5 research outputs found

    Realisability conditions for second order marginals of biphased media

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    16 pagesInternational audienceThis paper concerns the second order marginals of biphased random media. We give discriminating necessary conditions for a bivariate function to be such a valid marginal, and illustrate our study with two practical applications: (1) the spherical variograms are valid indicator variograms if and only if they are multiplied by a sufficiently small constant, which upper bound is estimated, and (2) not every covariance/indicator variogram can be obtained with a Gaussian level set. The theoretical results backing this study are contained in a companion paper

    Random Measurable Sets and Covariogram Realisability Problems

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    We provide a characterization of the realisable set covariograms, bringing a rigorous yet abstract solution to the S_2S\_2 problem in materials science. Our method is based on the covariogram functional for random mesurable sets (RAMS) and on a result about the representation of positive operators in a locally compact space. RAMS are an alternative to the classical random closed sets in stochastic geometry and geostatistics, they provide a weaker framework allowing to manipulate more irregular functionals, such as the perimeter. We therefore use the illustration provided by the S_2S\_{2} problem to advocate the use of RAMS for solving theoretical problems of geometric nature. Along the way, we extend the theory of random measurable sets, and in particular the local approximation of the perimeter by local covariograms.Comment: 35p

    The realization problem for tail correlation functions

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    For a stochastic process {Xt}tT\{X_t\}_{t \in T} with identical one-dimensional margins and upper endpoint τup\tau_{\text{up}} its tail correlation function (TCF) is defined through χ(X)(s,t)=limττupP(Xs>τXt>τ)\chi^{(X)}(s,t) = \lim_{\tau \to \tau_{\text{up}}} P(X_s > \tau \,\mid\, X_t > \tau ). It is a popular bivariate summary measure that has been frequently used in the literature in order to assess tail dependence. In this article, we study its realization problem. We show that the set of all TCFs on T×TT \times T coincides with the set of TCFs stemming from a subclass of max-stable processes and can be completely characterized by a system of affine inequalities. Basic closure properties of the set of TCFs and regularity implications of the continuity of χ\chi are derived. If TT is finite, the set of TCFs on T×TT \times T forms a convex polytope of T×T\lvert T \rvert \times \lvert T \rvert matrices. Several general results reveal its complex geometric structure. Up to T=6\lvert T \rvert = 6 a reduced system of necessary and sufficient conditions for being a TCF is determined. None of these conditions will become obsolete as T3\lvert T \rvert\geq 3 grows.Comment: 42 pages, 7 Table
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