Math 561. Theory of Probability I

Probability: Theory and Examples (2nd Edition) Durrett, 1996, Duxbury Publishing
Probability with Martingales, Williams, 1991, Cambridge University Press

1. Introduction and probabilistic framework
   Connections with measure theory; Convergence of random variables: a.s., convergence in L_p, convergence in probability, convergence in distribution

2. Information, Independence, and Conditioning
   Sigma-algebra manipulation: The calculus of sigma-algebras, Stopping times; Independence; Conditioning

3. Asymptotics: limit theorems
   Weak and strong law of large numbers; Large deviations; Central Limit Theorem

4. Martingales
   Optional Sampling; Maximal Inequalities; Crossing Lemmas; Convergence; Doob-Meyer Decomposition

5. Weak Convergence
   Weak convergence and the Prohorov metric; The topology of measure space

6. Construction of Wiener Measure