Lecture Notes on Probability Theory
DOWNLOAD FILE (updated Jan. 2, 2018)

TABLE OF CONTENTS

 Introduction
1. Probability Spaces.  
2. Random Variables.
3. Kolmogorov’s Consistency Theorem.

 Laws of Large Numbers
4. Inequalities for Sums of Independent Random Variables.
5. Laws of Large Numbers.
6. 0 – 1 Laws. Random Series. 
7. Stopping Times, Wald’s Identity, Markov Property. 

 Central Limit Theorem
8. Convergence of Laws.
9. Characteristic Functions.
10. The Central Limit Theorem.
11. Lindeberg’s CLT. Three Series Theorem.

 Conditional distributions and martingales
12. Conditional Expectations and Distributions.
13. Martingales. Uniform Integrability.
14. Stopping times, Optional Stopping Theorem.
15. Doob’s Inequalities and Convergence of Martingales.

 Metrics on probability measures
16. Bounded Lipschitz Functions.
17. Convergence of Laws on Metric Spaces.
18. Strassen’s Theorem.
19. Kantorovich-Rubinstein Theorem.
20. Brunn-Minkowski and Prekopa-Leindler Inequalities.

 Brownian motion
21. Stochastic Processes. Brownian Motion.
22. Donsker Invariance Principle.
23. Convergence of Empirical Process to Brownian Bridge.
24. Reflection Principles for Brownian Motion. 
25. Skorohod’s Imbedding and Laws of the Iterated Logarithm. 

Poisson processes.
26. Overview of Poisson Processes. 
27. Sums over Poisson Processes.
28. Infinitely Divisible Distributions: Basic Properties.
29. Canonical Representations.
30. α-stable Distribution and Processes.

Representations of exchangeable arrays
31. Moment Problem and de Finetti's Theorem for Coin Flips
32. The general de Finetti and Aldous-Hoover Representation.  
33.  The Dovbysh-Sudakov Representation.