## Admission requirements

We assume prior knowledge of elementary measure theory, in a probabilitistic context.

## Description

The course plans to form the basics of martingale theory and different conceptual results. The second part of the course will focus on applications in different parts of probability theory.

Revision- Absolute continuity and singularity of measures. Hahn-Jordan 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, 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. Infinite products of probability spaces, Hewitt-Savage 0-1 Law. Dirichlet problem. Finite and infinite exchangeable sequence of random variables, de Finetti's Theorem. SLLN for U-Statistics for exchangeable data.

Applications of martingale theory: Branching process: Galton watson trees, extinction probability, the Kesten-Stigum conditions.

Introduction to continuous parameter martingales: definition, examples and basic properties. Examples of Brownian motion. Characterisation of BM as a martingale.

6.Martingale Central Limit Theorem and applications, Azuma-Hoeffding Inequality and some applications to random graphs.

- Dirichlet problem solution using Brownian motion.

## Course Objectives

The objective of the course is to expose students to different applications of martingale theory in various areas.

## Mode of instruction

The course will be offered in Spring semester (2021-2022). The course will be mainly based on weekly (2 hours of lectures) and students will be provided with the lecture notes. There will be homework every two weeks.

## Assessment method

The final marks of the course consists of two parts:

1. 6 homeworks (25%)

2. written exam (75%)

## Literature

- David Williams: Probability with Martingales.
- Y. S. Chow and H. Teicher: Probability Theory
- Leo Breiman: Probability Theory
- Jacques Nevue: Discrete Parameter Martingales
- P. Hall and C. C. Heyde: Martingale Limit Theory and its Application
- R. Durret: Probability Theory and Examples
- P. Billingsley: Probability and Measures

## Contact

Rajat S. Hazra (email: r.s.hazra@math.leidenuniv.nl).