Random fields

Time and Place

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. 


4 hours lecture + 2 hours exercises
Credit points: 9


Probability and Calculus, Stochastics I

Intended Audience

Master students in Mathematics and Mathematical Economics, Mathematical Biometrics


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.


  • 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 reserve of the library.
(download: pdf)



Prof. Dr. Evgeny Spodarev

Mail: Evgeny.spodarev(at)uni-ulm.de


Office hours: on appointment

Teaching Assistant

Artur Bille

Mail: artur.bille(at)uni-ulm.de

Office hours: on appointment