Lectures on Probability Theory

TL;DR

This lecture notes provide a comprehensive introduction to probability theory, covering fundamental concepts, convergence, limit theorems, and advanced topics like martingales and Markov chains, suitable for students and researchers.

Contribution

It offers a structured, detailed exposition of probability theory with modern language and covers both foundational and advanced topics in a pedagogical manner.

Findings

Clear explanation of probability concepts and measures

Detailed coverage of convergence and limit theorems

Accessible introduction to martingales and Markov chains

Abstract

These are lecture notes written at the University of Zurich during spring 2014 and spring 2015. The first part of the notes gives an introduction to probability theory. It explains the notion of random events and random variables, probability measures, expectation, distributions, characteristic function, independence of random variables, types of convergence and limit theorems. The first part is separated into two different chapters. The first chapter is about combinatorial aspects of probability theory and the second chapter is the actual introduction to probability theory, which contains the modern probability language. The second part covers conditional expectations, martingales and Markov chains, which are easily accessible after reading the first part. The chapters are exactly covered in this order and go into some more details of the respective topic.