Math 230A / Stat 310A : Syllabus

The first half of the course covers the measure-theoretic foundations of probability theory.

The second half deals with the key limit theorems (law of large numbers, central limit theorems, notions of weak convergence).

Here is a more detailed syllabus (changes are possible, and the content will be adapted to the needs of the course). Numbers in parentheses refer to Amir Dembo's lecture notes.

Sep 25, 27

sigma-algebras, measure and probability spaces, generated sigma-algebras. Caratheodory extension theorem (statement and proof outline). Lebesgue measure (session) [1.1.1, 1.1.2, 1.1.3]

Oct 4

Random variables and their distribution. [1.2.1, 1.2.2, 1.2.3, 1.3.1, 1.3.2]

Oct 9, 11

Lebesgue integral and expectation. Almost sure convergence. Convergence in probability and in L_p. Monotone convergence theorem, dominated convergence theorem. [1.3.1, 1.3.2, 1.3.3]

Oct 16, 18

Change of variables. Independent random variables, Fubini theorem. [1.3.5,1.4.1, 1.4.3]

Oct 23, 25

Weak law of large numbers, Borel-Cantelli lemmas, Kolmogorov extension theorem. [2.1.1, 2.2.1, 1.4.2]

Oct 30, Nov 1

More on Borel-Cantelli, strong law of large numbers. [2.2.2, 2.3.1, 2.3.2]

Nov 6, 8

Lindeberg central limit theorem. Examples and applications. [3.1.1, 3.1.2]

Nov 13, 15

Weak convergence, convergence in distribution, relation with other forms of convergence. [3.2.1, 3.2.2]

Nov 27, 29

Characteristic functions and more on the central limit theorem. [3.3.1, 3.3.2, 3.3.3]

Dec 4, 6

Review.

Homeworks will be assigned every Wednesday, the first time on September 23, and due the following Wednesday (a total of 9 homeworks will be assigned).

An in-class midterm will be given on Friday, October 27, 3:30pm-6:30pm. You are allowed to bring and consult any of the listed textbooks (Dembo, Billingsley, Durrett, Williams), but no other sources.

The final will be on Wednesday, December 13, 3:30-6:30PM with the same rules as for midterm.