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A conditional limit theorem for high-dimensional ℓᵖ-spheres

Part of: General convexity Geometric probability and stochastic geometry Limit theorems

Published online by Cambridge University Press:  16 January 2019

Steven S. Kim  and
Kavita Ramanan
Steven S. Kim*
Affiliation:
Brown University
Kavita Ramanan*
Affiliation:
Brown University
*
* Postal address: Brown University, 182 George Street, Box F, Providence, RI 02912, USA.
* Postal address: Brown University, 182 George Street, Box F, Providence, RI 02912, USA.
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Abstract

The study of high-dimensional distributions is of interest in probability theory, statistics, and asymptotic convex geometry, where the object of interest is the uniform distribution on a convex set in high dimensions. The ℓp-spaces and norms are of particular interest in this setting. In this paper we establish a limit theorem for distributions on ℓp-spheres, conditioned on a rare event, in a high-dimensional geometric setting. As part of our proof, we establish a certain large deviation principle that is also relevant to the study of the tail behavior of random projections of ℓp-balls in a high-dimensional Euclidean space.

Keywords

Conditional limit theoremℓᵖ-spherelarge deviationGibbs conditioning principlecone measuresurface measurerelative entropyWasserstein topology

MSC classification

Primary: 60F10: Large deviations 52A20: Convex sets in $n$ dimensions (including convex hypersurfaces)
Secondary: 52A23: Asymptotic theory of convex bodies 60D05: Geometric probability and stochastic geometry
Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 4 , December 2018 , pp. 1060 - 1077
Copyright
Copyright © Applied Probability Trust 2018 

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