General course information:
This is a graduate introduction to probability theory. The topics include rigorous measure-theoretic overview of probability theory, from the initial definitions to the proofs of the
law of large numbers and central limit theorem. Stochastic processes part includes martingales and Markov chains in discrete time.
Students should refer to the Canvas website to all day-to-day class information.
Lecture notes (Fall 2025):
- Lecture 1
- Probability spaces and measures
- Lecture 2
- Construction of the Lebesgue measure
- Lecture 3
- Random variables
- Lecture 4
- Lebesgue integration
- Lecture 5
- Mathematical expectation, concentration inequalities
- Lecture 6
- Independence
- Lecture 7
- Convergence of random variables, Law of Large Numbers
- Lecture 8
- Central limit theorem - 1 (via characteristic functions)
- Lecture 9
- Central limit theorem - 2 (other proofs and related results)
- Lecture 10
- Conditioninal expectation
- Lecture 11
- Martingales - 1 (definition and convergence)
- Lecture 12
- Martingales - 2 (optimal stopping)
- Lecture 13
- Markov chains
Lecture notes above are based on the following recommended (and much more complete) texts:
- [1]
- R. Durrett, Probability: Theory and Examples (5th Edition), 2019
- [2]
- S. Chatterjee, Probability Theory lecture notes
- [3]
- A. Dembo, Probability Theory lecture notes, 2021
- [4]
- T. Tkocz, Probability lecture notes, 2020
- [5]
- E. Çınlar, Probability and Stochastics, 2011