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

Nicolas Chenavier, Christian Y. Robert

TL;DR

The paper analyzes limit theorems for squared increment sums of the pointwise maximum of two independent isotropic fractional Brownian fields with $H<1/2$, observed on a Poisson-$N$ set within $oldsymbol{C}$. It introduces a decomposition into two-trajectory and single-trajectory terms via a Poisson-Delaunay framework and proves that the dominant terms converge in probability to constants times the local time of the difference of the two fields, $L_{W^{(2\backslash 1)}}(0)$, with normalization $N^{-(2-oldsymbol{ abla})/4}$ where $oldsymbol{ abla}=2H= rac{ ext{alpha}}{1}$; the limiting constants $c_{V_2}$ and $c_{V_3}$ arise from integrals of auxiliary functions $F_2$ and $F_3$. Unlike the single-field case, the normalizations differ and the limit is a local-time object rather than a Gaussian limit, reflecting the irregular boundary where the two fields switch dominance. The results have implications for inference in max-stable field models, notably in composite likelihood constructions based on Delaunay-based increments. The work leverages Slivnyak–Mecke, Fourier-analytic representations, and occupation-time formulas to connect squared increments to the local time of the difference field.

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.

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

TL;DR

The paper analyzes limit theorems for squared increment sums of the pointwise maximum of two independent isotropic fractional Brownian fields with , observed on a Poisson- set within . It introduces a decomposition into two-trajectory and single-trajectory terms via a Poisson-Delaunay framework and proves that the dominant terms converge in probability to constants times the local time of the difference of the two fields, , with normalization where ; the limiting constants and arise from integrals of auxiliary functions and . Unlike the single-field case, the normalizations differ and the limit is a local-time object rather than a Gaussian limit, reflecting the irregular boundary where the two fields switch dominance. The results have implications for inference in max-stable field models, notably in composite likelihood constructions based on Delaunay-based increments. The work leverages Slivnyak–Mecke, Fourier-analytic representations, and occupation-time formulas to connect squared increments to the local time of the difference field.

Abstract

The pointwise maximum of two independent and identically distributed isotropic fractional Brownian fields (with Hurst parameter ) is observed in a family of points in the unit square . We assume that these points come from the 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). We investigate the asymptotic behaviors of the squared increment sums as . 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.

Paper Structure

This paper contains 15 sections, 4 theorems, 174 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Local time of the difference of two isotropic fractional Brownian fields
  4. Poisson-Delaunay graph
  5. Main results
  6. Proofs
  7. Proof of Proposition \ref{['prop:twotrajectories']}
  8. Proof for $V_{2,N}^{(2/1)}$
  9. Proof for $V_{3,N}^{(2/1)}$
  10. Proof of Theorem \ref{['cor:twotrajectories']}
  11. Proof for $V_{2,N}^{(W_{\vee })}$
  12. Proof for $V_{3,N}^{(W_{\vee })}$
  13. Appendix
  14. Asymptotic expansion for $1-\kappa _{x_{1},x_{3}}^{2}$
  15. Generalization of Proposition 3.1 in PodolskijRosenbaum18

Key Result

Proposition 1

Let $W^{(1)}$ and $W^{(2)}$ be two independent and identically distributed fractional Brownian random fields, with covariance given by eq:defcovariance, with $\sigma^2>0$ and $\alpha \in (0,1)$. Then, as $N\rightarrow \infty$,

Theorems & Definitions (4)

  • Proposition 1
  • Theorem 2
  • Lemma 3
  • Proposition 4