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

An isotropic fractional Brownian field (with Hurst parameter $H<1\mathrm{/}2$) is observed in a family of points in the unit square $\mathbf{C}=(-1\mathrm{/}2,1\mathrm{/}2{]}^2$% . 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 $N\to \mathrm{\infty }$.