## Probability theory - MA 361 (Spring 2010)

### Manjunath Krishnapur

Tuesday, Thursday 11:00 - 12:30 (PM), in LH III

This course is an introduction to the foundations of the mathematical theory of Probability. The intention is to learn the basic results as well as useful techniques of Probability theory. Probability is a branch of Analysis on the one hand, and has innumerable applications to real-life problems on the other. Hence, this course should be useful to students planning to work in any branch of Analysis as well as to engineers whose work has theoretical aspects to it.

### Prerequisites

It helps to have some acquaintance with discrete probability spaces and with measure theory. The material covered in the courses Probability Models and Analysis II are easily adequate for these two topics, respectively. You may also take Analysis II in parallel or read measure theory by yourself.

1. Weekly assignments (the two worst are not counted): 20%
2. Two class tests or best two of three tests: 30%
3. Final exam: 50%

### Topics expected to be covered

1. Measure theory Recap/Introduction to basic measure theory. Some proofs will be skipped (and can be picked up in Analysis II).
2. Independent random variables Weak and Strong laws of large numbers. Random series. Large deviations. Central limit theorems.
3. Exchangeable random variables Definetti's theorem. Polya's urn scheme.
4. Stationary and Ergodic sequences of random variables (tentative)
5. Martingales (discrete time) Conditional expectation, Optional sampling theorem, Martingale convergence theorem, Doob's inequalities, Applications.

### References

1. Rick Durrett Probability: Theory and examples Download the online version of the fourth edition before it disappears! This book has an excellent collection of material. Assumes measure theory and focuses on Probability.
2. Kallenberg Foundations of modern Probability The most succinct and streamlined presentation of all of basic Probability. Sometimes more suitable for a second reading to clear up muddled understanding of a first reading!
3. Billingsley Probability and Measure A well-written, slow-paced book with detailed explanations. Also has a detailed account of measue theory.
4. Kallenberg Foundations of modern Probability The most succinct and streamlined presentation of all of basic Probability. Sometimes more suitable for a second reading to clear up muddled understanding of a first reading!
5. Breiman Probability
6. Athreya and Lahiri Measure Theory and Probability Theory