Measure-theoretic Probability, 2013-2014
## AimsTo provide an introduction to the basic notions and results of measure theory and how these are used in probability theory. ## ContentsDuring 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. ## PrerequisitesKnowledge 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.
## LecturersShota Gugushvili (Leiden), Harry van Zanten (UvA). ## AssistantsDirk Erhard (erhardd@math.leidenuniv.nl), Jan van Waaij. ## ScheduleLocation: UvA, Science Park 904;
see the map
of Science Park and the travel directions.
## Reimbursement of travel costsStudents who are registered in a master program in mathematics at one of the Dutch universities can claim their travel expenses, see the rules. ## AssessmentWritten examination (E) and homework assignments (H).
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
## HomeworkHomework 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).
## EvaluationAt the end of the course you will be asked to evaluate it by filling in a questionnaire. ## ProgrammeCheck back regularly. Last updated:
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