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

Nicolas Chenavier, Christian Y. Robert

TL;DR

This work proves central limit theorems for squared increments of an isotropic fractional Brownian field observed at Poisson-Delaunay nodes in a fixed square, with $H<1/2$. The authors formulate two quadratic-variation-type statistics, one along Delaunay edges and another on pairs of edges within triangles, and establish Gaussian limits with explicit integral-form variances by adapting a Nourdin–Peccati extension of the Breuer–Major theorem to a marked Poisson framework. The analysis hinges on detailed mixing, ergodicity, and density bounds for the Poisson-Delaunay tessellation, as well as precise asymptotics for correlations between increments. The results support statistical inference for max-stable models built from fractional Brownian fields and offer a rigorous asymptotic foundation for random-node quadratic variations in irregular observation schemes.

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 $.

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

TL;DR

This work proves central limit theorems for squared increments of an isotropic fractional Brownian field observed at Poisson-Delaunay nodes in a fixed square, with . The authors formulate two quadratic-variation-type statistics, one along Delaunay edges and another on pairs of edges within triangles, and establish Gaussian limits with explicit integral-form variances by adapting a Nourdin–Peccati extension of the Breuer–Major theorem to a marked Poisson framework. The analysis hinges on detailed mixing, ergodicity, and density bounds for the Poisson-Delaunay tessellation, as well as precise asymptotics for correlations between increments. The results support statistical inference for max-stable models built from fractional Brownian fields and offer a rigorous asymptotic foundation for random-node quadratic variations in irregular observation schemes.

Abstract

An isotropic fractional Brownian field (with Hurst parameter ) is observed in a family of points in the unit square % . These points are assumed to come from a realization of a homogeneous Poisson point process with intensity . 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 .

Paper Structure

This paper contains 15 sections, 8 theorems, 167 equations.

Table of Contents

  1. Introduction
  2. Squared normalized increment sums
  3. Main result
  4. Preliminaries
  5. Proof of Theorem \ref{['th:CLTgaussian']} for $V_{2,N}^{(W)}$
  6. Checking Condition (a)
  7. Mixing of the marked point processes
  8. Ergodic means
  9. Deviations in $x_{3}.$
  10. Checking Condition (c)
  11. Proof of Theorem \ref{['th:CLTgaussian']} for $V_{3,N}^{(W)}$
  12. Technical lemmas
  13. Asymptotic correlations between pairs of normalized increments
  14. Bounds for the density functions of Delaunay neighbors
  15. Bounds for some exponential moments of a uniform distribution over a disc

Key Result

Theorem 1

Let $W$ be a fractional Brownian random field with covariance given by eq:defcovariance, with $H\in (0,1/2)$ and $\sigma ^{2}>0$. There exist finite constants $\sigma _{V_{2}}^{2}>0$ and $\sigma _{V_{3}}^{2}>0$ such that, as $N\rightarrow \infty$,

Theorems & Definitions (8)

  • Theorem 1
  • Theorem 2
  • Lemma 3
  • Proposition 4
  • Proposition 5
  • Lemma 6
  • Lemma 7
  • Lemma 8