Introductory description
This module runs in Term 2 and is available for students on a course where it is a listed option and as an Unusual Option to students who have completed one of the prerequisite modules.
Prerequisite(s): ST342 Mathematics of Random Events OR MA359 Measure Theory.
Leads to: ST401 Stochastic Methods in Finance, ST402 Risk Theory, ST403 Brownian Motion, ST411 Dynamic Stochastic Control
Module web page
Module 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.
Outline syllabus
This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.
 Independence and zeroone 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
Learning outcomes
By the end of the module, students should be able to:
 understand the different modes of convergence for sequences of random and the relationship between these different modes;
 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.
 understand the ideas relating to independence and zeroone laws and be able to apply these ideas in simple contexts;
Indicative reading list
View reading list on Talis Aspire
Subject specific skills
TBC
Transferable skills
TBC
Study time
Type 
Required 
Optional 
Lectures 
30 sessions of 1 hour (20%)

2 sessions of 1 hour

Tutorials 
5 sessions of 1 hour (3%)


Private study 
115 hours (77%)


Total 
150 hours 

Private study description
Weekly revision of lecture notes and materials, wider reading, practice exercises and preparing for examination.
Costs
No further costs have been identified for this module.
You must pass all assessment components to pass the module.
Students can register for this module without taking any assessment.
Assessment group B3

Weighting 
Study time 
Inperson Examination

100%


The examination paper will contain four questions, of which the best marks of THREE questions will be used to calculate your grade.
 Answerbook Pink (12 page)
 Students may use a calculator

Assessment group R2

Weighting 
Study time 
Inperson Examination  Resit

100%


The examination paper will contain four questions, of which the best marks of THREE questions will be used to calculate your grade.
 Answerbook Pink (12 page)
 Students may use a calculator

Feedback on assessment
Solutions and cohort level feedback will be provided for the examination.
Past exam papers for ST318
Courses
This module is Core for:

USTAG300 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics

Year 3 of
G30A Master of Maths, Op.Res, Stats & Economics (Actuarial and Financial Mathematics Stream)

Year 3 of
G30D Master of Maths, Op.Res, Stats & Economics (Statistics with Mathematics Stream)

USTAG301 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (with Intercalated

Year 3 of
G30E Master of Maths, Op.Res, Stats & Economics (Actuarial and Financial Mathematics Stream) Int

Year 3 of
G30H Master of Maths, Op.Res, Stats & Economics (Statistics with Mathematics Stream)

Year 4 of
G30E Master of Maths, Op.Res, Stats & Economics (Actuarial and Financial Mathematics Stream) Int

Year 4 of
G30H Master of Maths, Op.Res, Stats & Economics (Statistics with Mathematics Stream)
This module is Optional for:

UCSAG4G1 Undergraduate Discrete Mathematics

Year 3 of
G4G1 Discrete Mathematics

Year 3 of
G4G1 Discrete Mathematics

Year 3 of
UCSAG4G3 Undergraduate Discrete Mathematics

Year 4 of
UCSAG4G4 Undergraduate Discrete Mathematics (with Intercalated Year)

Year 4 of
UCSAG4G2 Undergraduate Discrete Mathematics with Intercalated Year

USTAG300 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics

Year 3 of
G300 Mathematics, Operational Research, Statistics and Economics

Year 4 of
G300 Mathematics, Operational Research, Statistics and Economics
This module is Core option list A for:

Year 3 of
USTAG1G3 Undergraduate Mathematics and Statistics (BSc MMathStat)

Year 4 of
USTAG1G4 Undergraduate Mathematics and Statistics (BSc MMathStat) (with Intercalated Year)
This module is Option list A for:

UMAAG105 Undergraduate Master of Mathematics (with Intercalated Year)

Year 3 of
G105 Mathematics (MMath) with Intercalated Year

Year 4 of
G105 Mathematics (MMath) with Intercalated Year

Year 5 of
G105 Mathematics (MMath) with Intercalated Year

UMAAG100 Undergraduate Mathematics (BSc)

Year 3 of
G100 Mathematics

Year 3 of
G100 Mathematics

Year 3 of
G100 Mathematics

UMAAG103 Undergraduate Mathematics (MMath)

Year 3 of
G100 Mathematics

Year 3 of
G103 Mathematics (MMath)

Year 3 of
G103 Mathematics (MMath)

Year 4 of
G103 Mathematics (MMath)

Year 4 of
G103 Mathematics (MMath)

UMAAG106 Undergraduate Mathematics (MMath) with Study in Europe

Year 3 of
G106 Mathematics (MMath) with Study in Europe

Year 4 of
G106 Mathematics (MMath) with Study in Europe

Year 4 of
UPXAGF14 Undergraduate Mathematics and Physics (with Intercalated Year)

USTAG1G3 Undergraduate Mathematics and Statistics (BSc MMathStat)

Year 3 of
G1G3 Mathematics and Statistics (BSc MMathStat)

Year 4 of
G1G3 Mathematics and Statistics (BSc MMathStat)

USTAG1G4 Undergraduate Mathematics and Statistics (BSc MMathStat) (with Intercalated Year)

Year 4 of
G1G4 Mathematics and Statistics (BSc MMathStat) (with Intercalated Year)

Year 5 of
G1G4 Mathematics and Statistics (BSc MMathStat) (with Intercalated Year)

USTAGG14 Undergraduate Mathematics and Statistics (BSc)

Year 3 of
GG14 Mathematics and Statistics

Year 3 of
GG14 Mathematics and Statistics

Year 4 of
USTAGG17 Undergraduate Mathematics and Statistics (with Intercalated Year)

Year 4 of
UMAAG101 Undergraduate Mathematics with Intercalated Year

USTAY602 Undergraduate Mathematics,Operational Research,Statistics and Economics

Year 3 of
Y602 Mathematics,Operational Research,Stats,Economics

Year 3 of
Y602 Mathematics,Operational Research,Stats,Economics

Year 4 of
USTAY603 Undergraduate Mathematics,Operational Research,Statistics,Economics (with Intercalated Year)
This module is Option list B for:

Year 1 of
TMAAG1PE Master of Advanced Study in Mathematical Sciences
This module is Option list C for:

USTAG302 Undergraduate Data Science

Year 3 of
G302 Data Science

Year 3 of
G302 Data Science

Year 3 of
USTAG304 Undergraduate Data Science (MSci)

Year 4 of
USTAG303 Undergraduate Data Science (with Intercalated Year)

Year 3 of
USTAG300 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics

USTAG301 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (with Intercalated

Year 3 of
G30F Master of Maths, Op.Res, Stats & Economics (Econometrics and Mathematical Economics Stream) Int

Year 4 of
G30F Master of Maths, Op.Res, Stats & Economics (Econometrics and Mathematical Economics Stream) Int
This module is Option list D for:

USTAG300 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics

Year 4 of
G30C Master of Maths, Op.Res, Stats & Economics (Operational Research and Statistics Stream)

Year 4 of
G30C Master of Maths, Op.Res, Stats & Economics (Operational Research and Statistics Stream)

Year 5 of
USTAG301 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (with Intercalated