General Information

• Time and place: MWF at 11 in MS 5217
• Instructor: Sebastien Roch
• Office hours: MWF 1-1:30 in MS 6228
• TA: No TA.
• Text: Probability: Theory and Examples, 3rd edition, by R. Durrett (Note: You can download a beta 4th edition of the book from Rick Durrett's website.)
• Topics: (a) Martingales, (b) Ergodic Theory, (c) Random Walks
• Grades will be based on homework to be assigned mostly from Durrett's book. There will be 8 regular assignments. In addition, a final homework will be somewhat longer than the others, and is to be done with no consultation (with the instructor, other students, or anyone else). The final grade will be based on the following scheme: final homework (50%) and regular homework (50%). The two lowest regular homework scores will be dropped.

Lectures

• Lec 1 [Jan 4]: Syllabus. Overview.
• Lec 2 [Jan 6]: Conditional expectation I: definition, existence, uniqueness. Sec 4.1.
• Lec 3 [Jan 8]: Conditional expectation II: examples, properties, (regular conditional probabilities). Sec 4.1.
• Lec 4 [Jan 11]: Martingales I: definition, examples. Sec 4.2.
• Lec 5 [Jan 13]: Martingales II: stopping times, betting systems. Sec 4.2.
• Lec 6 [Jan 15]: Martingale convergence theorem. Sec 4.2.
• No lecture on Jan 18 (Martin Luther King, Jr, holiday).
• Lec 7 [Jan 20]: Branching processes. Sec 4.3.
• Lec 8 [Jan 22]: Martingales in L2. Back to branching processes. Sec 4.4.
• Lec 9 [Jan 25]: Martingales in L2 (continued). Sec 4.4.
• Lec 10 [Jan 27]: Martingales in Lp. Sec 4.4.
• Lec 11 [Jan 29]: Azuma's inequality. McDiarmid's survey.
• Lec 12 [Feb 1]: Uniform Integrability. Sec 4.5.
• Lec 13 [Feb 3]: Levy's Upward Theorem. Sec 4.5.
• Lec 14 [Feb 5]: Levy's Downward Theorem. Sec 4.6.
• Lec 15 [Feb 8]: Optional Sampling Theorem. Sec 4.7.
• Lec 16 [Feb 10]: Stationary Markov Chains. Sec 5.4.
• Lec 17 [Feb 12]: Ergodic Theorem for Markov Chains. Sec 5.5.
• No lecture on Feb 15 (President's Day holiday).
• Lec 18 [Feb 17]: Stationary Processes. Sec 6.1.
• Lec 19 [Feb 19]: Birkhoff's Ergodic Theorem I. Sec 6.2.
• Lec 20 [Feb 22]: Birkhoff's Ergodic Theorem II. Sec 6.2.
• Lec 21 [Feb 24]: Subadditive Ergodic Theorem I. Sec 6.4.
• Lec 22 [Feb 26]: Subadditive Ergodic Theorem II. Sec 6.4.
• Lec 23 [Mar 1]: Topics in RWs and MCs: Wald's identities. Sec 3.1.
• Lec 24 [Mar 3]: Topics in RWs and MCs: Recurrence and transience in SRW. Sec 3.2.
• Lec 25 [Mar 5]: Topics in RWs and MCs: Convergence to stationarity, mixing time, coupling. Sec 5.5. See also Chapter 4, 5 of Levin et al..
• Lec 26 [Mar 8]: Topics in RWs and MCs: Eigenfunction techniques I. See Chapter 12, 13 of Levin et al..
• Lec 27 [Mar 10]: Topics in RWs and MCs: Eigenfunction techniques II. See Chapter 12, 13 of Levin et al..
• Lec 28 [Mar 12]: Topics in RWs and MCs: Shuffling genes. (Teaser for next quarter's MATH 285K.)

Assignments

• Hwk 1 [Due Jan 15]: Exercises 4.1.1, 4.1.5, 4.1.9, 4.1.13, 4.1.15 [4th edition!].
• Hwk 2 [UPDATE: Due Jan 25]: Exercises 4.2.3, 4.2.7, 4.2.9, 4.2.13, 4.2.14 [4th edition!].
• Hwk 3 [Due Jan 29]: Exercises 4.3.12, 4.3.13, 1.9.2, 1.9.3, 1.9.5 [4th edition!]. Note: Read Section 1.9.
• Hwk 4 [Due Feb 5]: Exercises 4.2.6, 4.4.4, 4.4.5, 4.4.6, 4.4.7 [4th edition! version of June 24, 2009].
• Hwk 5 [Due Feb 12]: Exercises 4.5.1, 4.5.2, 4.5.5, 4.5.6, 4.5.8 [4th edition! version of June 24, 2009].
• Hwk 6 [Due Feb 19]: Exercises 4.6.2, 4.6.5, 4.7.3, 4.7.5, 4.7.9 [4th edition! version of June 24, 2009].
• Hwk 7 [Due Feb 26]: Exercises 6.1.1, 6.1.6, 6.2.1, 6.2.2, 6.2.3 [4th edition! version of June 24, 2009].
• Hwk 8 [Due Mar 5]: Exercises 6.3.1, 6.4.1, 6.5.1, 6.5.2, 6.5.3 [4th edition! version of June 24, 2009].
• Take-Home Final [Due Mar 12 in class]: Final assignment will be distributed in class on Mar 5 (NOT POSTED); to be done without consultation. No late final will be accepted.