Time and place

Lecture:
Monday, 12–14, HeHo18 E60
Wednesday, 12–14, HeHo18 120

Exercise:
Monday, 14–16, HeHo18 E60

Length

4 hours lecture + 2 hours exercise
Credit points: 9

Theoretical background

Lectures: Probability Theory, Analysis

Target audience

Master Mathematics and Management, Mathematical Biometry, Teacher Education Mathematics

Contents

The lecture gives an introduction to the theory of random functions and fields. It looks at stochastic processes that are indexed by a spatial variable.

The lecture focusses on:

• basic model classes of random fields
• stationarity and isotropy
• Kolmogorov’s existence theorem
• correlation theory of stationary fields
• positive-semidefinite functions
• stochastic integration (random integrator)
• Gaussian random processes

The lecture will be held in English.

Lecture notes

The lecture notes can be found here; subject to change without notice.

Criteria for obtaining the prerequisite

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Examination

The criterion for admission to the examination is that the prerequisite has been passed (see above). The examination is an individual oral examination (in German or English as required); dates are to be arranged individually.

Exercise sheets

The exercise sheets and scores achieved will be published on Moodle.

Literature

• Adler, R. J., Taylor, J. E.: Random Fields and Geometry, Springer, 2007
• Azais, J.-M., Wschebor, M.: Level Sets and Extrema of Random Processes and Fields, Wiley, 2009
• Bogachev, V.I.: Gaussian Measures, AMS, 1998
• Brémaud, P.: Markov Chains, Gibbs Fields, Monte Carlo Simulation, and Queues, Springer, 1999
• Bulinski, A., Shashkin, A.: Limit Theorems for Associated Random Fields and Related Systems, World Scientific, 2007
• Dudley, R. M.: Uniform Central Limit Theorems, Cambridge Univ. Pr., 1999
• Fernique, X: Fonctions aléatoires gaussiennes vecteurs aléatoires gaussiens, CRM, Montreal, 1997
• Georgii, H.-O.: Gibbs Measures and Phase Transitions, de Gruyter, Berlin, 1988
• Guyon, X.: Random Fields on a Network, Springer, 1995
• Ivanov, A.V., Leonenko, N.N.: Statistical Analysis of Random Fields, Kluwer, 1989
• Ledoux, M., Talagrand, M.: Probability in Banach Spaces: Isoperimetry and Processes, Springer, 1991
• Leonenko, M.: Limit Theorems for Random Fields with Singular Spectrum, Kluwer, 1999
• Lifshits, M.A.: Gaussian Random Functions, Kluwer, 1995
• Khoshnevisan, D.: Multiparameter Processes: An Introduction to Random Fields, Springer, 2002
• Malyshev, V. A., Minlos, R. A.: Gibbs Random Fields: Cluster Expansions, Kluwer, 1991
• Piterbarg, V. I.: Asymptotic Methods in the Theory of Gaussian Processes and Fields, AMS, 1996
• Ramm, A.: Random Fields Estimation, World Scientific, 2005
• Yaglom, A. M.: Correlation Theory of Stationary and Related Random Functions, Volume I, Springer, 1987
• Yaglom, A. M.: Correlation Theory of Stationary and Related Random Functions, Volume II, Springer, 1987

The course reserves (Semesterapparate) can be found under the following link: Course reserves (Semesterapparate)