SI 537: Probability II

List of books for reading: 

1. William Feller                                                                Introduction to Probability and its applications Volume 2 **** (read it on your own risk)

2. Kai Lai Chung                                                                 A course in Probability Theory

3. D. W. Stroock                                                                 Mathematics of Probability 

4. Athreya and Lahiri                                                      Measure theory and Probability theory

5. Sidney Resnick                                                             A probability path***  (Strongly recommended for reading as text)

Some prerequisite materials can be found in the following books: 

1. Rohatgi and Saleh                                                  An introduction to probability and statistics

2. Hoel, Port and Stone                                           An introduction to probability theory

3. Grimmett and Strizeker                                    One Thousand Exercises in Probability 

4. Howie                                                                           Real analysis

5. Apostol                                                                       Mathematical analysis

6. Goldberg                                                                   Methods in real analysis

7. Ghorpure and Limaye                                       Two volumes on analysis and calculus

A handwritten notes on Real analysis can be found here. I shall use these results heavily. So, please be familiar with them and you may contact me in case you have any question. Please note that this note may contain some typos. So, please consult the books if you have any confusion. 

Advanced books: 

1. Durrett                                    Probability: Theory and Examples

2. Ash                                             Measure theory

3. Ash                                             Real analysis and probability

4. Dudley                                     Real analysis and Probability

5. Billingsley                             Probability and measure

6. Loeve                                       Probability theory

7. Laha and Rohatgi              Probability Theory

8. Stroock                                  Probability: An analytic view

9. Sinai                                        Probability: An introductory course 

10. Kallenberg                          Foundations of modern probability 

11. Venkatesh                           The Theory of Probability : Explorations and Applications (Most relevant book covering the most recent topics with some details)

12. B. Fristedt and L. Gray         A Modern Approach to Probability Theory 

I strongly suggest to pick up your favourite book or books and read the relevant sections for better understanding. As this is an elective, I will highly appreciate any discussion on advanced topics.  I may ask to present proofs of some Theorems from one of these books.

Course Policy: 

Midsem:  30

Endsem: 40

Quiz: 10 (two quizzes. One before midsem and one after the midsem.)

Class performance: 10

Advanced exercises: 10 (This is essential to be considered for the highest grade.)

Problem set related to Probability I  (Please solve all the problems)

I will post the handwritten notes here.

Lecture I    Overview of the course.

Lecture II   Construction of sigma-field                    Quick useful facts and theorems

Lecture III Construction of Probability

Lecture IV A measure-theoretic introduction to random variables, random vectors and random sequences

Lecture V      Expectation of a random variables and its analytic properties.                      Problem Set

Lecture VI     Transformation of random variables/vectors and expectation

Lecture VII   Probability on product spaces (finite, countable and uncoubtable*)

Lecture VIII Convergence concepts and fundamental limit theorems

(Consolidated handwritten lecture notes is available here)

Problem set

Additional Help: 

Video lecture: Functions of random vectors          Notes for the video lecture     handwritten notes

Video lecture: Functions of random vectors part II            Notes for the video lecture

Video lecture: Problem-solving session on functions of several random variables: part I  Notes used in the lecture

Video lecture: Problem-solving session on functions of several random variables: part II  Notes used in the lecture

Lecture IX    Exchangeable random variables

Lecture X      Limit theorems once again (The convergence notions in Probability                     Basic theorems related to convergence concepts with proofs)