Central limit theorems for squared increment sums of fractional Brownian fields based on a Delaunay triangulation in $2D$

TL;DR

This paper proves central limit theorems for squared increment sums of fractional Brownian fields observed on Delaunay triangulations generated by Poisson point processes in 2D, as the number of points increases.

Contribution

It establishes the first CLTs for squared increment sums of fractional Brownian fields on Delaunay triangulations based on Poisson points.

Findings

Central limit theorems are proven for normalized squared increment sums.

Results apply to fractional Brownian fields with Hurst parameter less than 1/2.

The theorems hold as the number of Poisson points tends to infinity.

Abstract

An isotropic fractional Brownian field (with Hurst parameter $H<1/2$) is observed in a family of points in the unit square $\mathbf{C}=(-1/2,1/2]^{2}$% . These points are assumed to come from a 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). Central limit theorems are established for the respective centered squared increment sums as $N\rightarrow \infty $.