# Necessary and sufficient conditions for the finiteness of the second moment of the measure of level sets

## TL;DR

This paper establishes necessary and sufficient conditions for the finiteness of the second moment of the measure of level sets in smooth Gaussian random fields, extending classical results to vector-valued fields and critical points.

## Contribution

It provides a comprehensive characterization of when the second moment of level set measures is finite for vector-valued Gaussian fields, using an adapted proof method.

## Key findings

- Derived necessary and sufficient conditions for second moment finiteness.
- Extended the method to critical points and higher-dimensional level sets.
- Connected classical Gaussian process results to vector-valued fields.

## Abstract

For a smooth vectorial stationary Gaussian random field $X : \Omega \times \mathbb{R}^d \to \mathbb{R}^d$, we give necessary and sufficient conditions to have a finite second moment for the number of roots of $X(t) - u$. The results are obtained by using a method of proof inspired on the one obtained by D. Geman for stationary Gaussian processes long time ago. Afterwards the same method is applied to the number of critical points of a scalar random field and also to the level set of a vectorial process $X : \Omega \times \mathbb{R}^D \to \mathbb{R}^d$ with $D > d$.