Peter Gracar

In this course we will study selected chapters from the broad field, starting with some theoretical foundations, such as unbiased estimation and Bayesian Inference. We will also study more modern applications of statistics, such as Neural Networks and Deep Learning.

Understanding of concepts from the course “Introductions to Stochastics“ is a requirement.

The link to the Zoom meeting for the Thursday lectures can be found on Ilias.

Literature

• R. M. Dudley. Real Analysis and Probability. Cambridge Studies in Advanced Mathematics. Cambridge UniversityPress, second edition, 2002.
• B. Efron and T. Hastie. Computer Age Statistical Inference: Algorithms, Evidence, and Data Science. Cambridge University Press, USA, first edition, 2016.
• H.-O. Georgii. Stochastics. Walter de Gruyter GmbH, Berlin, second edition, 2013.
• R. W. Keener. Theoretical Statistics: Topics for a Core Course. Springer Texts in Statistics. Springer New York, 2010.
• J. Shao. Mathematical Statistics. Springer-Verlag New York Inc, 2nd edition, 2003.

The seminal result in probability is the central limit theorem, a tool that tells us how ever larger sums of random variables fluctuate around their mean. A natural question that one might pose at this stage is: how can we describe events where this sum deviates from its mean by more than a “normal’’ amount? Answering this questions plays a crucial role in many fields such as probability theory, statistics, financial mathematics, operations research, ergodic theory, information theory, statistical physics and many more. In this seminar we will look at large deviation theory, by first looking at a toy example of sums of i.i.d. random variables. Once the basic concepts are understood, we will present the result in a more abstract/general way and look at a few more tailored statements.

Literature:

• Frank den Hollander. Large Deviations. American Mathematical Soc., 2008.

Useful links:

• Seminar details and assigned topics
• Collected synopses
• Klips 2.0: Large Deviations

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The lecture is a natural continuation of the course Probability Theory I. It is aimed at Masters students doing a degree in mathematics or economathematics and belongs to the area of stochastic and insurance mathematics.
The lecture will be held in English.

Examinations

Oral examinations will take place in Room 118 of the Mathematical insitute and be scheduled on a case per case basis. Please register for an examination slot by sending me an email at least a week ahead of time.

Material:

• Lecture notes
• Exercises and Further information

Literature:

• Achim Klenke. Probability theory: A comprehensive course. Universitext. Springer, London, second edition, 2014.
• Rick Durrett. Probability: theory and examples. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, fourth edition, 2010.
• R. M. Dudley. Real analysis and probability, volume 74 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2002. Revised reprint of the 1989 original.
• Daniel W. Stroock. Probability theory: An analytic view. Cambridge University Press, Cambridge, second edition, 2011.

Other useful links:

• Lecture notes - Probability 1
• Lecture notes - Introduction to Stochastics
• Klips 2.0: Probability Theory II