On a general Kac-Rice formula for the measure of a level set

TL;DR

The paper develops a general Kac-Rice framework for the expected $(D-d)$-dimensional measure of level sets $\\mathcal{L}_u(B)$ of random fields $X: T\\to\\mathbb{R}^d$ with $D\\ge d$, under a weak Bulinskaya-type regularity condition that ensures rectifiability. It proves a Kac-Rice formula for all levels, including non-Gaussian fields, by leveraging a push-forward construction via the co-area formula, and extends the result to $D>d$ using Crofton integral geometry. The work broadens the applicability of Kac-Rice beyond Gaussian settings, including extensions to manifolds and higher moments, and demonstrates utility through diverse examples such as Gaussian critical points, likelihood landscapes, gravitational microlensing, and shot-noise processes. Overall, the results provide a versatile, geometric-analytic toolkit for quantifying level-set measures in high-dimensional random fields with broad practical impact.

Abstract

Let $X(\cdot) $ be a random field $\mathbb{R}^D \to \mathbb{R}^d$, $D\geq d$. We first studied the level set $X^{-1}( u) $, $u \in \mathbb{R}^d$. In particular we gave a weak condition for this level set to be rectifiable. Then, we established a Kac-Rice formula to compute the $D-d$ Hausdorff measure. Our results extend known results, particularly in the non-Gaussian case where we obtained a very general result. We conclude with several extensions and examples of application, including functions of Gaussian random field, zeroes of the likelihood, gravitational microlensing, shot-noise.

Figures (1)

• Figure 1: Tapered functions

Theorems & Definitions (40)

• Lemma 2.1: Bulinskayabulinskaya
• Proposition 2.2: Bulinskaya-type Lemma
• Theorem 2.1: Rice formula for the expectation
• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Theorem 2.2
• Remark 5
• Remark 6