Department of Mathematics Syllabus

This syllabus is advisory only. For details on a particular instructor's syllabus (including books), consult the instructor's course page. For a list of what courses are being taught each quarter, refer to the Courses page.

MAT 235A: Probability Theory
Approved: 2010-05-01, Janko Gravner

Units/Lecture:

Fall, every year (alternating years, taught by Dept of Statistics); 4 units; lecture/term paper or discussion

Suggested Textbook: (actual textbook varies by instructor; check your instructor)
Probability-Theory and Examples, by Rick Durrett ($70)
Search by ISBN on Amazon: 0534424414

Prerequisites:

Courses MAT 125B and MAT 135A or STA 131A or consent of instructor.

Course Description:

Measure-theoretic foundations, abstract integration, independence, laws of large numbers, characteristic functions, central limit theorems. Weak convergence in metric spaces, Brownian motion, invariance principle. Conditional expectation. Topics selected from: martingales, Markov chains, ergodic theory.

Suggested Schedule:

Lectures Sections Topics/Comments
2 weeks
Probability spaces, measure-theoretic background
1 week
Random variables, distribution functions, examples of special distributions
1 week
Independence
1.5 weeks
Expected values
1.5 weeks
Weak and strong laws of large numbers
2 weeks
Gaussian distribution and Central Limit Theorem
Time permitting
Infinite series of independent random variables; the law of the iterated logarithm; Poisson convergence

Additional Notes:

The above topics cover chapters 1 and 2 of Durrett.

Measure theory is not assumed as a prerequisite, so some review (without longer proofs) is likely necessary.

A good supplementary reading is "Probability with Martingales," by David Williams.