including in signal processing, data aggregation and network sensing. Adding random noise to these scenarios, as is natural in the majority of applications, leads to a need for statistical analysis, the first step of which requires asymptotic distribution results for estimators. The first such result is provided in this paper, as a central limit theorem for the Euler integral of pure, Gaussian, noise fields.

A central limit theorem for Lipschitz–Killing curvatures of Gaussian excursions

“…Hence, G(F, t, ·) is continuous as a product of continuous functions. To apply dominated convergence for y ′ → y in (24), observe that Lemmata B.2 and B.3 yield…”

“…To apply dominated convergence for y ′ → y in (24), observe that Lemmata B.2 and B.3 yield for t ∈ W F v(F ) the estimates…”

“…This access to normal approximations is very popular and is used in various settings similar to ours, cf. [24], who show a central limit theorem for the Euler integration of random functions, [7], who investigate Gaussian excursions on the 2-sphere or [25], who studies critical points of random Fourier series on the m-dimensional torus.…”