Probability Theory: Math 605

Meeting : Tu-Th 1:00--2:15    LGRT 177

Instructor : Luc Rey-Bellet

 Office :  1423 K LGRT
 Phone :  545-6020
 E-Mail :  luc@math.umass.edu
 Office Hours :   Tu 3:00--4:00, Th 9:30--11:00, or by appointment

• 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. m

• Probability Essentials by Jean Jacod and Philip Protter, 2n edition. Universitext. Springer 2004. ISBN: 978-3-540-43871-7
  
  
• 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 weekly. 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 13):   Set #1      

Homework #2 (due on Friday September 20):   Set #2      

Homework #3 (due on Friday September 27):   Set #3      

Homework #4 (due on Monday October 7):   Set #4      

Homework #5 (due on Monday October 14):   Set #5      

Homework #6 (due on Friday October 25):   Set #6      

Homework #7 (due on Friday November 8):   Set #7      

Homework #8 (due on Friday November 15):   Set #8      

Homework #9 (due on Monday December 2):   Set #9      

Homework #10 (due on Friday December 9):   Set #10