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Stationary max-stable fields associated to negative definite functions. (English) Zbl 1208.60051
Summary: 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 $$\sigma^2(t)$$. Independently of $$W_i$$, let $$\sum^\infty_{i=1}\delta_{U_i}$$ 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 $$V_i(t)=U_i+W_i(t)-\sigma^2(t)/2$$, is translation invariant. In particular, the process $$\eta(t)=\bigvee^\infty_{i=1}V_i(t)$$ is a stationary max-stable process with standard Gumbel margins. The process $$\eta$$ 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 $$\eta$$ has a mixed moving maxima representation.

MSC:
 60G70 Extreme value theory; extremal stochastic processes 60G15 Gaussian processes
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