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

TL;DR

This paper studies the asymptotic behavior of squared increment sums of the maximum of two isotropic fractional Brownian fields over a fixed domain, revealing convergence to local time and differences from single-field cases.

Contribution

It introduces new limit theorems for squared increment sums of the maximum of two fractional Brownian fields, extending previous single-field results.

Findings

Normalized sums differ from single-field case

Sums converge to local time of the difference of fields

Asymptotic behavior depends on the maximum of two fields

Abstract

The pointwise maximum of two independent and identically distributed isotropic fractional Brownian fields (with Hurst parameter $H<1/2$) is observed in a family of points in the unit square $\mathbf{C}=(-1/2,1/2]^{2}$. We assume that these points come from the realization of a homogeneous Poisson point process with intensity $N$. We consider normalized increments (resp. pairs of increments) along the edges of the Delaunay triangulation generated by the Poisson point process (resp. pairs of edges within triangles). We investigate the asymptotic behaviors of the squared increment sums as $N\rightarrow \infty $. We show that the normalizations differ from the case of a unique isotropic fractional Brownian field as obtained in \cite{Chenavier&Robert25a} and that the sums converge to the local time of the difference of the two isotropic fractional Brownian fields up to constant factors.