# Mathematics

## ECMM742 - Advanced Probability Theory (2018)

MODULE TITLE CREDIT VALUE Advanced Probability Theory 15 ECMM742 Dr Dalia Terhesiu (Coordinator)
DURATION: TERM 1 2 3
DURATION: WEEKS 0 11 0
 Number of Students Taking Module (anticipated) 9
DESCRIPTION - summary of the module content

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

AIMS - intentions of the module

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.

INTENDED LEARNING OUTCOMES (ILOs) (see assessment section below for how ILOs will be assessed)

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

Module Specific Skills and Knowledge

1. recall and apply key definitions and theoretical results in real analysis, measure theory and the theory of integration
2. discuss the main concepts in probability theory and mathematical techniques required for proving theorems in this area

Discipline Specific Skills and Knowledge

3. extract abstract mathematical formulations from a diverse range of problems
4. apply abstract reasoning and rigorous analysis is to solve a large range of problems

Personal and Key Transferable / Employment Skills and Knowledge

5. show ability to think analytically and to use rigorous argument to formulate solutions as mathematical proofs
6. communicate results in a clear, correct and coherent manner

SYLLABUS PLAN - summary of the structure and academic content of the module

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.

LEARNING AND TEACHING
LEARNING ACTIVITIES AND TEACHING METHODS (given in hours of study time)
 Scheduled Learning & Teaching Activities Guided Independent Study Placement / Study Abroad 33 117 0
DETAILS OF LEARNING ACTIVITIES AND TEACHING METHODS
 Category Hours of study time Description Scheduled learning and teaching activities 33 Lectures including example classes Guided independent study 117 Lecture and assessment preparation

ASSESSMENT
FORMATIVE ASSESSMENT - for feedback and development purposes; does not count towards module grade
Form of Assessment Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Problem sheets problem sheets at regular intervals   In person, office hours

SUMMATIVE ASSESSMENT (% of credit)
 Coursework Written Exams 20 80
DETAILS OF SUMMATIVE ASSESSMENT
Form of Assessment % of Credit Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Written exam (closed books) 80 2 hours - Summer Examination Period All Exam mark, verbal/written feedback by request.
Coursework 20 10 hours All Coursework mark, feedback written on script.

DETAILS OF RE-ASSESSMENT (where required by referral or deferral)
Original Form of Assessment Form of Re-assessment ILOs Re-assessed Time Scale for Re-assessment
All above Written Exam (100%) All August Ref/Def period

RE-ASSESSMENT NOTES

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.

RESOURCES
INDICATIVE LEARNING RESOURCES - The following list is offered as an indication of the type & level of
information that you are expected to consult. Further guidance will be provided by the Module Convener

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

Web based and Electronic Resources:

Other Resources: