osnaDocs: Malliavin-Stein Method in Stochastic Geometry

Malliavin-Stein Method in Stochastic Geometry

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https://osnadocs.ub.uni-osnabrueck.de/handle/urn:nbn:de:gbv:700-2013031910717

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DC Element	Wert	Sprache
dc.contributor.advisor	Prof. Dr. Matthias Reitzner
dc.creator	Schulte, Matthias
dc.date.accessioned	2013-03-19T08:39:38Z
dc.date.available	2013-03-19T08:39:38Z
dc.date.issued	2013-03-19T08:39:38Z
dc.identifier.uri	https://osnadocs.ub.uni-osnabrueck.de/handle/urn:nbn:de:gbv:700-2013031910717	-
dc.description.abstract	In this thesis, abstract bounds for the normal approximation of Poisson functionals are computed by the Malliavin-Stein method and used to derive central limit theorems for problems from stochastic geometry. As a Poisson functional we denote a random variable depending on a Poisson point process. It is known from stochastic analysis that every square integrable Poisson functional has a representation as a (possibly infinite) sum of multiple Wiener-Ito integrals. This decomposition is called Wiener-Itô chaos expansion, and the integrands are denoted as kernels of the Wiener-Itô chaos expansion. An explicit formula for these kernels is known due to Last and Penrose. Via their Wiener-Itô chaos expansions the so-called Malliavin operators are defined. By combining Malliavin calculus and Stein's method, a well-known technique to derive limit theorems in probability theory, bounds for the normal approximation of Poisson functionals in the Wasserstein distance and vectors of Poisson functionals in a similar distance were obtained by Peccati, Sole, Taqqu, and Utzet and Peccati and Zheng, respectively. An analogous bound for the univariate normal approximation in Kolmogorov distance is derived. In order to evaluate these bounds, one has to compute the expectation of products of multiple Wiener-Itô integrals, which are complicated sums of deterministic integrals. Therefore, the bounds for the normal approximation of Poisson functionals reduce to sums of integrals depending on the kernels of the Wiener-Itô chaos expansion. The strategy to derive central limit theorems for Poisson functionals is to compute the kernels of their Wiener-Itô chaos expansions, to put the kernels in the bounds for the normal approximation, and to show that the bounds vanish asymptotically. By this approach, central limit theorems for some problems from stochastic geometry are derived. Univariate and multivariate central limit theorems for some functionals of the intersection process of Poisson k-flats and the number of vertices and the total edge length of a Gilbert graph are shown. These Poisson functionals are so-called Poisson U-statistics which have an easier structure since their Wiener-Itô chaos expansions are finite, i.e. their Wiener-Itô chaos expansions consist of finitely many multiple Wiener-Itô integrals. As examples for Poisson functionals with infinite Wiener-Itô chaos expansions, central limit theorems for the volume of the Poisson-Voronoi approximation of a convex set and the intrinsic volumes of Boolean models are proven.	eng
dc.subject	Malliavin calculus	eng
dc.subject	Malliavin-Stein method	eng
dc.subject	Poisson point process	eng
dc.subject	Stein's method	eng
dc.subject	Stochastic geometry	eng
dc.subject	Wiener-Itô chaos expansion	eng
dc.subject.ddc	510 - Mathematik
dc.title	Malliavin-Stein Method in Stochastic Geometry	eng
dc.type	Dissertation oder Habilitation [doctoralThesis]	-
thesis.location	Osnabrück	-
thesis.institution	Universität	-
thesis.type	Dissertation [thesis.doctoral]	-
thesis.date	2013-03-07	-
dc.contributor.referee	Prof. Dr. Giovanni Peccati
vCard.ORG	FB6
Enthalten in den Sammlungen:	FB06 - E-Dissertationen

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