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401-3601-00L 9 Credits BSC , MSC D-INFK , D-MATH , D-PHYS , D-ITET

Probability Theory

Lecturers & Examiners: Prof. Dr. Vincent Tassion
At most one of the three course units (Bachelor Core Courses) 401-3461-00L Functional Analysis I 401-3531-00L Differential Geometry I 401-3601-00L Probability Theory can be recognised for the Master's degree in Mathematics or Applied Mathematics. In this case, you cannot change the category assignment by yourself in myStudies but must take contact with the Study Administration Office ( ) after having received the credits. Moreover, 401-3601-00L Probability Theory can only be recognised for the Master Programme in Mathematics if neither 401-3642-00L Brownian Motion and Stochastic Calculus nor 401-3602-00L Applied Stochastic Processes has been recognised for the Bachelor Programme.
VVZ CR 4.2

Last Updated: 2026-06-03 00:07:37

Abstract

Basics of probability theory and the theory of stochastic processes in discrete time

Objective

The aim of this course is to develop a solid understanding of probabilistic reasoning and methods through some of the most fundamental results and proofs in probability theory. In particular, students should: - Develop the language of Probability: become comfortable with the abstract language of measure theory as the foundation of probability, and learn how to translate intuition into rigorous arguments. - Grasp probabilistic reasoning through key proofs: understand in detail the proof of the Law of Large Numbers and the Central Limit Theorem, and appreciate their central role in probability. - Distinguish modes of convergence: clearly differentiate between almost sure convergence, convergence in probability, Lp-convergence and convergence in distribution, and understand how these notions are related and where they differ. - Understand martingales: learn the basic properties of martingales, know how to recognize them in examples, and be able to use them to prove important theorems.

Content

Part I – Abstract Generalities Probability Space · Random Variables · Real Random Variables · Zero-One Laws Part II – Spatial Convergence of Random Variables Almost Sure Convergence, Convergence in Probability · Law of Large Numbers · Lp-Convergence Part III – Convergence in Distribution Characteristic Functions · Weak Convergence of Probability Measures · Convergence in Distribution · Central Limit Theorem · Relations Between the Different Types of Convergence Part IV – Conditional Expectation Discrete Theory · Abstract Definition · Properties Part V – Martingales Definition and Examples · Gambling Systems and Stopped Martingales · Almost Sure Convergence · Uniformly Integrable Martingales · Lp-Martingales · Backward Martingales

Resources

Lecture Notes

Lecture notes (in Latex) will be provided during the course.Handwritten lecture notes are available on the website from last year:https://metaphor.ethz.ch/x/2024/hs/401-3601-00L/

Literature

J.F. Le Gall, Probability Theory, Springer 2022 R. Durrett, Probability: Theory and examples, Duxbury Press 1996 H. Bauer, Probability Theory, de Gruyter 1996 J. Jacod and P. Protter, Probability essentials, Springer 2004 A. Klenke, Wahrscheinlichkeitstheorie, Springer 2006 D. Williams, Probability with martingales, Cambridge University Press 1991

General Information

Language
English
Levels
BSC , MSC
Frequency
Yearly recurring

Examination

Type
session examination
Mode
written 120 minutes
Aids
None

Course Components

Type Title Time & Place Hours
lecture Probability Theory No time listed 4 h weekly
exercise Probability Theory
Tue 14-15 or Tue 15-16 starting in the second week of the semester.
No time listed 1 h weekly

Offered In