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arXiv 0806.2780math.PR
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与负定函数相关的平稳极大稳定场

Stationary max-stable fields associated to negative definite functions

Zakhar Kabluchko, Martin Schlather, Laurens de Haan

备注：
Published in at http://dx.doi.org/10.1214/09-AOP455 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)
发表信息：
Annals of Probability 2009, Vol. 37, No. 5, 2042-2065
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摘要：

设$W_i,i\in{\mathbb{N}}$是具有平稳增量和方差$σ^2(t)$的零均值高斯过程$\{W(t),t\in{\mathbb{R}}^d\}$的独立副本。独立于$W_i$，设$\sum_{i=1}^{\infty}δ_{U_i}$是实直线上强度为$e^{-y} dy$的泊松点过程。证明了随机函数族$\{V_i(\cdot),i\in{\mathbb{N}}\}$的平移不变律，其中$V_i(t)=U_i+W_i(t)-σ^2(t)/2$是平移不变的.特别地，过程$η(t)=\bigvee_{i=1}^{\infty}V_i(t)$是具有标准Gumbel裕度的平稳最大稳定过程。过程$η$作为$n$ i.i.d.的适当归一化和重新缩放的逐点最大值的极限而出现。平稳高斯过程为$n\to\infty$当且仅当$W$是${\mathbb{R}}^d$上的（非各向同性）分数布朗运动。在$W$上的适当条件下，过程$η$具有混合移动极大值表示。

英文摘要：

Let $W_i,i\in{\mathbb{N}}$, be independent copies of a zero-mean Gaussian process $\{W(t),t\in{\mathbb{R}}^d\}$ with stationary increments and variance $σ^2(t)$. Independently of $W_i$, let $\sum_{i=1}^{\infty}δ_{U_i}$ be a Poisson point process on the real line with intensity $e^{-y} dy$. We show that the law of the random family of functions $\{V_i(\cdot),i\in{\mathbb{N}}\}$, where $V_i(t)=U_i+W_i(t)-σ^2(t)/2$, is translation invariant. In particular, the process $η(t)=\bigvee_{i=1}^{\infty}V_i(t)$ is a stationary max-stable process with standard Gumbel margins. The process $η$ arises as a limit of a suitably normalized and rescaled pointwise maximum of $n$ i.i.d. stationary Gaussian processes as $n\to\infty$ if and only if $W$ is a (nonisotropic) fractional Brownian motion on ${\mathbb{R}}^d$. Under suitable conditions on $W$, the process $η$ has a mixed moving maxima representation.

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