Lecture diary
Markov processes and martingales
2023 spring

week 1
27th Feb: review of measure theory
1st Mar: definition and basic properties of conditional expectation

week 2
6th Mar: properties of conditional expectation, problem solving about conditional expectation (problems 1, 3, 9)
8th Mar: martingales: definitions, examples, functions of Markov chains, Polya's urn

week 3
13th Mar: (super)martingale transform, stopping times, stopped (super)martingale, optional stopping, monkey at the typewriter (ABRACADABRA problem)
15th Mar: national holiday

week 4
20th Mar: regular conditional distribution, conditional characteristic function
22nd Mar: multivariate normal distribution, hitting time of simple random walk

week 5
27th Mar: hitting time distribution of the simple random walk, superharmonic functions of Markov chains, Doob's upcrossing lemma
29th Mar: Doob's forward convergence theorem, L^2 martingales, sum of independent zero mean random variables

week 6
3rd Apr: sum of independent zero mean random variables, Doob decomposition, angle bracket process
5th Apr: midterm test no. 1

week 7
10th Apr: Easter break
12th Apr: Easter break

week 8
17th Apr: convergence and finite angle bracket process, Cesaro's lemma, Kronecker's lemma, strong law for L2 martingales
19th Apr: Levy's extension of Borel-Cantelli lemmas, closed martingales, closed Lp convergence, noisy observations

week 9
24th Apr: reverse Fatou, bounded convergence, absolute continuity of integration, uniform integrability (UI), UI and L1 convergence, UI martingales
26th Apr: Levy's upward theorem, Kolmogorov's 0-1 law, Levy's downward theorem, martingale proof of the strong law of large numbers, Doob's submartingale inequality

week 10
1st May: break
3rd May: Kolmogorov's inequality, normal tail estimates, Kolmogorov's law of iterated logarithm
5th May: law of iterated logarithm, Doob's Lp inequality

week 11
8th May: Kakutani's theorem about product martingales, stationary processes, measure preserving transformations
10th May: solving problem 10, stationary processes and ergodicity

week 12
15th May: ergodic theorems: Neumann L2 and Birkhoff L1
17th May: consequences of ergodic theorems, Weyl's equidistribution theorem, central limit theorem for martingales, central limit theorem for Markov chains

week 13
22nd May: Markov chains and stopping times, reversible Markov chains
24th May: random walks on weighted graphs and electric networks

week 14
29th May: Pentecost
31st May: midterm test no. 2