Random tessellations associated with max-stable random fields

TL;DR

This paper introduces a method to associate random tessellations with max-stable random fields using Poisson point process representations, and characterizes their properties based on ergodic theory.

Contribution

It provides a novel geometric construction of tessellations linked to max-stable processes and relates their properties to the ergodic nature of the underlying flow.

Findings

Cells are bounded iff the process is generated by a dissipative flow.

Cells have positive density iff the process is generated by a positive flow.

Characterization of cell distribution via coverage and inclusion probabilities.

Abstract

With any max-stable random process $\eta$ on $\mathcal{X}=\mathbb{Z}^d$ or $\mathbb{R}^d$, we associate a random tessellation of the parameter space $\mathcal{X}$. The construction relies on the Poisson point process representation of the max-stable process $\eta$ which is seen as the pointwise maximum of a random collection of functions $\Phi=\{\phi\_i, i\geq 1\}$. The tessellation is constructed as follows: two points $x,y\in \mathcal{X}$ are in the same cell if and only if there exists a function $\phi\in\Phi$ that realizes the maximum $\eta$ at both points $x$ and $y$, i.e. $\phi(x)=\eta(x)$ and $\phi(y)=\eta(y)$. We characterize the distribution of cells in terms of coverage and inclusion probabilities. Most interesting is the stationary case where the asymptotic properties of the cells are strongly related to the ergodic properties of the non-singular flow generating the max-stable process. For example, we show that: i) the cells are bounded almost surely if and only if $\eta$ is generated by a dissipative flow, ii) the cells have positive asymptotic density almost surely if and only if $\eta$ is generated by a positive flow.