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

Jean-Marc Azaïs (1) , Jose R. Leon (2)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 IMERL - Instituto de Matemática y Estadística Rafael Laguardia [Montevideo]

For a smooth vectorial stationary Gaussian random field X : Ω × R d → 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 : Ω × R D → R d with D > d.

Mots clés: Level Sets; Kac-Rice formula; Moments; Random fields 2000 Mathematics Subject Classification: 60G15; 60G60

Domaines: Probabilités [math.PR]

Jean-Marc Azaïs, Jose R. Leon. Necessary and sufficient conditions for the finiteness of the second moment of the measure of level sets. 2019. ⟨hal-02142096⟩