Measure-theoretic Probability, 2013-2014

### Aims

To provide an introduction to the basic notions and results of measure theory and how these are used in probability theory.

### Contents

During the course the measure theoretic foundations of probability theory will be treated. Key words for the course are: limit theorems for Lebesgue integrals, product measures, random variables, distributions of random variables, convergence in probability, weak convergence, uniform integrability, conditional expectation, martingales in discrete time, convergence theorems for martingales, characteristic functions, central limit theorems. The course provides the necessary background for follow up courses like Stochastic Processes and Stochastic Integration, where in particular convergence theorems for martingales and characteristic functions are frequently used.

### Prerequisites

Knowledge at the level of for instance Richard T. Durrett, The Essentials of Probability and the first seven chapters of Walter Rudin, Principles of Mathematical Analysis.

### Literature

• The course is based on the set of lecture notes written by Peter Spreij.
• For the last part of the course we will use another set of lecture notes.
• Handout on a necessary condition for the SLLN is here.
• Handout on convolutions is here.
• Handout with a detailed proof of Theorem 4.19 (dominated convergence theorem) is here.

### Lecturers

Shota Gugushvili (Leiden), Harry van Zanten (UvA).

### Assistants

Dirk Erhard (erhardd@math.leidenuniv.nl), Jan van Waaij.

### Schedule

Location: UvA, Science Park 904; see the map of Science Park and the travel directions.

Autumn semester, Wednesdays 10.15-13.00 in room F1.02. (changes in the schedule will appear here).

### Reimbursement of travel costs

Students who are registered in a master program in mathematics at one of the Dutch universities can claim their travel expenses, see the rules.

### Assessment

Written examination (E) and homework assignments (H).

The final mark (F) is determined as F = max(E, 1/3 * H + 2/3 * E). See the announcements above for the dates of the written examination and the retake.

The exam is about all the material covered in the course. For a passing grade you need to know the results, know how to use them in exercises, understand the relations and know the main ideas behind proofs. You are expected to be able to reproduce the exact statements and detailed proofs of the following results (numbering refers to the old lecture notes):

• Lemma 3.14 (Borel-Cantelli lemma)
• Theorem 4.12 (Monotone convergence theorem)
• Lemma 4.15 (Fatou's lemma)
• Theorem 4.19 (Dominated convergence theorem)
• Everything in Section 10.1 (Upcrossings and Doob's martingale convergence theorem)
• Proposition 10.21 and Theorem 10.23 (Kolmogorov's 0-1 law and the strong law of large numbers)
• Characteristic function proof of the central limit theorem

### Homework

Homework assignments are due in one week, at the beginning of the lecture. It is allowed to work in couples (in fact it is strongly encouraged to do so). Homework may be submitted electronically as a pdf file. In that case please email it to the assistant (address above).

Brief solutions to selected exercises by Dirk Erhard.

Jan van Waaij has worked out some exercises to serve as additional practice material:

### Evaluation

At the end of the course you will be asked to evaluate it by filling in a questionnaire.

### Programme

Check back regularly. Last updated: 4 December 2013.

 Week Date Material covered Homework 1. 11/9 Chapter 1 1.5, 1.6, 1.9 2. 18/9 Chapter 3 3.1, 3.3, 3.4, 3.5 3. 25/9 Sections 4.1, 4,2 4.2, 4.3, 4.4 4. 2/10 Section 4.3, 4.4, 4.6, 4.7 4.9, 4.10, 4.11 5. 9/10 Chapter 5 5.2, 5.3, 5.7 6. 16/10 Sections 6.4, 6.5, 8.1 6.5, 6.7, 8.2 7. 23/10 no lecture! 8. 30/10 Sections 9.1, 9.2. Slides 9.3, 9.4, 9.6 9. 06/11 Sections 10.1, 7.2, 10.2. Slides 10.1, 7.4, 7.6 10. 13/11 Old notes: Section 10.4 (second part); New notes: Sections 1.1-1.4. Slides, part 1 and part 2 10.8 (old notes), 1.4 (new notes) 11. 20/11 New notes: Chapter 1 (the rest); Section 2.1. 1.10, 2.1, 2.2, 2.5 (all from new notes) 12. 27/11 New notes: Chapter 2 (the rest); Sections 3.1-3.2. 2.8, 2.9, 2.10 (all from new notes) 13. 04/12 New notes: Chapter 3 (the rest); handout on convolutions. 3.6, 3.7 and 3.8 (new notes; will not be graded) 14. 11/12 Office hour, practice exam, exercises.

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