Math 940- Advanced Probability
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
Brownian Motion; Morters and Peres
A course on Point processes; Reiss

Notes:
Additional exercises
Possible presentation topics


Presentations.


March 26,28. Tommy. Cramer-Rao inequality and a proof of the central limit theorem.

March 30, April 2, April 4. Sefika. The almost sure Ergodic theorem via maximal inequalites.

April 25. Raul. Hermite polynomials and proability theory.

April 27. Amanda. Iterating von Neumann's acceptance-rejection algorithm.

April 30. Neha. Stochastic integration.

May 2. Wenju. On choosing and bounding probability metrics and variations of autoencoders.

Final Exam Block: Tuesday May 8, 10:30 - 1:00 PM. Mehmet. Probabilistic number theory.
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
Martingales and the strong law
Applications of Hall's marriage theorem
Point processes I
Point processes II

Brownian Motion--Peter Morters lecture notes