## General course information:

This is a graduate introduction to probability theory.
It is an introduction to mathematical probability theory (measure-theoretic probability). The topics include mathematical measure-theoretic probability theory,
law of large numbers, central limit theorem, conditioning. Stochastic processes, martingales, filtrations and stopping times.
Time permitting, there is an introduction to Markov chains, Poisson processes, and Brownian motion.

Syllabus Fall 2023

**Please refer to the Canvas website to all day-to-day class information. **

## Approximate list of topics:

- # 1
- Probability spaces and measures
- # 2
- Construction of the Lebesgue measure, Caratheodory extension theorem
- # 3
- Random variables, their laws, generated sigma-algebras, distribution and density functions
- # 4
- Lebesgue integration (including monotone and dominated convergence theorems) and mathematical expectation
- # 5
- Properties of mathematical expectation, including Jensen's, Holder's, Minkowski's inequalities. Concentration inequalities (Markov's, Chebyshev's, Chernoff, Paley-Zygmund)
- # 5
- Further properties of mathematical expectation: independence and product measures, Fubini's theorem, convolution
- # 6
- Types of convergence of random variables
- # 7
- Weak and strong laws of large numbers, Borel-Cantelli lemma
- # 8
- Central limit theorem
- # 9
- Conditioninal expectation
- #10
- Martingales, stopping times
- #11
- Optional stopping theorem, applications
- #12
- Pólya urns, martingale convergence, Doob's decomposition theorem

## Some reference textbooks and materials:

- [1]
- A. Dembo, Probability Theory lecture notes, 2021
- [2]
- R. Durrett, Probability: Theory and Examples (5th Edition), 2019
- [3]
- E. Çınlar, Probability and Stochastics, 2011
- [4]
- S. Chatterjee, Probability Theory lecture notes