The first part of the course will be based on the text "Probability with Martingales" by David Williams, CUP 1991. Much of the material in Chapters 1-4 will be assumed. That is, you will need to know about sigma algebras, probability measures and Lebesgue measure, the Borel-Cantelli lemmas, random variables and independence.
The lecturer is Dr Owen Jones. Office hours are Tue 11-12 and Thu 10-12, in room 221, Richard Berry building. For administrative questions you may use email (odjones@unimelb.edu.au).
Assignment 1 will be due on Friday 31 August (end of week 6) at 4:00pm, and is worth 10%. Please deliver assignments to room 221, Richard Berry building. Slide them under the door if no one is in. You must read the rules on academic misconduct before commencing your assignment, and include a signed plagiarism declaration with your assignment. Assignments up to 1 week late will have their marks halved. Assignments more than 1 week late will not be marked.
The student representative is Giles Adams
Week 1: Kolmogorov's 0-1 law
Homework from the lectures: prove monotone convergence of measures for decreasing sequences of sets
prove Fatou's lemma for measures
show f^{-1} preserves set operations
show {X_k converges} is in the tail sigma algebra
Exercises: 4.1, 4.6, 4.7, 4.10 (at least)
Extension: go through the proof that measures are determined by their behaviour on pi-systems (Lemma 1.6)
Week 2: Monotone and dominated convergence
Homework from the lectures: for f measurable, show that f_+ and f_- are measurable
prove linearity for integrals of simple functions
prove that changing the function on a set of measure zero does not change the integral, for simple functions
prove that our elementary properties of integration hold for measurable functions in general
Exercises: 4.10 (again), 5.1, 5.2, S5.14(b) (from Chapter 5), and
show that a collection of r.v.s dominated by some X with finite integral, are uniformly integrable
Extension: read about the "standard machine" (S5.12) and use it to prove S5.14(c)
Week 3: Conditional expectation
Homework from the lectures: show that for p <= q, ||X||_p <= ||X||_q (see S6.7)
prove conditional forms of Fatou and Dominated convergence (see S9.7 and 9.8)
where did we use that X is in L^1 in our definition of conditional expectation?
Exercises: 5.2, S5.14(b) and UI exercise from last week, plus
9.1 and 9.2 from Williams,
conditional expectation w.r.t. a sigma field generated by a finite partition,
E(X|Y) for (X,Y) bivariate normal
Week 4: Martingales
Aihua Xia's notes may be useful.
Exercises: S10.6, 10.7 and 10.10(c) on previsible processes, then exercises 10.2, 10.4 and 10.6
Week 5: L^2 martingales
Exercises: an example of an L^1 bounded martingale that converges a.s. but not in L^1;
an example of a martingale that is not L^1 bounded and diverges to infinity a.s.;
a martingale that must decide whether or not to chose the road less travelled;
Williams E12.1 and 12.2
Week 6: UI martingales
Homework from the lectures: read S13.4;
show that absolute continuity Q w.r.t. P implies that
given e > 0 we can find d > 0 s.t. P(A) < d => Q(A) < e
Exercises: Williams E13.1
A: a finite collection of UI collections is UI
B: UI is equivalent to E G(|X|) bounded, where G(x) goes to infinity faster than x
C: L^p convergence is equivalent to convergence in probability plus UI of |X_n|^p
D: Radon-Nikodym derivatives on a filtration
Durrett (Probability: Theory and Examples) 5.5.2, 5.5.3, 5.5.5, 5.5.6
Extension: the "only if" part of the iff result on L^1 convergence