# 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 $\Phi=\{\phi_i, i\geq 1\}$. 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.
Keywords :
Type de document :
Pré-publication, Document de travail
36 pages. 2014
Domaine :

https://hal.archives-ouvertes.fr/hal-01073188
Contributeur : Clément Dombry <>
Soumis le : vendredi 10 octobre 2014 - 16:15:02
Dernière modification le : jeudi 7 janvier 2016 - 01:01:48
Document(s) archivé(s) le : dimanche 11 janvier 2015 - 11:11:16

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DombryKabluchko_final.pdf
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### Identifiants

• HAL Id : hal-01073188, version 2
• ARXIV : 1410.2584

### Citation

Clément Dombry, Z. Kabluchko. Random tessellations associated with max-stable random fields. 36 pages. 2014. <hal-01073188v2>

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