MATH 6710: Probability Theory I (Fall 2012)

Instructor: Nathaniel Eldredge

Prerequisites: Students should have previous experience with abstract measure theory (i.e. measure spaces), such as from MATH 6110 or 6210, and a solid grasp of undergraduate real analysis (epsilon-delta techniques and so on). An undergraduate probability course is recommended but not strictly required.

Textbook: R. Durrett, Probability: Theory and Examples, 4th edition. It can be bought from the Cornell bookstore; an e-book version is also available from Cambridge University Press.

This will be an introduction to measure-theoretic probability theory at the graduate level. We will begin with a brief review of abstract measure theory, with a view to its use in probability: random variables, expectation, measurability and sigma-fields, and modes of convergence. We will discuss the idea of independence and prove the strong law of large numbers and some related results. Next we will attack the central limit theorem, which will require some discussion of weak convergence (aka convergence in distribution) and characteristic functions (aka Fourier transforms). Finally, to begin a transition to stochastic processes, we will talk about random walk, conditional expectation and martingales, and perhaps Markov chains.