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.
-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]
Random variables and their distribution. Lebesgue integral and expectation. [1.2.1, 1.2.2, 1.2.3, 1.3.1, 1.3.2]
Almost sure convergence.
Convergence in probability and in 
.
Monotone convergence theorem, dominated convergence theorem. [1.3.1, 1.3.2, 1.3.3]
Change of variables. Independent random variables, Fubini theorem. [1.3.5,1.4.1, 1.4.3]
Weak law of large numbers, Borel-Cantelli lemmas, Kolmogorov extension theorem. [2.1.1, 2.2.1, 1.4.2]
More on Borel-Cantelli, strong law of large numbers. [2.2.2, 2.3.1, 2.3.2]
Lindeberg central limit theorem. Examples and applications. [3.1.1, 3.1.2]
Weak convergence, convergence in distribution, relation with other forms of convergence. [3.2.1, 3.2.2]
Characteristic functions and more on the central limit theorem. [3.3.1, 3.3.2, 3.3.3]
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 25, 3:00pm-6:00pm. You are allowed to bring and consult any of the listed textbooks (Dembo, Billingsley, Durrett, Williams), but no other sources.
The final will take place on Wednesday, December 11, 3:30pm–6:30pm