Functional limit theorems for certain intrinsic volumes of excursion sets of random fields [Elektronische Ressource] / vorgelegt von Daniel Meschenmoser

Functional Limit Theoremsfor Certain Intrinsic Volumesof Excursion Sets of Random FieldsDissertationzur Erlangung des Doktorgrades Dr. rer. nat.der Fakultät für Mathematik und Wirtschaftswissenschaftender Universität Ulmvorgelegt vonDaniel Meschenmoseraus Friedrichshafen2011Amtierender Dekan: Prof. Dr. Werner KratzErstgutachter: Prof. Dr. Evgeny SpodarevZweitgutachter: Prof. Dr. Ulrich StadtmüllerTag der Promotion: 28. 03. 2011iiContentsChapter 1. Introduction 11.1. Motivation 11.2. Overview of this Thesis 4Chapter 2. Basics of the Geometry of Random Fields 72.1. Integral Geometry 82.2. Random Fields 13Chapter 3. Computation of Intrinsic Volumes 213.1. Error Bound for a Classical Surface Area Estimator 233.2. Multigrid Convergent Computation of Intrinsic Volumes 463.3. Numerical Results 60Chapter 4. Functional Limit Theorems for Dependent Random Fields 674.1. Functional Limit Theorem for the Volume of Excursion Sets 684.2. Limitem for the Surface Area of Excursion Sets 864.3. Large Deviations and Statistical Applications 121Chapter 5. Conclusion 131Bibliography 135List of Figures 141Zusammenfassung 145iiiCHAPTER 1IntroductionCoincidences, in general, are great stumbling-blocks inthe way of that class of thinkers who have been educatedto know nothing of the theory of probabilities – thattheory to which the most glorious objects of humanresearch are indebted for the most glorious of illustration.