ST435      Half Unit
Advanced Probability Theory

This information is for the 2016/17 session.

Teacher responsible

Dr Luciano Campi COL.7.10


This course is available on the MSc in Risk and Stochastics, MSc in Statistics (Financial Statistics) and MSc in Statistics (Financial Statistics) (Research). This course is available with permission as an outside option to students on other programmes where regulations permit.

The course is offered as a regular examinable half-unit as well as a service to students and academic staff.


Analysis and algebra at the level of a BSc in pure or applied mathematics and basic statistics and probability theory with stochastic processes. Knowledge of measure theory is not required as the course gives a self-contained introduction to this branch of analysis.

Course content

The course covers core topics in measure theoretic probability and modern stochastic calculus, thus laying a rigorous foundation for studies in statistics, actuarial science, financial mathematics, economics, and other areas where uncertainty is essential and needs to be described with advanced probability models. Emphasis is on probability theory as such rather than on special models occurring in its applications. Brief review of basic probability concepts in a measure theoretic setting: probability spaces, random variables, expected value, conditional probability and expectation, independence, Borel-Cantelli lemmas Construction of probability spaces with emphasis on stochastic processes. Operator methods in probability: generating functions, moment generating functions, Laplace transforms, and characteristic functions. Notions of convergence: convergence in probability and weak laws of large numbers, convergence almost surely and strong laws of large numbers, convergence of probability measures and central limit theorems. If time permits and depending on the interest of the students topics from stochastic calculus might be covered as well.


20 hours of lectures and 10 hours of seminars in the MT.

Week 6 will be used as a reading/revision week.

Formative coursework

Exercises are set weekly and solutions are discussed in the lectures. There will be one set of compulsory written coursework in the MT which will be marked.

Indicative reading

Williams, D. (1991): Probability with Martingales. Cambridge University Press;

Kallenberg, O. (2002). Foundations of modern probability. Springer;

Billingsley, P. (2008). Probability and measure. John Wiley & Sons;

Jacod, J., & Protter, P. E. (2003). Probability essentials. Springer;

Dudley, R. M. (2002). Real analysis and probability (Vol. 74). Cambridge University Press


Exam (100%, duration: 2 hours) in the LT week 0.

Student performance results

(2012/13 - 2014/15 combined)

Classification % of students
Distinction 18.8
Merit 18.8
Pass 50
Fail 12.5

Key facts

Department: Statistics

Total students 2015/16: 1

Average class size 2015/16: 2

Controlled access 2015/16: No

Value: Half Unit

Guidelines for interpreting course guide information

Personal development skills

  • Specialist skills

Course survey results

(2012/13 - 2014/15 combined)

1 = "best" score, 5 = "worst" score

The scores below are average responses.

Response rate: %



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