Advanced Probability Theory - 2019 entry

The central objects of probability theory are random variables, stochastic processes and events. This course discusses how sums of independent random variables converge in distribution. The results covered are fundamental to statistical estimation and probability theory. We will study, in depth and using rigorous mathematical analysis, the weak and strong laws of large numbers, conditional expectation, central limit phenomena via characteristic functions, stable laws (followed by a succinct introduction to infinitely divisible laws), stationary processes and the ergodic theorem. We will also study renewal Markov chains with a focus on some celebrated renewal theorems. To achieve these goals, we will need to introduce various definitions and results from measure theory and theory of integration.

Prerequisites are: Required: Analysis, ECM2701, Preferred: Stochastic Processes, ECM3724, Preferred: Topology and Metric spaces, ECM3740

The aim of this course is to introduce you to advanced topics in rigorous analysis applied to modern probability theory, to provide you with a deep understanding of  the mathematical concepts in modern probability theory and to provide training in the techniques and manipulations most commonly used for proving theorems in this area.

On successful completion of this module you should be able to:

Module Specific Skills and Knowledge

Discipline Specific Skills and Knowledge

Personal and Key Transferable / Employment Skills and Knowledge

Measure theory. Theory of integration (Sigma algebras and measures; Probability spaces;  Integration and properties of the integral; Expected value and integration to the limit; Product measures)

Independence and laws of large numbers (Basic definitions and results, Weak law of large numbers and L^2 weak laws with proof, Strong law of large numbers with proof, Large deviation results)

Convergence in distribution and required theoretical tools (Distribution functions, Characteristic functions, Levy continuity theorem)

Central limit phenomena (Statement and Proof of the Central Limit theorem and related results)

Stable laws (Introduction to the domain of attraction of a stable law, Statement and proof for stable laws, Rough introduction to infinitely divisible laws)

Stationary processes and the ergodic theorem (Basic definitions, measure preserving transformations, invariant sets and ergodicity, Invariant random variables, the ergodic theorem)

If time permits we will study Renewal Markov chains with a focus on the celebrated renewal theorems.

If a module is normally assessed by examination or examination plus coursework, referred and deferred assessment will normally be by examination. For referrals, only the examination will count, a mark of 50% being awarded if the examination is passed. For deferrals, candidates will be awarded the higher of the deferred examination mark or the deferred examination mark combined with the original coursework mark.

Basic reading:

ELE: http://vle.exeter.ac.uk/

Web based and Electronic Resources:

Other Resources:

Reading list for this module: