Lecturer(s): Dr Paul Chleboun
Prerequisite(s): ST342 Mathematics of Random Events or MA359 Measure Theory.
Commitment: 3 lectures/week, 1 tutorial/fortnight. This module runs in Term 2.
- Independence and zero-one laws
- Modes of convergence for sequences of random variables
- Limit theorems: Law of Large Numbers (LLN) and Central Limit Theorems (CLT)
- Conditioning and discrete-time martingales
Aims: This course aims to give the student a rigorous presentation of some fundamental results in measure theoretic probability and an introduction to the theory of discrete time martingales. In so doing it aims to provide a firm basis for advanced work on probability and its applications.
Objectives: The objectives of the course are as follows: at the end of the course the student will:
- Understand the ideas relating to independence and zero-one laws and be able to apply these ideas in simple contexts.
- Understand the different modes of convergence for sequences of random variables and the relationship between these different modes.
- Be able to state and prove the Central Limit Theorem via the method of characteristic functions and understand how this result can be applied.
- Understand some basic results on discrete-time martingales including the martingale convergence theorem and optional stopping theorem, and show how these results can be used to obtain various characteristics of simple random walks.
Assessment: 100% by 2-hour examination.
Examination Period: Summer
Course Material: Lecture notes, example sheets, and other module material are to be found at the Module Resources page.