| Fall 2025 MAT 550: Introduction to Probability | ||
| Schedule | MW 3:30am-4:50pm Physics P127 | |
| Instructor | Robert Hough | |
| Office hours | Th 7-8 in MLC, F 9am-11pm in Math Tower 4-107. | |
| 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)
| Mon 8/25 | 1. | Random variables and integration | Durrett 1.1-1.5 | |
| Wed 8/27 | 2. | Expected value, Carathéodory extension theorem | Durrett 1.6-1.7, A.1 | |
| Mon 9/1 | No class - Labor day | |||
| Wed 9/3 | 3. | Kolmogorov extension theorem | Durrett A.2-A.5 | |
| Mon 9/8 | 4. | Weak law, Borel-Cantelli | Durrett 2.1-2.3 | |
| Wed 9/10 | 5. | Strong law | Durrett 2.4-2.7 | |
| Mon 9/15 | 6. | Characteristic functions | Durrett 3.1-3.3 | |
| Wed 9/17 | 7. | Central limit theorem, local limit theorem | Durrett 3.4-3.6 | |
| Mon 9/22 | 8. | Poisson process, stable laws | Durrett 3.7-3.10 | |
| Wed 9/24 | 9. | Martingales | Durrett 4.1-4.3 | |
| Mon 9/29 | 10. | Convergence of martingales, Doob's inequality | Durrett 4.4-4.6 | |
| Wed 10/1 | 11. | Backwards martingales, optional stopping theorem | Durrett 4.7-4.9 | |
| Mon 10/6 | 12. | The probabilistic method, 2nd moment method | Alon and Spencer, Chaps 2,4,5 | |
| Wed 10/8 | 13. | Correlation inequalities, Azuma's inequality, Chernoff's inequality | Alon and Spencer, Chaps 6,7,A | |
| Mon 10/13 | No class - Fall Break | |||
| Wed 10/15 | 14. | Markov chains, recurrence | Durrett 5.1-5.3 | |
| Mon 10/20 | 15. | Stationary measure, asymptotic behaviors | Durrett 5.4-5.6 | |
| Wed 10/22 | 16. | Tail behaviors | Durrett 5.7-5.8 | |
| Mon 10/27 | 17. | Birkhoff ergodic theorem | Durrett 6.1-6.3 | |
| Wed 10/29 | 18. | Subadditive ergodic theorem | Durrett 6.4-6.5 | |
| Mon 11/3 | 19. | Construction of Brownian motion | Durrett 7.1, Morters and Peres 1.1-1.4 | |
| Wed 11/5 | 20. | Strong Markov property | Durrett 7.2-7.3, Morters and Peres 2.1-2.4 | |
| Mon 11/10 | 21. | Martingales, Itô's formula | Durrett 7.5-7.6 | |
| Wed 11/12 | 22. | Harmonic functions, Dirichlet problem, occupation measure | Morters and Peres 3.1-3.4 | |
| Mon 11/17 | 23. | Hausdorff dimension, mass distribution principle | Morters and Peres 4.1-4.4 | |
| Wed 11/19 | 24. | Donsker's theorem, CLT for martingales and stationary sequences | Durrett 8.1-8.3 | |
| Mon 11/24 | 25. | Brownian bridge, law of iterated logarithm | Durrett 8.4-8.5 | |
| Wed 11/26 | No class - Thanksgiving | |||
| Mon 12/1 | 26. | Heat equation, Feynman-Kac formula | Durrett 9.1-9.4 | |
| Wed 12/3 | 27. | Occupation times, Schrödinger equation, local time | Durrett 9.5-9.8, Morters and Peres 6.1-6.2 | |
| Mon 12/8 | 28. | Ray-Knight Theorem, equilibrium measure | Morters and Peres 6.3-6.4, 8.1-8.2 |
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