Stationary max-stable fields associated to negative definite functions.

By Zakhar Kabluchko, Martin Schlather, Laurens de Haan. 2008

452 citations recorded by the source corpus.

Let Wi, i∈ℕ, be independent copies of a zero-mean Gaussian process {W(t), t∈ℝd} with stationary increments and variance σ2(t). Independently of Wi, let ∑i=1∞δUi 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 {Vi(⋅), i∈ℕ}, where Vi(t)=Ui+Wi(t)−σ2(t)/2, is translation invariant. In particular, the process η(t)=⋁i=1∞Vi(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→∞ if and only if W is a (nonisotropic) fractional Brownian motion on ℝd. Under suitable conditions on W, the process η has a mixed moving maxima representation.

Subfields: Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/555dd69456d87ab0f04cb648885a34f0b79fc44f

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Finance

In Mathematical Physics

Citations in the corpus

Generalized Pickands constants and stationary max-stable processes (2016, methodology): “The Brown–Resnick process ξW is both max-stable and stationary (Kabluchko et al., 2009; Kabluchko, 2009, 2011; Molchanov and Stucki, 2013; Molchanov et al., 2014).”
Mapping extreme snowfalls in the French Alps using max‐stable processes (2013, unknown): “Using the general expression of the bivariate probability [Brown and Resnick, 1977; Kabluchko et al., 2009; Davison et al., 2012],”
Conditional sampling for spectrally discrete max-stable random fields (2010, background): “[16], Wang and Stoev [28], among many others.”
Non-Stationary Dependence Structures for Spatial Extremes (2014, background): “Another interesting stationary sub-model is the Brown–Resnick process (Brown and Resnick, 1977; Kabluchko et al., 2009), which arises as a limiting case of (6) as df →∞ (Davison et al., 2012); its storm profiles may be expressed as W (s) = exp{ε(s)−γ(s)}, where ε(s) is a Gaussian random field with…”
Factor Copula Models for Replicated Spatial Data (2015, methodology): “One can generalize this result and show that the process described by model (1) is in the max-domain of attraction of the (maxstable) Brown-Resnick process (Kabluchko, Schlather, and de Haan 2009).”
Spatial extremes: Max-stable processes at work (2013, background): “Although the process Y is not stationary, the associated max-stable process is stationary Kabluchko et al. (2009).”
Bayesian Inference from Composite Likelihoods, with an Application to Spatial Extremes (2009, methodology): “Although the general methodology we propose could be applied with any max-stable model [Smith, 1990; Schlather, 2002; Kabluchko et al., 2009], we focus here on the Gaussian extreme value process of Smith [1990], where d ζ ds, and ϕ is the zero mean d -variate normal density with covariance matrix Σ.”
Asymptotic models and inference for extremes of spatio-temporal data (2010, background): “The process η(s) = ∨∞i=1 ( Ui + Wi (s) − σ 2(s)/2 ) , (4) where Wi (s) are independent copies of an intrinsically stationary Gaussian process with variance σ 2(s) and variogram γ (s), and Ui are the points of a Poisson point process on the real line with intensity e−ydy, independent of W (s), is a s”
Statistical inference for max‐stable processes in space and time (2012, methodology): “The following proposition is a space–time extension of a result in Kabluchko et al. (2009), which was also considered in Davis et al. (2011).”
A note on stable point processes occurring in branching Brownian motion (2011, methodology): “Immediately after the article [19] was published on the arXiv, the author was informed by Ilya Molchanov that the representation (1.1) could be obtained from a classic result known as the LePage decomposition of a stable point process , which holds true in much more general settings.”