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Rates of convergence for random approximations of convex sets

Published online by Cambridge University Press:  01 July 2016

Lutz Dümbgen*
Affiliation:
Universität Heidelberg
Günther Walther*
Affiliation:
Stanford University
*
Postal address: Institut für Angewandte Mathematik, Universität Heidelberg, Im Neuenheimer Feld 294, D-69120 Heidelberg, Germany. E-mail: [email protected]
∗∗ Postal address: Department of Statistics, Stanford University, Stanford CA 94305, USA. e-mail: [email protected]

Abstract

The Hausdorff distance between a compact convex set K ⊂ ℝd and random sets is studied. Basic inequalities are derived for the case of being a convex subset of K. If applied to special sequences of such random sets, these inequalities yield rates of almost sure convergence. With the help of duality considerations these results are extended to the case of being the intersection of a random family of halfspaces containing K.

Type
Stochastic Geometry and Statistical Applications
Copyright
Copyright © Applied Probability Trust 1996 

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