On the finiteness of the second moment of the number of critical points of Gaussian random fields

TL;DR

This paper proves that the second moment of the number of critical points in sufficiently regular Gaussian random fields on compact stratified manifolds is finite, without requiring stationarity, extending previous results.

Contribution

It establishes the finiteness of the second moment for critical points of non-stationary Gaussian fields on stratified manifolds, generalizing prior stationary assumptions.

Findings

Second moment of critical points is finite for regular Gaussian fields.

Results apply without stationarity assumptions.

Under stationarity, conditions imply the generalized Geman condition.

Abstract

We prove that the second moment of the number of critical points of any sufficiently regular random field, for example with almost surely $ C^3 $ sample paths, defined over a compact Whitney stratified manifold is finite. Our results hold without the assumption of stationarity - which has traditionally been assumed in other work. Under stationarity we demonstrate that our imposed conditions imply the generalized Geman condition of Estrade 2016.