Random tessellations associated with max-stable random fields
Random tessellations associated with max-stable random fields
复制标题
与最大稳定随机场相关的随机镶嵌
DOI：
10.3150/16-bej817
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发表时间：
2014
期刊：
影响因子：
--
通讯作者：
Z. Kabluchko
中科院分区：
文献类型：
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作者：
C. Dombry;Z. Kabluchko
对于$\mathcal{X}=\mathbb{Z}^d$或$\mathbb{R}^d$上的任何极大稳定随机过程$\eta$，我们联系参数空间$\mathcal{X}$的一个随机镶嵌.这种构造依赖于极大稳定过程$ETA$的Poisson点过程表示，它被视为函数$\Phi={\Phi\i，iGeq1\$的随机集合的逐点最大值。这种镶嵌构造如下：数学{X}$中的两个点$x，y在同一个单元中当且仅当存在一个函数$\Phi\in\Phi$，该函数在两个点$x$和$y$处实现最大的$\eta$，即$\Phi(X)=\eta(X)$和$\Phi(Y)=\eta(Y)$.我们用复盖率和包含率来表征细胞的分布。最有趣的是平稳情形，其中胞元的渐近性质与产生极大稳定过程的非奇异流的遍历性质密切相关。例如，我们证明了：i)胞元几乎必然有界当且仅当$\eta$是由耗散流生成的；ii)胞元几乎必然具有正的渐近密度当且仅当$\eta$是由正流生成的。
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.