# 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

-algebras, measure and probability spaces, generated -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 . 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.