Advanced Probability I University of Sao Paulo
The general treatment of Probability Theory requires its formulation in abstract spaces, in the framework introduced by Kolmogorov.
Introduce the basics of Probability Theory into abstract spaces, including the necessary elements of Measure Theory, in the framework formulated by Kolmogorov.
Exam and exercises, with the possibility of collecting an article at the end of the course.
1. Probability Spaces: (a) Lebesgue-Stieltjes Measure, Carath_dory Extension Theorem; (b) Measures of Probability, Random Variables; (c) Integration, Expectation, Convergence Theorems; (d) Product measures, Fubini's theorem; (e) Independence; (f) Kolmogorov Extension Theorem; (g) Radon-Nikodym Theorem, Conditional Expectation. 2. Laws of Large Numbers: (a) Convergence in Probability and Almost Sure Convergence; (b) Weak Law of Large Numbers; (c) Borel-Cantelli lemmas; (d) Strong Law of Large Numbers. 3. Central Limit Theorem: (a) Convergence in Distribution; (b) Characteristic Functions; (c) TLC for Random Variables I.I.D; (d) TLC for Triangular Arrangements.
Online Course Requirement
Vladimir Belitsky, Miguel Natalio Abadi, Anatoli Iambartsev
Site for Inquiry
Please inquire about the courses at the address below.
Email address: https://www.ime.usp.br/en