MATH 671, FALL 2004
Probability Theory I  
MWF 11:15-12:05, 406 Malott
Greg Lawler, 567 Malott
TA, Brigitta Vermesi

Office Hours:


Greg Lawler, Tues 3:15 - 4:30, Wed 1:30 - 2:30, 567 Malott
Brigitta Vermesi, Mondays 6:00 - 8:00. Place 218 Malott

This is a course in measure theoretic probability. I will assume that students have seen meaasure theory and some probability. Students with good analysis backgrounds who have had no probability can probably survive with a little extra reading. Here are some notes on probability (for students who know measure theory). The first five sections outline the prerequisite material for this course. The main topics of the course are martingales, random walk, and Brownian motion.

NEWS ABOUT THE FINAL

Problem 3 is miswritten. Here is the corrected version.

Text

Durrett, Probability: Theory and Examples, 3rd edition.
We will cover (in approximate order) Chapter 4, the second half of Chapter 2, Chapter 3, and Chapter 7.

Extra reading: Lawler, Introduction to Stochastic Processes (on reserve in Mathematics Library). I will hand out copies of some chapters from this book.

Homework and Exams

Regular HW assignments will be given in the lectures. Solutions will be handed in on Wednesday of the following week (unless mentioned otherwise). The assignments will be listed on this page. In lieu of an in-class final exam, there will be an long final problem set due on the day the final exam is scheduled.

Here are some notes on strong approximation and modulus of continuity of Brownian motion.

Problem set 1 (due 9/1): p. 22, 4.1; p. 33, 4.13, 4.14

Problem set 2 (due 9/8): p. 225, 1.7; p. 226, 1.11; p. 234, 2.4; p.235, 2.9 (in part iii of this problem, do we need to allow c=-infinity as a possible value? How about if P(Y_j = 0) > 0?);
ISP , 5.5, 5.10
Suppose X_1,X_2,.. are iid random variables with finite mean m. Let S_n = X_1 + ... + X_n. What is E(X_1 | S_n)?

Problem set 3 (due 9/15): p. 235, 2.11; p. 237, 3.1, 3.4; p. 248, 4.5; EP3-1, EP3-2

Problem set 4 (due 9/22): EP4-1, EP4-2 ; p. 260, 5.2; p. 262, 5.8.

Problem set 5 (due 9/29) p. 272 -- 273, 7.3 (ii) (Part (i) was done in class; you may assume this); 7.4, 7.5, 7.6, ; ISP , 5.14 (The fact from exercise 5.13 was done in class and you may assume this)

Problem set 6 (due 10/6) ISP 2.6, 2.7, 2.9,; EP6-1 .

Problem set 7 (due 10/20) p. 170, 9.8; EP7-1, EP 7-2, EP 7-3 (Note: EP7-1 has been corrected.)

Problem set 8 (due 10/27) p. 179, 1.12; p. 180, 1.15; EP8-1,8-2

Problem set 9 (due 11/03) p. 160, 8.2, 8.3; EP9-1; ISP 8.1,8.6 Problem set 10 (due 11/10), EP10-1 ; ; ISP 8.7, 8.8

Problem set 11 (due 11/17), p. 393, 4.2; p.397, 5.1; p. 398, 5.2 EP11-1, EP11-2

Problem set 12 (due 12/1), p. 399, 5.6, 5.7; ISP 8.9, 8.10 (Note: in 8.10, there is a misprint --- the 4/5 should be 3/5.)

EXCEPT FOR THE FINAL PROBLEM SET, students may discuss homeworks with each other (and others) but should write solutions themselves. The final problem set should be done without consulting others (but any books or other written material may be consulted).

Last modified: 3 December 2004