| Fall 2021 MAT 550: Introduction to Probability | ||
| Schedule | TTh 11:30am-12:50pm Math P131 | |
| Instructor | Robert Hough | |
| Office hours | Tu 7-8 in MLC, F 9am-11pm in Math Tower 4-118. | |
| Description | Introduction to probability theory: independence, laws of large numbers, central limit theorems, martingales, Markov chains, and a selection of other topics such as ergodic theory, Brownian motion, random walks on graphs and groups, percolation, mixing times, randomized algorithms. | |
| Textbook | Rick Durrett. Probability: theory and examples. Cambridge University Press. 5th Edition. | |
| Supplementary Textbooks | Alon and Spencer. The probabilistic method. John Wiley and Sons (2016). Morters and Peres. Brownian Motion. Cambridge University Press (2010). | |
| Grading | The course grade is based upon the written homework. The problem numbers are from Durrett. | |
Syllabus/schedule (subject to change)
| Tues 8/24 | 1. | Random variables and integration | Durrett 1.1-1.5 | Homework 1: 1.1.3, 1.1.5, 1.2.6, 1.4.4, 1.5.2, 1.5.3, 1.6.2, 1.6.6, 1.7.2 |
| Thurs 8/26 | 2. | Expected value, Carathéodory extension theorem | Durrett 1.6-1.7, A.1 | |
| Tues 8/31 | 3. | Kolmogorov extension theorem | Durrett A.2-A.5 | Homework 2: 2.1.1, 2.1.2, 2.1.5, 2.2.6, 2.3.4, 2.3.5, 2.3.10 |
| Thurs 9/2 | 4. | Weak law, Borel-Cantelli | Durrett 2.1-2.3 | |
| Tues 9/7 | 5. | Strong law | Durrett 2.4-2.7 | Homework 3: 2.4.2, 2.5.3, 2.5.6, 2.5.11, 2.6.7, 2.7.4, 3.1.3, 3.2.2, 3.2.4, 3.2.16, 3.3.3, 3.3.9, 3.3.12 |
| Thurs 9/9 | 6. | Characteristic functions | Durrett 3.1-3.3 | |
| Tues 9/14 | 7. | Central limit theorem, local limit theorem | Durrett 3.4-3.6 | Homework 4: 3.4.5, 3.4.6, 3.4.12, 3.7.6, 3.7.7, 3.8.6, 3.9.3, 3.10.5 |
| Thurs 9/16 | 8. | Poisson process, stable laws | Durrett 3.7-3.10 | |
| Tues 9/21 | 9. | Martingales | Durrett 4.1-4.3 | Homework 5: 4.1.1, 4.1.4, 4.2.1, 4.3.4, 4.3.8, 4.4.5, 4.4.10, 4.6.2 |
| Thurs 9/23 | 10. | Convergence of martingales, Doob's inequality | Durrett 4.4-4.6 | |
| Tues 9/28 | 11. | Backwards martingales, optional stopping theorem | Durrett 4.7-4.9 | Homework 6: 4.7.1, 4.7.2, 4.8.4, 4.8.5, 4.8.6, 4.9.1 |
| Thurs 9/30 | 12. | The probabilistic method, 2nd moment method | Alon and Spencer, Chaps 2,4,5 | |
| Tues 10/5 | 13. | Correlation inequalities, Azuma's inequality, Chernoff's inequality | Alon and Spencer, Chaps 6,7,A | |
| Thurs 10/7 | 14. | Markov chains, recurrence | Durrett 5.1-5.3 | |
| Tues 10/12 | No class - Fall Break | Homework 7: 5.1.1, 5.1.5, 5.2.4, 5.2.7, 5.2.11, 5.3.7, 5.4.4, 5.5.1, 5.5.5, 5.6.5, 5.6.6 | ||
| Thurs 10/14 | 15. | Stationary measure, asymptotic behaviors | Durrett 5.4-5.6 | |
| Tues 10/19 | 16. | Tail behaviors | Durrett 5.7-5.8 | Homework 8: 5.8.1, 6.1.2, 6.1.3, 6.1.4, 6.2.2, 6.2.3, 6.3.3 |
| Thurs 10/21 | 17. | Birkhoff ergodic theorem | Durrett 6.1-6.3 | |
| Tues 10/26 | 18. | Subadditive ergodic theorem | Durrett 6.4-6.5 | Homework 9: 6.5.5, 7.1.2, 7.1.6, 7.2.1, 7.2.2, 7.3.2, 7.3.3, 7.3.6 |
| Thurs 10/28 | 19. | Construction of Brownian motion | Durrett 7.1, Morters and Peres 1.1-1.4 | |
| Tues 11/2 | 20. | Strong Markov property | Durrett 7.2-7.3, Morters and Peres 2.1-2.4 | Homework 10: 7.4.1, 7.4.2, 7.4.4, 7.5.1, 7.5.2, 7.5.6, 7.6.2 |
| Thurs 11/4 | 21. | Martingales, Itô's formula | Durrett 7.5-7.6 | |
| Tues 11/9 | 22. | Harmonic functions, Dirichlet problem, occupation measure | Morters and Peres 3.1-3.4 | |
| Thurs 11/11 | 23. | Hausdorff dimension, mass distribution principle | Morters and Peres 4.1-4.4 | |
| Tues 11/16 | 24. | Donsker's theorem, CLT for martingales and stationary sequences | Durrett 8.1-8.3 | Homework 11: 8.4.1, 8.4.2, 8.4.4, 8.5.1 |
| Thurs 11/18 | 25. | Brownian bridge, law of iterated logarithm | Durrett 8.4-8.5 | |
| Tues 11/23 | 26. | Heat equation, Feynman-Kac formula | Durrett 9.1-9.4 | Homework 12: 9.1.1, 9.5.1, 9.5.2, 9.7.1 |
| Thurs 11/25 | No class - Thanksgiving | |||
| Tues 11/30 | 27. | Occupation times, Schrödinger equation, local time | Durrett 9.5-9.8, Morters and Peres 6.1-6.2 | |
| Thurs 12/2 | 28. | Ray-Knight Theorem, equilibrium measure | Morters and Peres 6.3-6.4, 8.1-8.2 |
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