Mathematics – Probability
Scientific paper
2008-06-17
Annals of Probability 2009, Vol. 37, No. 5, 2042-2065
Mathematics
Probability
Published in at http://dx.doi.org/10.1214/09-AOP455 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Scientific paper
10.1214/09-AOP455
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 $\sum_{i=1}^{\infty}\delta_{U_i}$ 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 $\eta(t)=\bigvee_{i=1}^{\infty}V_i(t)$ 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.
Haan Laurens de
Kabluchko Zakhar
Schlather Martin
No associations
LandOfFree
Stationary max-stable fields associated to negative definite functions does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Stationary max-stable fields associated to negative definite functions, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Stationary max-stable fields associated to negative definite functions will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-408873