## Math 6040: Mathematical Probability, Fall 2015

**Time and place:** 3:00--4:30 MW, 8/24 to 12/9 except for 9/7, 10/12, and 10/14, in JTB 110.

**Instructor:** S. Ethier (Prof.), 581-6148, ethier@math.utah.edu. Office hours are 2:00--2:50 and 4:35--5:00 MW. Other times are available by appointment.

**Text:** Rick Durrett's *Probability: Theory and Examples*, available free online at https://www.math.duke.edu/~rtd/PTE/PTE4_1.pdf. Also available in hardcover at Amazon for $68.93.

**Prerequisite:** Math 6210 or familiarity with same.

**Grades:** Based on problem sets and a take-home final exam.

**Expected learning outcomes:** This is a first *rigorous* course in probability. The student who completes it successfully will understand how to prove basic results in probability related to the strong law of large numbers, the central limit theorem, conditional expectations, martingales, and Brownian motion.

- Week 1: Aug. 24, 26. We covered probability spaces and random variables through Theorem 1.2.2 of the text, omitting some topics.
- Week 2: Aug. 31, Sep. 2. We finished Chapter 1 (omitting some measure theory topics) and continued through Section 2.1.1. Assignment 1, due Sep. 14. Exercises from Durrett: 1.2.1, 1.2.4, 1.3.9, 1.6.3, 1.6.6, 1.6.10.
- Week 3: Sep. 9. We continued in Chapter 2 through an L^2 version of the WLLN.
- Week 4: Sep. 14, 16. We finished the proof of the SLLN. Assignment 2, due Sep. 30. Exercises from Durrett: 2.1.16, 2.1.18, 2.3.2, 2.3.18, 2.4.3, 2.4.4.
- Week 5: Sep. 21, 23. We began weak convergence, getting through Helly's theorem.
- Week 6: Sep. 28, 30. We continued with weak convergence, proving the continuity theorem. Assignment 3, due Oct. 21. Exercises from Durrett: 3.2.14, 3.2.15, 3.3.8, 3.3.13, 3.4.2, 3.4.4.
- Week 7: Oct. 5, 7. We proved the central limit theorem and started conditional expectation.
- Week 8: Oct. 19, 21. We finished conditional expectation and started martingales. Assignment 4, due Nov. 18. Exercises from Durrett: 5.1.10, 5.1.11, 5.2.7, 5.2.9, 5.2.11, 5.2.12.
- Week 9: Oct. 26, 28. We proved the martingale convergence theorem.
- Week 10: Nov. 2, 4. We covered the optional stopping theorem.
- Week 11: Nov. 9, 11. We finished martingales and began Brownian motion.
- Week 12: Nov. 16, 18. We finished Section 8.1 of Brownain motion. Assignment 5, due Dec. 9. Exercises from Durrett: 8.1.3, 8.2.4, 8.3.6, 8.4.1, 8.5.4, 8.7.2.
- Week 13: Nov. 23, 25. We studied the strong Markov property of Brownian motion.
- Week 14: Nov. 30, Dec. 2. We nearly finished Donsker's theorem.
- Week 15: Dec. 7, 9. The last week will be devoted to reviewing old probability prelims.

Take-home final exam, corrected 12/11.