Random Fields

Prof. Dr. Evgeny Spodarev 

Teaching Assistant
Dr. Vitalii Makogin

Time and Place


Tuesday, 10:15 - 11:45  He18 Room E60

Thursday, 10:15 - 11:45  He18 Room E20

Exercise Session

Friday 12:15 - 13:45  He18 Room 2.20


May, 18th One-time room exchange. He22 room 142,
May, 25th No exercise session,
May, 29th  Exercises instead of lecture,
June, 1st Lecture instead of exercises,
June, 15th Lecture instead of exercises,
June, 19th Exercises instead of lecture,
June, 21st Exercises instead of lecture,
July, 6th Lecture instead of exercises,
July, 10th Exercises instead of lecture.


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 to write the final exam

Successful work out of at least 50% of the exercises in the exercise sheets.

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)



Office hours: Wednesday, 4 - 5 pm
Phone: +49 (0)731/50-23530

Teaching Assistant


Office hours: on appointment


  • The lecture will be canceled because of illness today, 17.07.2018 at 10:15.
  • On 18.05.2015 the exercise session will take place in He22-142
  • The first lecture will take place on Tuesday, 17.04.2018 at 10:15 in the He18-E60.
  • On 20.04.2018 at 12:15 there will be an additional lecture in He18 room 220.
  • Exercise sessions start on 27.04.2018