Course Description:

Continuation of MA/STAT 538 (which provides a mathematically rigorous and measure-theoretic introduction to probability spaces and measures). This second course touches upon properties of stochastic processes, in particular their limit theorems and long time behaviors. Topics include conditional probability, martingale theory, random walk, Brownian Motions and invariance principle (Chapters 6 and 7 of Billinsgley's book). Depending on time and students' interests, additional topics such as Gaussian, stationary, and Levy Processes might be covered.

Instructor:

Aaron Nung Kwan Yip

Department of Mathematics

Purdue University

Contact Information:

Office: MATH 432

Office phone: (765) 494-1941

Fax: (765) 496-1551

Email: click here

Mailing address:

Department of Mathematics

Purdue University

150 N. University Street

West Lafayette, IN 47907-1395

Lecture Time and Place:

MWF 12:30pm-1:20pm, REC 226

Office Hours:

T, Th: 12:00pm-1:30pm or by appointment

Textbook:

Probability and Measure, by Patrick Billingsley, third edition

Reference:
Theory of Probability and Random Processes, by L. B. Koralov, Y. G. Sinai (also available online from library page using Purdue web-address)
Probability: theory and examples, by Richard Durrett.
Probability, by Leo Breiman
An introduction to probability theory and its applications, v.1 and 2, by William Feller

Prerequisites:

The prerequisite is MA/STAT 538. But anyone with good backgrounds in probability and measure theory (at the level of MA/STAT 519 and MA 544) can also benefit. The main knowledge I will assume is a good understanding of the Weak and Strong Law of Large Numbers, and the Central Limit Theorem (as covered in Chapters 4, 5 of Billingsley's book).

You can also refer to the MA/STAT 538 web-page for the Spring, 09 for some general background and information.

Homework:

Homeworks will be assigned (probably) bi-weekly (and usually due on Wednesday, 3pm, unless otherwise specified). They will be gradually assigned as the course progresses. Please refer to the course announcement below.

• Steps must be shown to explain your answers. No credit will be given for just writing down the answers, even if it is correct.
  
  
• Please staple all loose sheets of your homework to prevent 5% penalty.
  
  
• Please resolve any error in the grading (hws and tests) within ONE WEEK after the return of each homework and exam.
  
  
• No late homeworks are accepted (in principle).
  
  
• You are encouraged to discuss the homework problems with your classmates but all your handed-in homeworks must be your own work.

Examinations:

Midterm: TBA (evening exam, late Oct?)
Final examination: presentation of some research paper.

Grading Policy:

Homeworks (50%)

Midterm (25%)

Final Presentation (25%)

You are expected to observe academic honesty to the highest standard. Any form of cheating will automatically lead to an F grade, plus any other disciplinary action, deemed appropriate.

Course Outline:

Core Topics:

Conditional Probability and Expectation ([BI Chapters 6], [BR Chapter 4])

Martingales ([BI Chapter 6], [BR Chapter 5])

Stochastic Processes ([BI Chapter 7.36])

Poisson Process ([BI Chapter 4.23])

Random Walk and Brownian Motion ([BI Chapter 7.37], [BR Chapter 12])

Invariance Principle ([BR Chapter 13])

Additional Topics:

Stationary Processes and Ergodic Theory

Gaussian Processes

Levy Processes

Course Progress and Announcement:

(You should consult this section regularly, for homework assignments, additional materials and announcements.)

A recent survey paper on large deviation (Varadhan, 2008)

Homework One. Due: 3pm, Fri, Sept. 11

Homework Two. Due: 3pm, Fri, Oct. 2

Homework Three. Due: 3pm, Fri, Oct. 30

Excerpt from Probability Theory, on Markov Chain, Varadhan

Introductory Chapter from Martingale Limit Theory and Its Application (Hall and Heyde, 1980) (book on reserve in math library)
Review Paper on Central Limit Theorems for Martingales (Helland, 1982)
Martingale Central Limit Theorem (Brown, 1971)

Test: Thu Nov, 5, 8-10pm, REC 226
Open notes but closed books and No calculator. You can bring in your homeworks and solutions and any notes produced by you and me. Practice Exam I Practice Exam II
Hw 1 Solution Hw 2 Solution Hw 3 Solution

Information about Final Exam Presentation