Remarks on Pickands theorem
نویسنده
چکیده
In this article we present Pickands theorem and his double sum method. We follow Piterbarg’s proof of this theorem. Since his proof relies on general lemmas we present a complete proof of Pickands theorem using Borell inequality and Slepian lemma. The original Pickands proof is rather complicated and is mixed with upcrossing probabilities for stationary Gaussian processes. We give a lower bound for Pickands constant.
منابع مشابه
Bivariate extension of the Pickands–Balkema–de Haan theorem
We prove a two-dimensional version of the famous Pickands–Balkema–de Haan theorem of extreme value theory. The bivariate random variables are generated using the copula language. This representation of dependence structures allows to derive asymptotic results for bivariate excess distributions. 2003 Elsevier SAS. All rights reserved. Résumé Une version en dimension 2 du célèbre théorème de Pi...
متن کاملAn entropic view of Pickands' theorem
It is shown that distributions arising in Rényi-Tsallis maximum entropy setting are related to the Generalized Pareto Distributions (GPD) that are widely used for modeling the tails of distributions. The relevance of such modelization, as well as the ubiquity of GPD in practical situations follows from Balkema-De Haan-Pickands theorem on the distribution of excesses (over a high threshold). We ...
متن کاملGeneralized Pickands Constants
Pickands constants play an important role in the exact asymptotic of extreme values for Gaussian stochastic processes. By the generalized Pickands constant Hη we mean the limit Hη = lim T→∞ Hη(T ) T , where Hη(T ) = IE exp ( maxt∈[0,T ] (√ 2η(t)− σ2 η(t) )) and η(t) is a centered Gaussian process with stationary increments and variance function σ2 η(t). Under some mild conditions on σ2 η(t) we ...
متن کاملRemarks on the Paper ``Coupled Fixed Point Theorems for Single-Valued Operators in b-Metric Spaces''
In this paper, we improve some recent coupled fixed point resultsfor single-valued operators in the framework of ordered $b$-metricspaces established by Bota et al. [M-F. Bota, A. Petrusel, G.Petrusel and B. Samet, Coupled fixed point theorems forsingle-valued operators in b-metric spaces, Fixed Point TheoryAppl. (2015) 2015:231]. Also, we prove that Perov-type fix...
متن کاملProbability, Networks and Algorithms Some remarks on properties of generalized Pickands constants
We study properties of generalized Pickands constants H that appear in the extreme value theory of Gaussian processes and are de ned via the limit H lim T H T T where H T IE exp maxt T p t Var t and t is a centered Gaussian process with stationary increments We give estimates of the rate of convergence of H T T to H and prove that if n t weakly converges in C to t then under some weak condition...
متن کامل