Enveloppes convexes des processus gaussiens |
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| Authors: | Youri Davydov |
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| Institution: | Laboratoire de statistique et probabilités, bât. M2, FRE-CNRS 2222, Université des sciences et technologies de Lille, 59655, Villeneuve d'Ascq cedex, France |
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| Abstract: | Let X={X(t), t 0,1]} be a process on 0,1] and VX=Conv{(t,x) t0,1], x=X(t)} be the convex hull of its path.The structure of the set ext(VX) of extreme points of VX is studied. For a Gaussian process X with stationary increments it is proved that: • The set ext(VX) is negligible if X is non-differentiable. • If X is absolutely continuous process and its derivative X′ is continuous but non-differentiable, then ext(VX) is also negligible and moreover it is a Cantor set. It is proved also that these properties are stable under the transformations of the type Y(t)=f(X(t)), if f is a sufficiently smooth function. |
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| Keywords: | Mots-clé : Processus gaussiens Enveloppes convexes Ensemble de CantorMots-clé : Gaussian processes Convex hulls Cantor sets |
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