Gaussian processes - Einzelansicht

This is an advanced course for students with very strong mathematical ability and very strong interest in theoretical mathematics, especially analysis and probability theory.

This is an advanced course that will survey some parts of the theory of Gaussian processes. Gaussian processes are used extensively in mathematics, mathematical statistics, and also in applications such as statistical regression for machine learning or the design of compressed sensing matrices. We will not consider applications in this course. We will focus on mathematics.

The course will be organised around 4-hour lectures and a 2-hour seminar.

Evarist Gine and Richard Nickl, "Mathematical foundations of infinite-dimensional statistical models", Cambridge Series in Statistical and Probabilistic Mathematics (2016)

This course will be online only. The lectures will be given as asynchronous video lectures. The seminar will be synchronous on Zoom.

This course requires that participants know the following:

1) real analysis, for example epsilon-delta proofs, series and sequences, measure theory, Lebesgue integration, Fatou's lemma, monotone convergence theorem, dominated convergence theorem, Fubini-Tonelli theorem, differentiation and integration, totally bounded sets, precompact sets, compact sets, Bolzano-Weierstrass theorem, limit inferior and limit superior of sequences, Taylor expansions, uniformly continuous functions;

2) functional analysis, including linear spaces, Hilbert spaces, Banach spaces, dual spaces, topological spaces, metric spaces, linear operators;

3) measure-theoretic probability theory, including sigma-algebras, probability spaces, random variables, conditional expectations, different types of convergence of random variables, limit superior and limit inferior of events, Borel-Cantelli lemmas, Chebyshev's inequality.

Students who have read through ”Real Analysis” by Gerald Folland (Wiley, 1999) and ”Real analysis and probability” by Richard Dudley (Cambridge University Press, 2002) will be sufficiently prepared for this course.

Below are some titles of books that cover some of the topics above and that are available through the University of Potsdam library.

Christian Clason, "Einführung in die Funktionalanalysis", Cham: Birkhäuser (2019)

Achim Klenke, "Probability theory: a comprehensive course", Springer (2014)

Achim Klenke, "Wahrscheinlichkeitstheorie", Springer (2013)

Norbert Henze, "Stochastik: Eine Einführung mit Grundzügen der Maßtheorie", Springer Spektrum (2019)

Christopher Heil, "Introduction to Real Analysis" Cham: Springer (2019)

Sergei Ovchinnikov, "Functional Analysis: An introductory course", Cham: Springer (2018)

Satish Shirali, "Measure and Integration", Cham: Springer (2019)

To be eligible to take the final exam, students must give a 90-minute seminar presentation.

The final exam will be an oral exam of 30 minutes.

The plan is to present some parts of the following topics in Gaussian processes.

• Definitions and basic concepts
• Isoperimetric inequalities and concentration
• Metric entropy bounds for sub-Gaussian processes
• Comparison theorems and Sudakov’s lower bound
• Reproducing kernel Hilbert spaces

This course is targeted at students who:

• have very strong mathematical background and satisfy the prerequisites;
• have very good English and/or German language skills;
• enjoy rigorous proof-based mathematics; and
• are interested in analysis and probability theory.