Random fields: spectral representation, smoothness, excursions. Fall 2014

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First Lecture: Wednesday, Oct 29 13:00 – 14:45 in MVL15

Lectures and problem sessions: Tuesdays 13:00 – 14:45 and Wednesdays 13:00 – 14:45


The course will cover the basics theory for random fields, with emphasis on stationary and differentiable Gaussian fields. It will consist of lectures,   and problem sessions where participants present solution to selected problems from the course book.

Course literature:

An unfinished manuscript “Applications of RANDOM FIELDS AND GEOMETRY: Foundations and Case Studies” by Robert Adler, Jonathan Taylor, and Keith Worsley. A copy will be sent to all on the course address list.


Complementary literature:

“Level sets and extrema of random processes and fields” by Jean-Marc Azais and Mario Wschebor, Wiley, 2009

“Asymptotic Methods in the Theory of Gaussian Processes” by Vladimir Piterbarg, American Mathematical Society, ser. Translations of Mathematical Monographs, Vol. 148, 1995 

“Random fields and Geometry” by Robert Adler and Jonathan Taylor, Springer 2007

"Spectral domain" by Monserrat Fuentes and Brian Reich, in Handbook of Spatial Statistics, eds. Gelfand, Diggle, Fuentes, Guttorp, p. 57-78, CRC press, 2010


Course content:

         Kolmogorov existence theorem, separable processes, measurable processes

        Stationarity and isotropy

        Spectral representations

        Exceedance sets

        Rice formula

        Slepian models



        Oral/written exam: individual discussion of solutions to a couple of problems

        Solutions of exercises during the course








Wednesday 29/10


introduction, notation, Kolmogorov existence theorem, separability, measurability

ATW Ch 1 (read on your own), Ch 2, pp. 23-30

Slides: RF1

Tuesday        4/11



Gaussian fields, stationarity, isotropy, cosine fields, orthogonal expansions

ATW  Ch 2, pp. 30-39

Slides: RF1

Wednesday   5/11


Spectral representation of covariance function,  differentiability,  white noise

ATW  Ch 2, pp. 36-47

Slides: RF1, RF2

Tuesday      11/11


 Excercise session

 2.8.1, 2.8.2, 2.8.3, 2.8.4,  2.8.5

Wednesday 12/11


differentiability, white noise,  spectral representation of fields


ATW  Ch 2, pp. 42-56

Slides:  RF2

Tuesday      18/11


spectral representation for isotropic fieds, continuity

ATW  Ch 2, pp. 53-67

Slides:  RF2, RF3

Wednesday 19/11


 continuity, differentiability

 ATW  Ch 2, pp. 60-70

Slides:  RF3

Tuesday      25/11


 Excercise session

 2.8.7, 2.8.9, 2.8.11, 2.8.12

Wednesday 26/11


 Spectral estimation

"Spectral domain",  Fuentes & Reich


Tuesday        2/12


 Excercise session


Wednesday   3/12



 Borel,  Slepian inequality

ATW  Ch 2, pp.  70-78


Tuesday        9/12




Wednesday 10/12

 MVL22 Note!

 Excercise session


Tuesday      16/12




Wednesday 17/12




  Thursday 18/12,
  13:15 -15:00
  Non-stationary spatial structures
 Guest lecture by Peter Guttorp
 nonstationary, nonstationary_references

  Tuesday 13/1

  Rice's formula, Slepian models

 ATW Ch 2, pp. 78-82, Ch 4, pp.  194-214


Wednesday  15/1

 (starts after intro to  GRF course), MVL15

Rice's formula, Slepian models

ATW Ch 2, pp. 78-82, Ch 4, pp.  194-214


  Monday 19/1, 13-15   Exercise session

Wednesday, 28/1, 13-15


Guest  lecture & excercise session, by Igor Rychlik    Bring your own PC!




RF 1

RF 2

RF 3

RF 4

RF 5



nonstationary references