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Foundations of Modern Probability

Unit code: MATH20722
Credit Rating: 10
Unit level: Level 2
Teaching period(s): Semester 2
Offered by School of Mathematics
Available as a free choice unit?: N



Additional Requirements

See above


The course unit unit aims to

  • provide the basic knowledge of facts and methods needed to state and prove the law of large numbers and the central limit theorem;
  • introduce fundamental concepts and tools needed for the rigorous understanding of third and fourth level course units on probability and stochastic processes including their applications (e.g. Financial Mathematics).


The law of large numbers and the central limit theorem are formulated and proved. These two results embody the most important results of classical probability theory having a large number of applications. 

Learning outcomes

On completion of this unit successful students will be able to

  • state and use fundamental inequalities (Markov, Jensen, Holder, Minkowski) and modes of convergence (almost sure, in probability, in distribution, in mean);
  • state and use Fatou's lemma, monotone convergence theorem, and dominated convergence theorem;
  • state and prove the law of large numbers and the central limit theorem in a variety of theoretical and applied settings;
  • apply the methods of proof developed to related problems in classical/modern probability and its applications.


Assessment methods

  • Other - 20%
  • Written exam - 80%

Assessment Further Information

  • Mid-semester coursework: weighting 20%
  • End of semester examination: two hours weighting 80%


1. Probability measures. Probability spaces. Random variables. Random vectors. Distribution functions. Density functions. Laws. The two Borel-Cantelli lemmas. The Kolmogorov 0-1 law. [4 lectures]

2. Inequalities (Markov, Jensen, Holder, Minkowski). Modes of convergence (almost sure, in probability, in distribution, in mean). Convergence relationships. Fatou's lemma. Monotone/dominated convergance theorem. [4 lectures]

3. Expectation of a random variable. Expectation and independence. The Cesaro lemma. The Kronecker lemma. The law of large numbers (weak and strong). [5 lectures]

4. Fourier transforms (characteristic functions). Laplace transforms (moment generating functions). Uniqueness theorems for Fourier and Laplace transforms. Convergence of characteristic functions: the continuity theorem. The central limit theorem. [6 lectures]

5. Brownian motion as the weak limit of a random walk. Donsker's Theorem. [3 lectures]

Recommended reading

  • D Williams, Probability with Martingales, Cambridge University Press, 1991.
  • A N Shiryaev, Probability, Springer-Verlag, 1996.
  • G R Grimmett and D R Stirzaker, Probability and Random Processes, Oxford University Press, 1992.

Feedback methods

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

Study hours

  • Lectures - 22 hours
  • Tutorials - 11 hours
  • Independent study hours - 67 hours

Teaching staff

Goran Peskir - Unit coordinator

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