Spring 2018

MWF 11:00--11:50 SNOW 564

Terry Soo, Snow 610

Office hours: Drop in or email me.

Course description.

This is a measure theoretic probability course covering some standard topics in advanced probabilty such as limit theorems, conditional expectation, random walks, Markov chains, martingales, point processes, Palm measure, random graphs, and Brownian motion. We will cover topics according to our interests. In addition, in order to avoid overlap with Math727/Math728/865/866, we will try approach these topics from a different perspective. For example, in Math727, the typical proof of the central limit theorem is by characteristic functions. Rather than revisiting this and merely giving more details of the proof, I'm hoping we will cover an entropy proof of the central limit theorem. We will also make an effort to connect our studies to present day research in probability theory.

Prerequisites.

Measure theory and Math 727. Please see me if you have doubts about background. This is a high level graduate course.

Grading

Subject to revision

Presentations (of topics or important lemmas and exercises)

Final project/presentation

No textbook is required. We will not follow any particular textbook, but will draw material from a variety of sources, including papers.

Suitable references

Probability: theory and examples; Durrett

Foundations of modern probability; Kallenberg

Probability with martingales; Williams

Real analysis and probability; Dudley

A course on Borel sets; Srivastava

Notes:

Additional exercises

Possible presentation topics

Measure theory refresher I

Integration and expectation

Etemadi's Strong Law

Introduction to the probabilistic method

Conditional expectation I

Conditional expectation II

Conditional expectation III

Introduction to martingales

Uniform integrability and convergence in the mean