Academic Year/course:
2023/24
453 - Degree in Mathematics
27032 - Probability Theory
Syllabus Information
Academic year:
2023/24
Subject:
27032 - Probability Theory
Faculty / School:
100 - Facultad de Ciencias
Degree:
453 - Degree in Mathematics
ECTS:
6.0
Year:
4
Semester:
Second semester
Subject type:
Optional
Module:
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1. General information
In contrast to the probability matters seen in previous courses, an approach to probability theory based on measure theory is introduced in this course. The different concepts are rigorously studied, paying special attention to the convergence in law and the central limit theorem. The theoretical character of the subject is compensated by means of various specific assignments to the students throughout the course.
The approaches and objectives of this module are aligned with the Sustainable Development Goals (SDGs) of the United Nations 2030 Agenda; the learning activities could contribute to some extent to the achievement of the goals 4 (quality education), 5 (gender equality), 8 (decent work and economic growth), and 10 (reducing inequality).
2. Learning results
- To understand and handle the fundamental notions of probability theory (probability spaces, random variables, expected values, types of convergence of random variables,...) in a rigorous way, based on measure theory.
- To understand the notions and results on conditional calculus, namely, conditional probabilities and expectations, and their applications, particularly to mixtures of random variables.
- To know, handle and apply the notions of independence of random variables and the different types of convergence of random variables, in particular, the strong and weak law of large numbers.
- To understand and handle the notion of characteristic function, as a fundamental tool in different areas of probability theory.
- To understand the meaning of central limit theorems, their proof based on characteristic functions, and their various applications to statistics and other sciences.
3. Syllabus
Topic 1. Probability, random variables, and expected values. Events and sigma-algebras: Dynkin`s theorem. Probability spaces: Carathéodory`s extension theorem. Random variables: distribution function and image probability. Expected values: definition and limit theorems. Moments and inequalities. Discrete and absolutely continuous random variables: Radon-Nikodym`s theorem. Independent random variables.
Topic 2. Conditional probabilities and expectations. Stochastic kernels and examples: the discrete and absolutely continuous cases. Construction of product probabilities. Inverse problem: probability disintegration theorem. Independence and Fubini`s theorem: applications. Mixtures of random variables and examples.
Topic 3. Convergence of random variables. Convergence in law. Convergence of random variables: almost sure, in probability, in mean of order p, and in law. Skorohod`s theorem and its consequences: coupling methods. Helly-Bray`s theorem. Slutsky`s theorem. Applications: probabilistic methods in approximation theory.
Topic 4. Characteristic functions. Definition and examples. Derivatives of characteristic functions and moments. Uniqueness theorem. Inversion theorem: the discrete and absolutely continuous cases. Kolmogorov and total variation distances: Sheffé`s lemma. Lévy`s continuity theorem.
Topic 5. Central limit theorems and Poisson approximation. Classical version of the central limit theorem and examples. Moment convergence. Poisson approximation: classical approach and coupling methods. Lévy-Lindeberg`s theorem for triangular arrays. Necessary conditions: Feller`s theorem. Rates of convergence: Berry-Esseen bounds and Edgeworth expansions.
4. Academic activities
Master classes: 30 hours.
Problem solving: 30 hours.
Project: 24 hours.
Study: 60 hours.
Assessment tests: 6 hours.
5. Assessment system
There will be a mid-term exam between weeks 8 and 9 on topics 1 and 2 of the course. This mid-term exam will count for 40% of the final grade. There will also be a final exam on the date indicated by the Faculty, which will include the rest of the material covered, if the student has passed the previous exam, or all the material if this is not the case. The student may choose to take the exam with the whole subject, even if he/she has passed the intermediate exam, if he/she wishes to improve his/her mark.
However, students may substitute the type of assesment described above for the completion of 4 (optionally 5) assignments throughout the course, either individually or in a group of two. The final mark will be the average mark of these assignments.
Such assignments may include connections of probability theory with other disciplines, such as mathematical analysis, combinatorics or number theory, among others.