Limit theorems for squared increment sums of the maximum of two isotropic fractional Brownian fields under fixed-domain asymptotics

Abstract

We study squared increment sums of the pointwise maximum of two independent and identically distributed isotropic fractional Brownian fields over a fixed two-dimensional domain. The fields are observed at the points of a homogeneous Poisson point process with intensity \(N\), and increments are computed along the edges of the associated Delaunay triangulation. In contrast with the case of a single fractional Brownian field, where centered squared increment sums satisfy a central limit theorem after the usual normalization, the pointwise maximum exhibits a different asymptotic regime. The dominant contribution comes from Delaunay edges located in a shrinking neighborhood of the random interface where the two fractional Brownian fields exchange the role of the maximizer. For Hurst parameter \(H<1/2\), we prove that the properly normalized squared increment sum converges in probability to a deterministic constant times the local time at zero of the difference between the two fields. This shows that the asymptotic behavior is governed by the geometry of the random contact set rather than by Gaussian fluctuation effects. The result provides a key ingredient for fixed-domain asymptotic inference in Brown--Resnick type models based on randomly located observations.