A multivariate Gnedenko law of large numbers

We show that the convex hull of a large i.i.d. sample from an absolutely continuous log-concave distribution approximates a predetermined convex body in the logarithmic Hausdorff distance and in the Banach-Mazur distance. For log-concave distributions that decay super-exponentially, we also have approximation in the Hausdorff distance. These results are multivariate versions of the Gnedenko law of large numbers, which guarantees concentration of the maximum and minimum in the one-dimensional case. We provide quantitative bounds in terms of the number of points and the dimension of the ambient space.Comment: Published in at http://dx.doi.org/10.1214/12-AOP804 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Comments on the floating body and the hyperplane conjecture

We provide a reformulation of the hyperplane conjecture (the slicing problem) in terms of the floating body and give upper and lower bounds on the logarithmic Hausdorff distance between an arbitrary convex body $K\subset \mathbb{R}^{d}$\ and the convex floating body $K_{\delta}$ inside $K$.Comment: 8 page

Variations and extensions of the Gaussian concentration inequality, Part II

Pisier's version of the Gaussian concentration inequality is transformed and used to prove deviation inequalities for locally Lipschitz functions with respect to heavy tailed product measures on Euclidean space. The approach is, in our opinion, more direct than much of the modern theory of concentration of measure (i.e. Poincar\'{e} and log-Sobolev inequalities, estimating moments etc.).Comment: 35 pages. The original paper, which was uploaded as a separate submission (arXiv:1812.10938), has been split into two papers, Part I and Part II; this is Part I

Geometric and nonlinear limit theorems in probability theory

Title from PDF of title page (University of Missouri--Columbia, viewed on August 28, 2012).The entire thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file; a non-technical public abstract appears in the public.pdf file.Dissertation advisors: Professor Nigel Kalton, Professor Alexander Koldobsky, Professor Mark RudelsonIncludes bibliographical references.Vita.Ph. D. University of Missouri--Columbia 2012"May 2012"The concentration of measure phenomenon is a nonlinear equivalent of the law of large numbers that deals with real valued Lipschitz functions and includes linear functionals such as the sample mean. In the first part of this dissertation we study functions that take values in more general metric-like spaces and have the property that they are invariant under coordinate permutations. In Chapter 1 we study functions that take values in the space of convex bodies, in Chapter 2 we study order statistics and in Chapter 3 we prove abstract concentration inequalities for functions taking values in an arbitrary metric space. In the second part of the dissertation we study the central limit theorem. We show that if one conditions on certain tail events then convergence to the normal distribution can be achieved without having to take a large number of summands. In fact 2 summands is enough.Includes bibliographical reference

An Integrated Management Approach in a Higher Education Technology Support Unit

This case study is located in the Department for Education Innovation (EI), a teaching and learning support unit at the University of Pretoria in South Africa. The initial problem was the need to apply project management and quality management principles to the services offered by the department to faculty members. The authors describe the implementation of a formal, online, process-based Quality Management System (QMS) designed to self evaluate, document, and improve the Instructional Design (ID) process that guides the development of educational technology solutions in EI. The project was completed in 2005 and was included in a CEN (European Committee for Standardization) Good Practice Guide for outstanding implementations of quality approaches in e-learning. The QMS provides a mechanism to support a consistent project management approach, and the case illustrates successful integration between three cycles: Project Management (PM), Quality Management (QM), and the ADDIE (Analysis, Design, Development, Implementation, and Evaluation) instructional design process