Seminar Supervisor

Prof. Dr. Evgeny Spodarev
Prof. Dr. Volker Schmidt

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Seminar Advisor

Stefan Roth

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Date and Place

Time: Monday 2-4 pm, Place: Helmholtzstraße 22, E18

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Prerequisites

The level of difficulty in this seminar is varying between the different topics. The audience is at least supposed to be familiar with basic probability and statistics. There are a few talks that require the lecture Stochastics II or beyond. We ensure the participants, that most of the 'beyond' knowledge will be learned during the seminar.

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Intended Audience

Bachelor and Master Students of any mathematical study course.

The "(Bachelor)" in the list of talks, means that this talk is supposed to be given by a Bachelor's Student.

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Content

Stein's Method: Central limit theorems for sums of independent random variables play a crucial role in statistics. Most of the classical proofs are based on Fourier analysis that give no ideas on the rates of convergence. It was Charles Stein who proposed a general method to estimate the approximation error in central limit theorems systematically. The idea of Stein's method and applications for different classes of limiting distributions will be given in this seminar. In this context we will see how to prove limit theorems for sums of dependent random variables as well as for stochastic processes.

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Criterias to pass the seminar:

Each student is supposed to give a talk and to attend the seminar on a regular basis. Those who give a (good) talk and attend the seminar regularely will pass the seminar.

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List of Talks

1. 11th of April, Normal approximations for smooth functions. (Bachelor) assigned, Advisor: Matthias Neumann
   
2. 18th of April, Uniform Berry-Esseen bounds. (Bachelor)
3. 25th of April, Non-uniform Berry-Esseen bounds. (Bachelor) assigned, Advisor: Stefan Roth
   
4. 9th of May, canceled!
5. 23rd of May, The general procedure of Stein and Poisson approximation. assigned, Advisor: Alexander Nerlich
   
6. 6th of June, Multidimensional Stein's method. (Bachelor) 
7. 13th of June, Compound Poisson approximation. assigned 
8. 20th of June, Poisson process approximation. assigned
9. 27th of June, Malliavin operators in the one-dimensional case. (Bachelor)
10. 04th of July, Malliavin operators and isonormal Gaussian processes.
11. 11th of July, Normal approximations on Wiener chaos.

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Literatur

A.D. Barbour, Louis H. Y. Chen: An Introduction to Stein's Method, Singapore University Press, Singapore, 2005.

I. Nourdin, G. Peccati: Normal Approximations with Malliavin Calculus - From Stein's Method to Universality, Cambridge University Press, New York, 2012