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 21, 23
-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. [1.2.1, 1.2.2, 1.2.3, 1.3.1, 1.3.2]
Oct 5, 7
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 12, 14
Change of variables. Independent random variables, Fubini theorem. [1.3.5,1.4.1, 1.4.3]
Oct 19, 21
Weak law of large numbers, Borel-Cantelli lemmas, Kolmogorov extension theorem. [2.1.1, 2.2.1, 1.4.2]
Oct 26, 28
More on Borel-Cantelli, strong law of large numbers. [2.2.2, 2.3.1, 2.3.2]
Nov 2, 5
Lindeberg central limit theorem. Examples and applications. [3.1.1, 3.1.2]
Nov 9, 11
Weak convergence, convergence in distribution, relation with other forms of convergence. [3.2.1, 3.2.2]
Nov 16, 18
Characteristic functions and more on the central limit theorem. [3.3.1, 3.3.2, 3.3.3]
Nov 30, Dec 2
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 Monday, October 26. 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 Monday, December 7 with the same rules as for midterm.