Random Fields I

Lecturer
Prof. Dr. Evgeny Spodarev

Teaching Assistant
Daniel Meschenmoser

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Time and Place

Lecture
Thursday 12:30 - 14:00 (without break) in E60 (Helmholtzstrasse 18)

Exercise Session
Thursday 16 - 18 in E60 (Helmholtzstrasse 18)

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Type

2 hours lecture + 2 hours exercises

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Prerequisites

Probability Calculus

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Intended Audience

Bachelor / Master students in Mathematics and Business Mathematics

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Content

This course provides an introduction to random surfaces (so-called fields) which are random processes with spatial index.

The main topics are:

• Kolmogorov's existence theorem
• Stationarity and isotropy
• Correlation theory
• Basic models of random fields

The course will be taught in English.

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Requirements to obtain the certificate (Übungsschein)

In order to obtain the certificate, one has to earn 50% of all homework credits.

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Exercise sheets

• Sheet 1, due May 7, 2009
• Sheet 2, due May 15, 2009
• Sheet 3, due May 28, 2009
• Sheet 4, due June 18, 2009
• Sheet 5, due June 25, 2009
• Sheet 6, due July 2, 2009
• Sheet 7, due July 9, 2009
• Sheet 8, due July 16, 2009

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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

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