## AISSP 2023

**Brief description about the school:** Probability theory is now considered an integral part of mathematics. The mathematics programmes of most Indian Universities, however, do not include probability theory in much detail. The aim of this school is to give a comprehensive training in probability and stochastic processes to students of the undergraduate and postgraduate programmes. Besides, giving them a chance to interact with researchers in these topics is also a goal of this instructional school.

Three courses -- 1) Measure theoretic probability, 2) Conditional probability and martingales , 3) Brownian motion will be taught here.

**Who can Apply?**

Any Master degree students and PhD students can apply.

**To Apply:**

Please go to NCM, Mumbai website and complete the application process by registering there.

**Last date for receiving online applications:** 14th May 2023 (Link for application will be provided here shortly.)

**Instructors: **

- Anish Sarkar, ISI, Delhi
- Soumendu Sudar Mukherjee, ISI, Kolkata
- Arijit Chakrabarty, ISI, Kolkata
- Manjunath Krishnapur, IISc, Bengaluru
- Suprio Bhar, IIT, Kanpur
- Arup Bose, ISI, Kolkata

**Detailed Syllabus:**

*Measure theoretic Probability:* Caratheodory extension theorem , Monotone class theorem, Dynkin’s pi-lambda theorem, MCT, Fatou’s Lemma, DCT, Fubini’s theorem. Probability spaces, random variables and random vectors, expected value and its properties. Independence. Various modes of convergence and their relation. The Borel-Cantelli lemmas. Weak Law of large numbers for i.i.d. finite mean case. Kolmogorov 0-1 law, Kolmogorov’s maximal inequality. Statement of Kolmogorov’s three-Series theorem (proof if time permits). Strong law of large numbers for i.i.d. case. Characteristic functions and its basic properties, inversion formula, Levy’s continuity theorem. Lindeberg CLT, CLT for i.i.d. finite variance case, Lyapunov CLT.

*Conditional probability and martingales:* Absolute continuity and singularity of measures. Hahn-Jordon decomposition, Radon-Nikodym Theorem, Lebesgue decomposition. Conditional expectation – Definition and Properties. Regular conditional probability, proper RCP. Regular conditional distribution. Discrete parameter martingales, sub-and super-martingales. Doob’s Maximal Inequality, Upcrossing inequality, martingale convergence theorem, Lp inequality, uniformly integrable martingales, reverse martingales, Levy’s upward and downward theorems. Stopping times, Doob’s optional sampling theorem. Discrete martingale transform, Doob’s Decomposition Theorem. Applications of martingale theory: SLLN for i.i.d. random variables.

*Brownian Motion:* Introduction to Brownian Motion, Kolmogorov Consistency theorem, Kolmogorov Continuity theorem, Construction of BM. Basic Martingale Properties and path properties – including Holder continuity and non-differentiability. Quadratic variation. Markov Property and strong Markov property of BM, reflection principle, Blumenthal’s 0-1 law. Distributions of first passage time and of running maximum of BM.