STA 447/2006: Stochastic Processes
Wenlong Mou, Department of Statistical Sciences, University of Toronto, Winter 2026
Schedule
| lecture | topic | book chapteres | notes |
| 01/07 | introduction, Markov definitions and transition probabilities, recurrent and transient states, recurrent state theorem, recurrence of (multi-dimensional) random walks | Rosenthal 1.1 – 1.5 | Lecture 1 |
| 01/14 | Proof of recurrence state theorem, f-expansion, relation between recurrence and transience of different states, equivalent characterization of recurrent/transient Markov chains | Rosenthal 1.6, 1.7 | Lecture 2 |
| 01/21 | Recurrence equivalence continued, reducible Markov chains, stationary distributions and stationary measures, detailed balance condition, vanishing transition probabilities and non-existence results, periodicity | Rosenthal 1.6, 2.1 – 2.3, Lawler 1.3 | Lecture 3 |
| 01/28 | Markov chain convergence theorem, coupling, average convergence for periodic chains, positive recurrence and null recurrence, strong law of large numbers for Markov chains | Rosenthal 2.4, 2.5, 2.8, Durrett 1.7 | Lecture 4 |
| 02/04 | Midterm exam #1 | | |
| 02/11 | Metropolis–Hastings algorithm, concepts of martingales and stopping times, optional stopping theorem, application to gambler's ruin | Rosenthal 2.6 , Rosenthal 3.1, 3.2, Lawler 5.3 | Lecture 5 |
| 02/25 | | | |
| 03/04 | | | |
| 03/11 | Midterm exam #2 | | |
| 03/18 | | | |
| 03/25 | | | |
| 04/02 | | | |
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Recommended practice questions
Week 1: Rosenthal 1.3.9, 1.4.8, 1.5.4, 1.5.9
Week 2: Rosenthal 1.5.13, 1.5.14, 1.6.20, 1.6.21, 1.6.25, 1.7.12
Week 3: Rosenthal 2.1.3, 2.4.13, 2.4.14, 2.4.15, 2.4.16, Lawler 1.7, Durrett 1.70, 1.72
Week 4: Rosenthal 2.4.18, 2.4.20, Lawler 1.15, Durrett 1.73, 1.74, 1.75, (see also a book, Chapter 4 and 5 for more information about coupling methods)
Week 5: (midterm exam 1)
Week 6: Rosenthal 3.1.7, 3.2.3, 3.2.8, 3.2.9, 3.2.10, 3.2.11, Lawler 5.4, 5.8, 5.9 (correction)
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