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A large deviation principle for Minkowski sums of heavy-tailed random compact convex sets with finite expectation

Part of: Limit theorems General convexity

Published online by Cambridge University Press:  14 July 2016

Thomas Mikosch ,
Zbyněk Pawlas  and
Gennady Samorodnitsky
Thomas Mikosch
Affiliation:
University of Copenhagen, Department of Mathematics, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark. Email address: [email protected]
Zbyněk Pawlas
Affiliation:
Charles University in Prague, Department of Probability and Mathematical Statistics, Faculty of Mathematics and Physics, Charles University in Prague, Sokolovská 83, Praha 8, Czech Republic. Email address: [email protected]
Gennady Samorodnitsky
Affiliation:
Cornell University, School of Operations Research and Information Engineering, Cornell University, 220 Rhodes Hall, Ithaca, NY 14853, USA. Email address: [email protected]
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Abstract

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We prove large deviation results for Minkowski sums Sn of independent and identically distributed random compact sets where we assume that the summands have a regularly varying distribution and finite expectation. The main focus is on random convex compact sets. The results confirm the heavy-tailed large deviation heuristics: ‘large’ values of the sum are essentially due to the ‘largest’ summand. These results extend those in Mikosch, Pawlas and Samorodnitsky (2011) for generally nonconvex sets, where we assumed that the normalization of Sn grows faster than n.

Keywords

Minkowski sumrandom compact setlarge deviationregularly varying distribution

MSC classification

Primary: 60F10: Large deviations
Secondary: 52A22: Random convex sets and integral geometry
Type
Part 3. Heavy Tail Phenomena
Information
Journal of Applied Probability , Volume 48 , Issue A: New Frontiers in Applied Probability (Journal of Applied Probability Special Volume 48A) , August 2011 , pp. 133 - 144
Copyright
Copyright © Applied Probability Trust 2011 

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