Random fields

Time and Place

Lecture:
Monday, 14:15 - 15:45, Heho 18, E60.
Friday, 08:15 - 09:45, Heho 18, E60.

Exercise Session:
Wednesday, 14:15 - 15:45, Heho 18, E20.

Type

4 hours lecture + 2 hours exercises
Credit points: 9

Prerequisites

Probability and Calculus, Stochastics I

Intended Audience

Master students in Mathematics and Mathematical Economics, Mathematical Biometrics

Content

This is an introductory course in the theory of random functions and fields. It provides an extension of some topics treated in the course "Stochastic II", by studying random processes with a spatial index.

The main topics are:

Kolmogorov's existence theorem
Stationarity and isotropy
Basic models of random fields
Correlation theory of stationary random fields
positive semi-definite functions
orthogonally scattered measures
stochastic integration

The course will be taught in English.

Requirements for the final exam

There will be individual oral exams at the end of the term.
Requirement: Get at least 50% of all exercise points.

Lecture notes

The lecture notes for “Random fields” can be found here.

Exercise sheets

The exercise sheets and scores 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