Manjunath Krishnapur

Department of Mathematics

Indian Institute of Science

Probability theory (Aug-Dec 2022)

Classes: Tue, Thu, 2:00-3:30, LH-5, Mathematics department

Teaching assistant: Biltu Dan
Tutorial timings:To be announced
MS teams link: Click and request if you wish to attend the class

Description: This is a first course in measure theoretical probability theory. It is optional for everyone, but undergraduates, int-phd and phd students (including PhD students from CSA, ECE, etc.) are all welcome to credit or audit. The prerequisites are basic real analysis, at least a first course in discrete probability. Measure theory is required for the course, but will be not covered in class and will be de-emphasized in general. It is possible to learn it concurrently or make do with some familiarity if you have good intuition for probability.

Grading: Final exam (50%), Two mid-terms (15% each), Homeworks (20%)

Texts and other resources: My lecture notes (Part-1 and Part-2) are from previous versions of this course and will continue to be edited as and when required. The problem set will also be updated occasionally. Previous versions of the course, including homeworks, exams etc. can be accessed from the course pages under "Teaching" on my home page. There are many good books on the subject, some of which I mention below.
  1. Rick Durrett Probability: theory and examples. Click for a copy!
  2. John Walsh Knowing the odds, AMS (2012) (available in Indian edition)
  3. Patrick Billingsley Probability and measure, 3rd ed. Wiley India. Available in Tata book house
  4. Richard Dudley Real analysis and probability, Cambridge university press
  5. David Williams Probability with martingales, Cambridge University Press (available in Indian edition)
  6. J V Uspensky Introduction to mathematical probability, McGraw-Hill (1937)
  7. William Feller An Introduction to Probability Theory and its Applications, Vol 2, 2ed, Wiley (Indian ed.)(1971)
  8. Amir Dembo Lecture notes from a Stanford course
  9. Olav Kallenberg Foundations of Modern Probability 3rd ed. (in two volumes), Springer (2021)
Some comments: Durrett's book has both theory and interesting examples, but it is a graduate level text and many have found it difficult or at least terse. Billingsley's book is well-written, very detailed and has elaborate explanations. Dudley's book is concise and has both measure theory and probability theory and the relevant functional analysis but perhaps lacks interesting examples of interest in modern probability theory. David Williams book is exceedingly well-written (and easy compared to the others, except perhaps Walsh's book). The last two books are much older. Uspensky's book contains certain things that are not usually found in modern books. So does Feller's book, but it is not well-suited for the beginning of the course. For specialized topics like characteristics functions, there is no better resource that I know of. Kallenberg's book has no parallel. It covers a great deal of probability (about five times what we will) but considered better for a second reading.
Notes and homeworks/links: Lecture notes (Part-1 and Part-2) and problem set.

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