Random fields: spectral representation,
smoothness, excursions. Fall 2014
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 JeanMarc 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. 5778, CRC
press, 2010
Course content:
•
Kolmogorov existence theorem, separable
processes, measurable processes
• Stationarity and isotropy
• Spectral representations
• Exceedance sets
• Rice formula
• Slepian models
Examination:
• Oral/written exam: individual
discussion of solutions to a couple of problems
• Solutions of exercises during the
course
Date 
Content 
Literature 



Wednesday 29/10 MVL15 
introduction,
notation, Kolmogorov existence theorem, separability,
measurability 
ATW Ch 1 (read on your own), Ch 2,
pp. 2330 Slides: RF1 
Tuesday 4/11 MVL14 

Slides: RF1 
Wednesday 5/11 MVL15 
Spectral representation of covariance function, differentiability, white noise 
Slides: RF1, RF2 
Tuesday 11/11 MVL14 
Excercise session 
2.8.1, 2.8.2, 2.8.3, 2.8.4, 2.8.5 
Wednesday 12/11 MVL15 
differentiability, white noise, spectral representation of fields 
Slides: RF2 
Tuesday 18/11 MVL14 
spectral representation for isotropic fieds, continuity 
Slides: RF2, RF3 
Wednesday 19/11 MVL15 
continuity, differentiability 
Slides: RF3 
Tuesday 25/11 MVL14 
Excercise session 
2.8.7, 2.8.9, 2.8.11, 2.8.12 
Wednesday 26/11 MVL15 
Spectral estimation 
"Spectral domain", Fuentes & Reich RF4 
Tuesday 2/12 MVL14 
Excercise session 

Wednesday 3/12 MVL15 
Borel, Slepian inequality 
RF5 
Tuesday 9/12 MVL14 
Cancelled 

Wednesday 10/12 MVL22 Note! 
Excercise session 

Tuesday 16/12 MVL14 
Cancelled 

Wednesday 17/12 MVL15 
Cancelled 

Thursday 18/12, 13:15 15:00 
Nonstationary spatial structures 
Guest lecture by Peter Guttorp nonstationary, nonstationary_references 
Tuesday 13/1 
Rice's formula, Slepian models 
ATW Ch 2, pp. 7882, Ch 4, pp. 194214 
Wednesday 15/1 (starts after intro to GRF course), MVL15 
Rice's formula, Slepian models 
ATW Ch 2, pp. 7882, Ch 4, pp. 194214 RF6 
Monday 19/1, 1315  Exercise session 

Wednesday, 28/1, 1315 
WAFO 
Guest lecture & excercise session, by Igor Rychlik Bring your own PC! 
Slides: