Syllabus for Math 605, Probability Theory (Fall2022)

Instructor

Luc Rey-Bellet
LGRT 1423K
luc@math.umass.edu

Office hours:

• Monday 11:00 AM-12 PM on ZOOM
  
• Friday 1:00 PM-2:00 2PM in LGRT 1423 K or on ZOOM
• By appointment is always possible, and/or ask your questions be email.

Class Meeting

Tu-Th, 11:30AM--12:45PM in LGRT 141

Class homepage

On Moodle https://umass.moonami.com/course/view.php?id=33138

Syllabus

This is the first part of a (newly renamed) 2-semester graduate sequence Math605-Math606 which leads to the Stochastics qualifying exam. Prerequisites are

1. a solid working knowledge of undergaduate probablilty (at least STAT 515 or STAT 607 or equivalent).
2. a solid working knowedge of analysis (at least Math 523 or equivalent)
3. some mathematical maturity

In Math 605 we will cover some of the foundations of probability

1. Axioms of probability and the construction of probability spaces.
   
2. Random variable, integration, and convergence of random variables. The law of large numbers. Simulation of random variables.
3. Gaussian random variables, characteristic and moment generation functions, the central limit theorem, large deviation and concentration inequalities.
   
4. Conditional expectation, the radon-nikodym theorem and martingales.

In the second semester (Math 606) we will get to stochastic processes, Poisson processes, Markov processes, Branching process, Renewal processes, stochastic control, and Brownian motion.

Learning Objectives

One can do many interesting applications with undergaduate probability (i.e. Probability without measure theory) but at some point you will need more. The goal of the class is to bring your probability knowledge to the next level so that you can tackle more sophisticated problems and are able to take classes (or read by yourself) on, for exmaple, stochastic processes, stochastic differential equations, statistical learning theory (an important component of machine learning), large deviation theory, optimal transport, information theory, ergodic theory and so on....

Grade/assignment

Weekly homework, one midterm and one final exam, each valued 1/3 of your grade.

Textbooks

The official text book for the class is

• Probability and Stochastics by Erhan Cinlar. Graduate Texts in Mathematics 261. Springer 2011 ISSN: 0072-5285 ISBN: 978-0-387-87859-1. DOI:101007/978-0-387-87859-1

This books makes a very good job introducing the main ideas of probability without getting lost in too much technical details.

Another excellent textbook for Math 605

• Probability Essentials by Jean Jacod and Philip Protter, 2n edition. Universitext. Springer 2004. ISBN: 978-3-540-43871-7

It is not as comprehensive as other texts and it is closer to a set of classnotes. Very well written and very well suited to learn a new subject.

Other classical and much used textbooks are

1. A first look at rigorous probability by Jeffrey Rosenthal, 2n edition. World Scientific 2006 ISBN-10: 9812703713 ISBN-13: 978-9812703712

2. A Probability Path by Sidney Resnick. Birkhauser 2014

3. Probability: Theory and Examples by Rick Durrett, 4th edition. Cambridge University Press ISBN-10: 0521765390 ISBN-13: 978-0521765398

4. Real Analysis and Probability by R.M Dudley, 2nd edition. Cambridge University Press 2004

5. Probability by A. Shiryaev, 2nd edition. Springer 1995