Extremes of independent Gaussian processes

Kabluchko, Zakhar

doi:10.1007/s10687-010-0110-x

Cited by 51 publications

(82 citation statements)

References 16 publications

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“…Its law depends only on the variogram γ(t) = Var(B(t) − B(0)) and we can therefore assume without loss of generality that W (0) = 0; see Kabluchko (2011) for details. The generalized Pickands constant can also be defined for non-Gaussian processes.…”

Section: Generalized Pickands Constantsmentioning

confidence: 99%

“…We denote by ε x the unit Dirac measure at x ∈ R. The Brown-Resnick process ξ W is both max-stable and stationary Kabluchko, 2009Kabluchko, , 2011Molchanov and Stucki, 2013;Molchanov et al, 2014). The stationarity means that the processes {ξ W (t), t ∈ R} and {ξ W (t + h), t ∈ R} have the same distribution for any h ∈ R. Moreover, the process ξ W arises naturally as the limit of suitably normalized pointwise maxima of independent copies of stationary Gaussian processes (Kabluchko et al, 2009, Theorem 17).…”

Section: Introductionmentioning

confidence: 99%

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Generalized Pickands constants and stationary max-stable processes

2017

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Abstract. Pickands constants play a crucial role in the asymptotic theory of Gaussian processes. They are commonly defined as the limits of a sequence of expectations involving fractional Brownian motions and, as such, their exact value is often unknown. Recently, Dieker and Yakir (2014) derived a novel representation of Pickands constant as a simple expected value that does not involve a limit operation. In this paper we show that the notion of Pickands constants and their corresponding Dieker-Yakir representations can be extended to a large class of stochastic processes, including general Gaussian and Lévy processes. We furthermore provide a link to spatial extreme value theory and show that Pickands-type constants coincide with certain constants arising in the study of max-stable processes with mixed moving maxima representations.

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Section: Generalized Pickands Constantsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Generalized Pickands constants and stationary max-stable processes

2017

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“…In the finitedimensional setting, the asymptotic behavior of the maxima of Gaussian random vectors was first investigated by Huesler and Reiss [13]. Recently, Kabluchko, Schlather and de Haan [15] and Kabluchko [14] consider the functional setting and prove convergence of the maxima of i.i.d. centered Gaussian processes under suitable conditions on their covariance structure.…”

Section: Gaussian Casementioning

confidence: 99%

“…Our purpose here is to revisit their results and apply the present framework: we put the emphasis on regular variations and point processes. In the sequel, we follow the approach by Kabluchko, see Theorems 2 and 6 in [14].…”

Section: Gaussian Casementioning

confidence: 99%

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Extremes of independent stochastic processes: a point process approach

Éyi-Minko

Dombry

2016

Extremes

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For each n ≥ 1, let {X in , i1} be independent copies of a nonnegative continuous stochastic process X n = (X n (t)) t∈T indexed by a compact metric space T . We are interested in the process of partial maximãwhere the brackets [ · ] denote the integer part. Under a regular variation condition on the sequence of processes X n , we prove that the partial maxima processM n weakly converges to a superextremal processM as n → ∞. We use a point process approach based on the convergence of empirical measures. Properties of the limit process are investigated: we characterize its finite-dimensional distributions, prove that it satisfies an homogeneous Markov property, and show in some cases that it is max-stable and self-similar. Convergence of further order statistics is also considered. We illustrate our results on the class of log-normal processes in connection with some recent results on the extremes of Gaussian processes established by Kabluchko.

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Max‐Stable Processes

Padoan

2012

Encyclopedia of Environmetrics

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Environmental problems such as floods require statistical analysis that takes into account the complex nature of the data, namely, the observations that are sampled at different spatial points in a given region for a certain time. Thus the spatial dependence structure cannot be ignored. Extreme statistics for the design of structures for flood protection, for the study of the structural failures such as those in bridges, dams, and so on, for the prediction of heat waves, and other spatial extremes should be based on a solid theoretical framework. Max‐stable processes provide such a theory and in the last decade have emerged as fertile ground for research and have become a common tool for the statistical modeling of spatial extremes. This article provides a summary of max‐stable processes.

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Extremes of independent Gaussian processes

Cited by 51 publications

References 16 publications

Generalized Pickands constants and stationary max-stable processes

Generalized Pickands constants and stationary max-stable processes

Extremes of independent stochastic processes: a point process approach

Max‐Stable Processes

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