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)
Prerequisites:
MAT 127B; (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.
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.