3105 (Probability)

							                               3105 (Probability)
Year:                    2009–2010
Code:                    MATH3105
Old Code:                MATHC393
Value:                   Half unit (= 7.5 ECTS credits)
Term:                    2
Structure:               3 hour lectures per week
Assessment:              100% examination
Normal Pre-requisites:   3101
Lecturer:                Prof KM Ball

Course Description and Objectives

This course follows on from earlier courses in real analysis and measure theory, and describes
what is perhaps the most important application of measure theory in mathematics; the rigorous
theory of probability. The course material is focused on the two most fundamental principles
of the theory: the strong law of large numbers and the central limit theorem.

Recommended Texts

A recommended text is D Williams, Probability with Martingales.

Detailed Syllabus

Basic definitions. Events, random variables. Probability laws, distribution functions, densities.

Independence. The Borel-Cantelli lemmas. IID random variables. Tail σ-algebras and the
Kolmogorov 0-1 law.

Expectation. Markov’s inequality. Covariance of independent random variables in L1. Weak
Law of large numbers for L2 random variables. Strong law for e.g. bounded RVs.

Conditional expectations. Basic properties. Filtrations and Martingales. Examples and inter-
pretations. Stopping times. The optional stopping theorem with applications.

The Martingale convergence theorem. Sums of independent square integrable variables. The
strong law.

Characteristic functions: the Parseval formula. Convergence in distribution and its relation to
characteristic functions. The central limit theorem.


                                                                                    May 2008 3105