KIT - Fakultät für Mathematik - AG Stochastische Geometrie (Sommersemester 2019)

Termine
Seminar:	Freitag 9:45-11:15	SR 2.58

Lehrende
Seminarleitung	Prof. Dr. Daniel Hug
	Sprechstunde: Nach Vereinbarung.
	Zimmer 2.051 Kollegiengebäude Mathematik (20.30)
	Email: daniel.hug@kit.edu
Seminarleitung	Prof. Dr. Günter Last
	Sprechstunde: nach Vereinbarung.
	Zimmer 2.001, Sekretariat 2.056 Kollegiengebäude Mathematik (20.30)
	Email: guenter.last@kit.edu

Studierende und Gäste sind jederzeit herzlich willkommen. Wenn nicht explizit anders unten angegeben, finden alle Vorträge im Seminarraum 2.58 im Mathebau (Gebäude 20.30) statt.

Freitag, 3.5.2019

9.45 Uhr László Györfi (Budapest University of Technology and Economics)

The limit distribution of the probability content of the Voronoi cell with application for residual estimation in nonparametric regression

Abstract: Our aim was to construct an estimate of the minimum mean squared error (called residual variance) for nonparametric regression such that without any regularity assumption the variance of the estimate is of order $O(1/n)$ . We proved a CLT of a first nearest neighbor based estimate. Interestingly, the variance of the probability content of the Voronoi cells plaid a crucial role in the analysis and the asymptotic variance of the estimate does not increase with the dimension of the feature vector.

Dienstag, 7.5.2019 (im Rahmen der AG Stochastik, Raum 2.58)

15.45 Uhr Ilya Molchanov (Universität Bern)

Sieving random iterative function systems

Abstract: It is known that backward iterations of a contractive random Lipschitz function converge almost surely under mild assumptions. By a sieving (or thinning) procedure based on adding to the functions time and space components, it is possible to construct a scale invariant stochastic process. We study its distribution and paths properties. In particular, we show that it is càdlàg and has finite total variation. We also provide examples of particular sieved iterative function systems including perpetuities and infinite Bernoulli convolutions, iterations of maximum, and continued fractions.
(Joint work with Alexander Marynych, Kiev)

Dienstag, 21.5.2019 (im Rahmen der AG Stochastik, Raum 2.58)

15.45 Uhr Clément Dombry (Université de Franche-Comté, Besançon)

The coupling method in extreme value theory

Abstract: One of the main goals of extreme value theory is to infer probabilities of extreme events for which only limited observations are available and require extrapolation of the tail distribution of the observations. One major result is Balkema-de Haan-Pickands theorem that provides an approximation of the distribution of exceedances above high thresholds by a generalized Pareto distribution. We revisit these results with coupling arguments and provide quantitative estimates for the Wasserstein distance between the empirical distribution of exceedances and the limit Pareto model. In a second part of the talk, we extend the results to the analysis of a proportional tail model for quantile regression closely related to the heteroscedastic extremes framework developed by Einmahl et al. (JRSSB 2016). We introduce coupling arguments relying on total variation and Wasserstein distances for the analysis of the asymptotic behavior of estimators of the extreme value index and the integrated skedasis function.

Freitag, 24.5.2019

9.45 Uhr Felix Herold (KIT)

Does a central limit theorem hold for Poisson hyperplanes in hyperbolic space?

Freitag, 14.6.2019

9.45 Uhr Christoph Thäle (Ruhr-Universität Bochum)

Monotonicity for random polytopes

Abstract: Random polytopes are classical objects studied at the crossroad of convex geometry and probability. In this talk we discuss several monotonicity questions for random polytopes. As a special case we consider the expected f-vector of random projections of regular polytopes.

Dienstag, 18.6.2019 (im Rahmen der AG Stochastik, Raum 2.58)

15.45 Uhr Alexander Koldobsky (University of Missouri, Columbia)

An estimate for the distance from a convex body to subspaces of $L_p$

Abstract: For $p\geq 1,\, n \in\mathbb{N}$ , and an origin-symmetric convex body $K$ in $\mathbb{R}^n$ , let

$d_{ovr}(K,L_p^n)=\inf\left\{\left(\frac{|D|}{|K|}\right)^{1/n}: K\subset D, D\in L_p^n\right\}$

be the outer volume ratio distance from $K$ to the class $L_p^n$ of the unit balls of $n$ -dimensional subspaces of $L_ p$ . We show that there exist absolute constants $c_1,c_2 > 0$ so that

$c_1 \sqrt{\frac np}\leq \sup_K d_{ovr}(K,L_p^n)\leq c_2 \sqrt{\frac{n+p}{p}}.$

Freitag, 28.6.2019

9.45 Uhr Ilya Molchanov (Universität Bern und KIT)

Random convex sets unleashed

Dienstag, 2.7.2019 (im Rahmen der AG Stochastik, Raum 2.58)

15.45 Uhr François Bacelli (INRIA and ENS Paris)

Dynamics on Unimodular Random Graphs

Abstract: The talk will discuss deterministic dynamics on infinite random graphs. Such a dynamic can be seen as a set of navigation rules on the nodes of the graph, which are deterministic functions of the local geometry of the rooted graph. We will focus on random graphs that are unimodular (i.e., satisfy the mass transport equations) and on navigation rules that are covariant (invariant by isomorphisms of rooted graphs).

We will first give a classification of these dynamics based on the properties of their stable manifolds. Each foil in the stable manifold is a generation in some unimodular infinite random family tree. The classification is in term of the number of ends and the cardinality of the generations in this family tree.

In order to analyze these trees and their covariant subgraphs, we will then discuss two new notions of dimension, namely the unimodular Minkowski and Hausdorff dimensions. These dimensions are of general interest in that they can be extended to all unimodular random graphs.

These notions will be illustrated by examples from the theory of point processes, branching processes, random walks, and self-similarity.

Work in collaboration with M.-O. Haji-Mirsadeghi and A. Khezeli.

Freitag, 19.7.2019

9.45 Uhr Christian Hirsch (Universität Mannheim)

Optimal stationary markings

Abstract: Optimal stationary markings are a generalization of maximal-volume hard-core thinnings that provide a general framework for dealing with point-process based optimization problems. Examples include minimum matching and travelling salesman from combinatorial optimization as well as caching and throughput from telecommunication. We highlight intensity-optimal and locally optimal markings as two possible formalizations of this concept. We discuss general conditions for existence, and provide novel examples for uniqueness and non-uniqueness.

This talk is based on joint work with Bartek Błaszczyszyn.