Lecturer
Dr. Patricia Alonso-Ruiz

Teaching Assistant
Dr. Vitalii Makogin

---

Time and Place

Lecture

Monday 08:15 - 09:45  He18 Room E20

Exercise Session

Tuesday 12:15 - 13:45  He22 Room E18

---

Type

2 hours lecture + 2 hours exercises (each 15 days)

---

Prerequisites

Probability and Calculus 

---

Intended Audience

Master students in Mathematics and Business Mathematics

---

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 to write the final exam

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

Scoring:  use SLC login.

Dates: Final exam: 12.07.2016

---

Exercise sheets

 Exercise sheet 1, till 19th of April 2016

 Exercise sheet 2 (revised), till 3d of May 2016

 Exercise sheet 3, till 17th of May 2016

 Exercise sheet 4, till 31st of May 2016

 Exercise sheet 5, till 14th of June 2016

 Exercise sheet 6, till 28th of June 2016

---

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

(download: pdf)