## MAST90063 Probability and Mathematical Statistics II

### Part I: Probability with Martingales

The first part of the course will be based on the text "Probability with Martingales" by David Williams, CUP 1991. The material in Part A (Chapters 1-8) will be assumed. For example, you will need to know about measure spaces, the Borel-Cantelli lemmas, monotone and dominated convergence, Schwarz Hoelder and Jensen inequalities, and the strong law of large numbers.

The lecturer is Dr Owen Jones. Office hours are Mon 2-3 and Thu 11-1, in room 221, Richard Berry building. For administrative questions you may use email (odjones@unimelb.edu.au).

Assignment 1 will be due on Friday 6 September (end of week 6) at 5:00pm, 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.

Assignment 1 solutions.

The student representative is Lachlan McIntosh

Week 0: Revision/practice

Exercises: 4.6, 4.7, 5.1

Extension: read about the "standard machine" (S5.12) and use it to prove S5.14(c)

Week 1: Conditional expectation

Completeness of L^p spaces, existence of orthogonal projection in L^2, defn, existence and a.s. uniqueness of conditional expectation (as limit of L^2 projections)

Conditional expectation was also tackled in Probability and Statistical Inference (mast30020). You can find the slides here. Conditional expectation is on slides 90 to 109.

Homework from the lectures: show that for p <= q, ||X||_p <= ||X||_q (see S6.7)

Week 2: Martingales

Definitions, stopping times, Doob's optional-stopping thm

Aihua Xia's notes may be useful.

Homework from the lectures: read 9.7 and 9.8 (j) on conditions required to "take out what is known".

Week 2 tute questions

Week 3: The martingale convergence theorem

Wald's identity, upcrossing lemma, Martingale convergence theorem

Homework from the lectures: derive the generating function of Z_n, the population of a branching process at generation n.

Week 3 tute questions

Week 4: L^2 martingales

Pythagoras' thm and orthogonality of increments, L^2 cvg thm, Cesaro's lemma, Kronecker's lemma, SLLN (via L^2 martingales and a truncation argument)

Homework from the lectures: show L^2 convergence implies ||M_n|| -> ||M_inf||; find an example of a sequence of i.i.d. r.v.s whose partial sums do not converge a.s.

Week 4 tute questions (note questions 1 and 2 were also on last week's sheet)

Week 5: UI martingales

Defn of UI and examples, L^1 cvg iff cvg in prob and UI, L^1 cvg for UI mart