The aim of this course is to give a solid background on measure
theoretic probability and random processes for PhD students in
information theory, signal processing, learning and control.
The main motivation behind developing the course is that to appreciate
some, both classic and new, work on theory for signals and systems, it
is necessary to have at least some basic background on measure theory
and the language used in results that build on it. One important class
of problems where this holds in particular is achievability proofs in
information theory based on ergodic theory. Another important area is
decision and estimation theory.
The course is registered as FEO3230 and is worth 12 cu's.
Material
The main text for the course is Robert Gray (Stanford):
Probability, random processes and ergodic properties (1st
edition available from Gray's webpage, 2nd edition printed by
Springer). As a complement, parts of the first half of the course will
be based on lecture notes only. The lecture notes essentially follow
McDonald and Weiss: A course in real analysis, and students
who plan to dig deeper are advised to acquire this textbook too, as a
complement.
Other texts, useful as complements, are Klenke:
Probability Theory, Springer 2008; Kallenberg: Foundations of
Modern Probability, Springer 1997; Shiryaev: Probability,
Springer 1996; Wong/Hajek: Stochastic Processes in Engineering
Systems, Springer 1985; and, Aliprantis/Border: Infinite
Dimensional Analysis, Springer 2006. (For the books published by
Springer, note that KTH has access to Springer Link via the library
website.)
All meetings, except meetings 2 and 13, are held on Fridays at 9:3012:00.
Meeting 1 is held in
 Room V22, Teknikringen 72, floor 4
meeting 2 in
 Room Q2, Malvinas väg 10, floor 2: Starting at 10:00 on Monday March 7,
while the location for meetings 3 through 9 is
 Room Harry Nyquist, Malvinas väg 10, floor 7
Meeting 10 is held in
 Room Gustaf Dahlander, Teknikringen 31, floor 3
while for meetings 11 through 13 we return to Harry Nyquist.
Schedule 2022
 Lecture 1 (Feb 18): Lebesgue measure on the real line
 Lecture 2 (Mar 7): The Lebesgue integral on the real line
 Lecture 3 (Mar 11): General measure theory
 Measure spaces and measurable functions
 Convergence in measure
 Lecture 4 (Mar 18): General integration theory
 The abstract Lebesgue integral
 Distribution functions and the LebesgueStieltjes integral
 Lecture 5 (Mar 25): Probability and expectation
 Probability spaces
 Expectation
 The law of large numbers for i.i.d. sequences
 Lecture 6 (Apr 1): Differentiation
 Functions of bounded variation
 Absolutely continuous functions
 The RadonNikodym derivative
 Probability distributions and pdf's; absolutely continuous
random variables
 Lecture 7 (Apr 8): Conditional probability and expectation
 Conditional probability/expectation
 Decomposition of measures; continuous, mixed and discrete
random variables
 Lecture 8 (Apr 22): Topological and metric spaces
 Topological and metric spaces
 Completeness and separability, Polish spaces
 Standard spaces
 Lecture 9 (Apr 29): Extensions of measures and product measure
 Extension theorems
 Product measure
 Lecture 10 (May 13): Random processes
 Process measure, Kolmogorov's extension theorem
 Lecture 11 (May 20): Dynamical systems and ergodicity
 Random processes and dynamical systems
 The ergodic theorem
 Lecture 12 (June 3): Applications
 Detection and estimation
 Information and coding
Downloads

lecture 0,
lecture 1,
homework 1

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lecture 3,
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lecture 4,
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lecture 5,
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homework 6

lecture 7,
homework 7

lecture 8,
homework 8

lecture 9,
homework 9

lecture 10,
homework 10

lecture 11,
homework 11

lecture 12 (no HW)
