Bicovariograms and Euler characteristic of 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 = {x ∈ R2 : f(x) λ} an upper level set of f, with λ ∈ R. We give a new expression of the Euler characteristic of F in terms of the three-points indicator functions of the set, related to its bicovariograms. We also derive a bound on the number of connected components of F in terms of the values of f and its gradient, valid in higher dimensions. In dimension 2, this bound allows to pass this identity to expectations if f’s partial derivatives have Lipschitz constants with finite moments of sufficiently high order, without the requirement of a bounded conditional density. This approach provides an expression of the mean Euler characteristic in terms of the field’s third order marginal. We give sufficient conditions and explicit formulas for Gaussian fields, relaxing the usual C2 Morse hypothesis.

Mots clés: Random fields; Euler characteristic; Gaussian processes; covariograms; intrinsic volumes; C11 functions

Domaines: Probabilités [math.PR]

Raphaël Lachièze-Rey. Bicovariograms and Euler characteristic of random fields excursions. 2015. ⟨hal-01207503v2⟩