**ICTS Course no.: **MTH 124.4

**Venue:** Online

**Class timings:** Tue/Th 11:00-12:30

**First meeting:** Jan 11, 2022

**Course description:**

- Review of discrete Probability
- Review of Measure theoretic facts (distribution of a random variable, expectation, product measures, Fubini’s theorem)
- Laws of Large numbers: independence, sums of an independent random variable, weak law of large numbers, Borel-Cantelli theorems, strong law of large numbers, random series
- Central limit theorems: weak convergence of probability measures, characteristic functions, central limit theorem, infinitely divisible distributions
- Conditional expectation, Martingales (uniform integrability, Doob's up crossing lemma, martingale convergence and related topics)
- Introduction to Brownian motion
- Additional topics (if time permits): Random walks, Markov chains, ergodic theory.

Course Evaluation: The final grade will be based on the following three components:

- Final exam: The final exam will count towards 40% of the final grade.
- Homework: There will be four homework assignments throughout the semester and they will count towards 40% of the final grade.
- Class performance: The students will be required to make short (~30-45 min) in-class presentations approximately once every two weeks on pre-specified topics. These will count towards 20% of the final grade.

- Teacher: Anirban Basak
- Teacher: Riddhipratim Basu