Department of Mathematics and Statistics
McGill University
MATH 589 Advanced Probability Theory II
Reading Course


MATH 589 Advanced Probability Theory (4 credits). (Prerequisite: MATH 587 or equivalent.) Martingales and martingale convergence theorems (if not covered in 587). Weak convergence of measures. Characteristic functions: elementary properties, inversion formula, uniqueness and continuity theorems. Lindeberg-Feller Central Limit Theorem. If time permits, a selection from: infinitely divisible laws, stable laws, Brownian motion and its properties.

Textbook: Billingsley, P. (1995). Probability and Measure 3rd Ed. Wiley-Interscience, New York. (Sections 25-30, 35, extra topics)

The Syllabus: We won't work very closely out of Billingsley. Rather, you should read and be responsible for the following chapters from the 587 and 589 notes.

  1. Section 11 "Martingales" at the end of the 587 notes. It's not very big, but it has most of the well-known results for discrete-time parameter martingales (including supermartingales and submartingales). Section 11.1 on stopping times and stochastic processes is some background you need for reading the martingale stuff.
  2. Chapter 1 "Weak convergence of Probability Measures" from the 589 notes. Leave out the section on total variation measure. The important stuff is in sections 1.3, 1.4, and 1.5. You can leave out proposition 1.3.4, the discussion called "background" on page 9. The standard reference is Billingsley below if you need it.
  3. Chapter 2 "Characteristic Functions" from the 589 notes. All of this chapter.
  4. Chapter 3 "The Central Limit Theorem". Section 2.1 (The CLT) is absolutely important. Read the statement of the Berry-Esseen Theorem 3.2.3 in section 3.2 (but not the proof, which is overwhelming), and the binomial example in the remark which follows the proof. Do section 3.3 on Poisson convergence, but leave out proof 2 and the preceding remark. Sections 3.4 and 3.6 are fun but you don't have to do them. They could form the basis of a project, though.
  5. Chapter 4 "Weak Convergence in C[0,1] and Weiner Measure" and Chapter 5 "Brownian Motion". As far as present day applications of stochastic processes are concerned, Brownian Motion is extremely important. Another name for it is "Weiner Process". Chapter 4 establishes (a) the existence of the Brownian motion process using the weak convergence methods of chapter one, and also (b) a strengthened version of the CLT called the Invariance Principle. But the proofs are very technical and somewhat overwhelming. So skim it and satisfy yourselves that we are using the theory from chapter 1. The stuff I have in chapter 5 is very brief, and easier to read. It could be developed into a nice project.

References:
Ash, R. B. and C.A. Doleans-Dade. (2000). Probability and Measure Theory, 2nd Ed. Academic Press. (Chapters 6,7,9 approximately)
Billingsley, P. (2000). Convergence of Probability Measures, 2nd Ed., Wiley, New York.
Chow, Y. S. and H. Teicher (1997). Probability Theory: Independence, Interchangeability, Martingales, 3rd Ed. Springer-Verlag, New York.
Chung, K. L. (2001). A Course in Probability Theory, 3rd Ed. Academic Press, San Diego.
Dudley, R. M. (1989). Real Analysis and Probability. Wadsworth, Pacific Grove, California.
Durrett, Richard (1996). Probability: Theory and Examples 2nd Ed. Duxbury, Belmont, California.
Feller, William (1971). An Introduction to Probability Theory and Its Applications, Volume II 2nd Ed. Wiley, New York.
Hoffmann-Jorgensen, J. (1994). Probability with a View Toward Statistics, Volumes I and II. Chapman-Hall, New York.
Resnick, S. I. (2001). A Probability Path. Birkhauser, Boston.
Stoyanov, J. (1997). Counterexamples in Probability, 2nd Ed. Wiley, New York.
Stromberg, K. (1994). Probability for Analysts. Chapman-Hall, New York.

Weekly Meetings: These take place on Thursdays from 11:30 to 12:30. They are compulsory.

Assignments: #1 ,#2, #3 , #4

Solutions to Assignments: #1 ,#2 , #3 ,#4

Marking Scheme: There are three categories, of which you must choose two: They are (1) a project, (2) a final exam, and (3) the assignments. For the two that you choose, each is worth 50%.

W. J. Anderson
Burnside Hall 1245
Tel: 398-3848
email:bill@math.mcgill.ca