Unit information: Probability 34 in 2013/14

Unit name	Probability 34
Unit code	MATHM0700
Credit points	10
Level of study	M/7
Teaching block(s)	Teaching Block 1B (weeks 7 - 12)
Unit director	Dr. Leslie
Open unit status	Not open
Pre-requisites	None
Co-requisites	None
School/department	School of Mathematics
Faculty	Faculty of Science

Description including Unit Aims

This course deals with various modes of convergence of random variables (almost surely, weak, in probability, and in Lp) and the connections between them. We also discuss and prove weak laws and strong laws of large numbers, prove the Borel-Cantelli lemmas, the Kolmogorov 0-1 law, and the three series theorem. We study the properties and applications of characteristic functions. Central Limit Theorems, Lindeberg conditions, Local limit theorems, Barry-Essen inequality will proved / discussed.

Aims

To outline, discuss, and prove some of the key results in probability theory and their applications to statistics.

Syllabus

1. Properties of conditional expectations.
2. Modes of convergence of random variables.
3. Borel-Cantelli lemmas, 0-1 laws.
4. Weak and Strong law of large numbers.
5. Characteristic functions.
6. Central Limit Theorems and related topics.

Relation to other units

This unit develops the rigorous theoretical background to much of probabilistic (and partly statistical) methodology covered in probability/statistics units at levels 1, 2, 3, and M.

Intended Learning Outcomes

To gain a (better) understanding of:

• Conditional expectations;
• Types of convergence of random variables;
• Strong and Weak laws and Central Limit Theorems;
• The ways to establish the above results rigorously.

Transferable Skills:

Self-assessment by working examples sheets and using solutions provided.

Teaching Information

Lectures, assignments, and exercises to be done by students.

Assessment Information

20% of the assessment mark for Probability 34 is based on the assignment done during the course.

80% of the assessment mark is calculated from a 1½-hour written examination in April consisting of THREE questions. A candidate's best TWO answers will be used for assessment. Calculators are NOT permitted for this examination.

Reading and References

Each of the following texts will be useful:

• R. Durrett, Probability: Theory and Examples, 2nd edition, Duxbury Press.

• S.R.S. Varadhan, Probability Theory, AMS

• G.R. Grimmett and D.R. Stirzaker, Probability and Random Processes, Oxford Univ. Press.