Description

Math 275 is an introduction to rigorous probability at the graduate level. The Fall quarter will focus on foundations and sequences of independent random variables, including: measure theory background; independence; laws of large numbers; weak convergence and characteristic functions; central limit theorems.

While you may have encountered some of these topics in an undergraduate probability course, we will take a much deeper look at them here. This course will be followed by (and required) for Math 275B (Winter 2011) and 275C (Spring 2011) which develop the theory of stochastic processes in discrete and continuous time. It should appeal both to students interested in pure mathematics (esp. analysis) and in applications (esp. physics, engineering, biology, economics).

Prerequisites: Although prior or concurrent coursework on measure theory (typically Math 245A) will be useful, all required measure-theoretic material will be covered in class.

General Information

• Instructor: Sebastien Roch (Office hours: MWF 3-3:30 in MS 6228)
• TA: Alexander Vandenberg-Rodes (Office hours: R 3-5 MS 6153)
• Time and place: Lectures MWF 2 in MS 6221; Discussion R 2 in MS 6201
• Required Text: Probability: Theory and Examples (4th Edition) by Durrett (available online)
• Grades will be based on homework to be assigned mostly from Durrett's book. There will be 9 regular assignments and a take-home final. The take-home final will be somewhat longer than the regular assignments, and is to be done with no consultation (with the instructor, TA, other students, or anyone else). The final grade will be based on the following scheme: take-home final (50%) and regular homework (50%). The lowest regular homework score will be dropped.

News

• [Oct 1]: My office hours on Friday, Oct 8 are cancelled.
• [Sep 22]: The first discussion section will take place on Sep 23 (before the first lecture). Alexander will give a brief review of undergraduate probability.

Lectures

• Lec 1 [Sep 24]: Syllabus. Overview.
• Lec 2 [Sep 27]: Probability spaces. Section 1.1 in [D].
• Lec 3 [Sep 29]: Random variables. Sections 1.2, 1.3 in [D].
• Lec 4 [Oct 1]: Independence. Section 2.1 in [D].
• Lec 5 [Oct 4]: Expectations. Sections 1.4-1.6 in [D].
• Lec 6 [Oct 6]: More on expectations. Product measures. Sections 1.6, 1.7 in [D].
• Lec 7 [Oct 8]: First Borel-Cantelli lemma. Section 2.3 in [D].
• Lec 8 [Oct 11]: Second Borel-Cantelli lemma. Section 2.3 in [D].
• Lec 9 [Oct 13]: Applications of Chebyshev's inequality. Section 2.2 in [D].
• Lec 10 [Oct 15]: Weak law of large numbers. Section 2.2 in [D].
• Lec 11 [Oct 18]: More on the weak law of large numbers. Section 2.2 in [D].
• Lec 12 [Oct 20]: Strong law of large numbers. Section 2.4 in [D].
• Lec 13 [Oct 22]: Random series. Section 2.5 in [D].
• Lec 14 [Oct 25]: More on random series. Section 2.5 in [D].
• Lec 15 [Oct 27]: Large deviations. Section 2.6 in [D].
• Lec 16 [Oct 29]: Law of the iterated logarithm.
• Lec 17 [Nov 1]: Weak convergence. Section 3.2 in [D].
• Lec 18 [Nov 3]: Helly's selection theorem and tightness. Section 3.2 in [D]
• Lec 19 [Nov 5]: Characteristic functions: defintion, properties. Section 3.3 in [D].
• Lec 20 [Nov 8]: Characteristic functions and weak convergence. Section 3.3 in [D].
• Lec 21 [Nov 10]: Lindeberg-Feller CLT. Section 3.4 in [D].
• Lec 22 [Nov 12]: Poisson approximation by coupling. Section 3.6 in [D].
• Lec 23 [Nov 15]: Stable laws and infinitely divisible distributions. Sections 3.7 and 3.8 in [D]. Quick background on Levy processes can be found here.
• Lec 24 [Nov 17]: Stable laws: domain of attraction. Section 3.7 in [D].
• Lec 25 [Nov 19]: Method of moments. Section 3.3 in [D]. See Terry Tao's blog for an application in random matrix theory.
• Lec 26 [Nov 22]: Chen-Stein method. See Chapter 14 in Lange's book. See Terry Tao's blog for the Gaussian case.
• Lec 27 [Nov 24]: Lindeberg's proof of the Lindeberg-Feller CLT. See Terry Tao's blog.
• No lecture on Nov 26. Thanksgiving holiday.
• Lec 28 [Nov 29]: Random walks: stopping times. Section 4.1 in [D].
• Lec 29 [Dec 1]: Random walks: recurrence. Section 4.2 in [D].
• Lec 30 [Dec 3]: Random walks: arcsine laws. Section 4.3 in [D].

Assignments

• Hwk 1 [Due in class Oct 1]: Exercises 1.1.2, 1.1.3, 1.1.4, 1.1.5, 1.1.6. Read Appendix.
• Hwk 2 [Due in class Oct 8]: Exercises 1.2.3, 1.2.4, 1.3.8, 2.1.9, 2.1.10. For 1.3.8, you may assume 1.3.7
• Hwk 3 [Due in class Oct 15]: Exercises 1.6.3, 1.6.4, 1.6.6, 1.7.2, 1.7.4.
• Hwk 4 [Due in class Oct 22]: Exercises 2.3.2, 2.3.8, 2.3.14, 2.3.15, 2.3.18.
• Hwk 5 [Due in class Oct 29]: Exercises 2.2.2, 2.2.3, 2.2.8, 2.2.9, 2.4.4.
• Hwk 6 [Due in class Nov 5]: Exercises 2.5.5, 2.5.9, 2.5.10, 2.5.11, 2.6.5.
• Hwk 7 [Due in class Nov 12]: Exercises 3.1.1, 3.2.6, 3.2.13, 3.3.17 (i) and (ii), 3.3.22. Read Section 3.3.4.
• Hwk 8 [Due in class Nov 19]: Exercises 3.4.5, 3.4.6, 3.4.9, 3.4.12, 3.4.13.
• Hwk 9 [Due in class Nov 29. Note this is a MONDAY.]: Exercises 3.3.16, 3.6.3, 3.7.5, 3.7.8, 3.9.7 (Hint: You may want to do Exercise 3.9.6 first.).
• Take-Home Final [Due Dec 3 in class]: Final assignment will be distributed in class on Monday Nov 29 (NOT POSTED); to be done without consultation. No late final will be accepted.