Probability Theory: Math 605

Meeting : Tu-Th 1:00--2:15    LGRT 1322

Instructor : Luc Rey-Bellet

 Office :  1423 K LGRT
 Phone :  545-6020
 E-Mail :  luc@math.umass.edu
 Office Hours :   Tu 3:00--4:00, F 11--12:30, or by appointment

• Probability Essentials by Jean Jacod and Philipp Protter, 2nd edition. Springer 2004 ISBN-10: 3540438718 ISBN-13: 978-3540438717
  
  This book has been chosen since it a good guide for the reader to go straight into the heart of probability without getting lost in technicalities. It is also adequate for people with various backgrounds. The next two books on the lists by Rosenthal and Resnick have the same spirit and are highly recommended.

• A first look at rigorous probability by Jeffrey Rosenthal, 2n edition. World Scientific 2006 ISBN-10: 9812703713 ISBN-13: 978-9812703712
  
  
• A Probability Path by Sidney Resnick. Birkhauser 2014
  
  
• Probability: Theory and Examples by Rick Durrett, 4th edition. Cambridge University Press ISBN-10: 0521765390 ISBN-13: 978-0521765398
  
  
• Real Analysis and Probability by R.M Dudley, 2nd edition. Cambridge University Press 2004
  
  
• Probability by A. Shiryaev, 2nd edition. Springer 1995
  
  

Syllabus: This is the first part of a 2-semester introduction to probability (Semester 1) and stochastic process (Semester 2). The prerequisites for this class is a good working knowledge of undergraduate Probability (without measure theory), such as Stat 515 or STAT 607. Also required are some mathematical maturity and familiarity with mathematical reasoning and of curse a fearless adventurous spirit! During the first semester 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 variable and the law of large numbers.

• 3) Gaussian random variables, characteristic and moment generation functions, the central limit theorem and concentration inequalities.

• 4) Conditional expectation, the radon-nikodym theorem and martingales.

Grade: There will be a midterm exam and a final exam. Homework will be assigned regularly. Each of the midterm, final and the combined homework will be worth 1/3 of your grade.

Exams:

• Midterm: TBA         Room: TBA        
  
  
• Final: TBA                
  

Homework :

Homework #1 (due on Friday September 15):   Set #1      

Homework #2 (due on Friday September 22):   Set #2      

Homework #3 (due on Friday September 29):   Set #3      

Homework #4 (due on Friday October 6):   Set #4      

Homework #5 (due on Friday October 20 ):   Set #5      

Homework #6 (due on Friday October 27):   Set #6      

Take-Home midterm Exam (due on Friday November 3):   Mid-term exam      

Homework #7 (due on Friday November 10):   Set #7      

Homework #8 (due on Friday November 17):   Set #8      

Homework #9 (due on Friday December 1):   Set #9      

Homework #10 (due on Friday December 8):   Set #10