Random tessellations associated with max-stable random fields

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 $\Phi=\{\phi_{i},i\geq1\}$. 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 $y$, that is, $\phi(x)=\eta(x)$ 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 and mixing properties of the max-stable process $\eta$ and to its conservative/dissipative and positive/null decompositions.