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 (minor 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 24, 26
-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 1, 3
Random variables and their distribution. Almost sure convergence. Lebesgue integral and expectation. [1.2.1, 1.2.2, 1.2.3, 1.3.1, 1.3.2]
Oct 8, 10
Convergence in probability and in . Monotone convergence theorem, dominated convergence theorem. Change of variables. [1.3.3, 1.3.5]
Oct 15, 17
Independent random variables, Kolmogorov extension theorem, Fubini theorem. [1.4.1, 1.4.2, 1.4.3]
Oct 22, 24
Weak law of large numbers, Borel-Cantelli lemmas. [2.1.1, 2.2.1]
Oct 29, 31
More on Borel-Cantelli, strong law of large numbers. [2.2.2, 2.3.1, 2.3.2]
Nov 5, 7
Lindeberg central limit theorem. Examples and applications. [3.1.1, 3.1.2]
Nov 12, 14
Weak convergence, convergence in distribution, relation with other forms of convergence. [3.2.1, 3.2.2]
Nov 26, 28
Characteristic functions and more on the central limit theorem. [3.3.1, 3.3.2, 3.3.3]
Dec 3, 5
Homeworks will be assigned every Wednesday, the first time on September 26, and due the following Wednesday (a total of 9 homeworks will be assigned).
An in-class midterm will be given on Friday, October 26. The exam will be open-book.
The final will be on Tuesday, December 11 with the same rules as for midterm.