## Math 230A / Stat 310A : SyllabusThe 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. |