## 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:**

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.

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.