Asymptotic formula for the conjunction probability of smooth stationary Gaussian fields

Abstract

. Let { X i ( t ) : t ∈ S ⊂ R d } i =1 , 2 ,...,n be independent copies of a stationary centered Gaussian field with almost surely smooth sample paths. In this paper, we are interested in the conjunction probability defined as P ( ∃ t ∈ S : X i ( t ) ≥ u, ∀ i = 1 , 2 , . . . , n ) for a given threshold level u . As u → ∞ , we will provide an asymptotic formula for the conjunction probability. This asymptotic formula is derived from the behaviour of the volume of the set of local maximum points. The proof relies on a result of Azaïs and Wschebor (2014) describing the shape of the excursion set of a stationary centered Gaussian field. Our result partially confirms the validity of the Euler characteristic method.