Bicovariograms and Euler characteristic II. Random fields excursions

Raphaël Lachièze-Rey (1)

1 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5

Let f be a C1 bivariate function with Lipschitz derivatives, and F its level set at elvel lambda. We give an expression of the Euler characteristic of F in terms of the three-points indicator functions of the set. If f is a two-dimensional C1 random field and the derivatives of F have Lipschitz constants with finite moments of sufficiently high order, taking the expectation provides an expression of the mean Euler characteristic in terms of the third order marginal of the field. We provide sufficient conditions and explicit formulas for Gaussian fields, relaxing the usual C2 Morse hypothesis.

Mots clés: C1; 1 functions; intrinsic volumes; Euler characteristic; Random fields; Gaussian processes; covariograms

Domaines: Probabilités [math.PR]

Raphaël Lachièze-Rey. Bicovariograms and Euler characteristic II. Random fields excursions. 2015. ⟨hal-01207503v1⟩