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ST318 Probability Theory

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.

Content:

  • 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.