A conditional limit theorem for high-dimensional ℓ^p spheres

2015/09/17 by Steven Soojin Kim, Kim, Steven Soojin, Kavita Ramanan +1
Mathematics · #52A20 (Primary) #52A23 #60D05 (Secondary) #60F10 #FOS: Mathematics #Probability (math.PR) #math.PR #msc:52A20 #msc:52A23 #msc:60D05 #msc:60F10

paper · pdf · doi:10.48550/arxiv.1509.05442

17 pages; formerly titled "A Sanov-type theorem for empirical measures associated with the surface and cone measures on $\ell^{p}$ spheres"

arxiv created 2018/06/20 · arxiv updated 2018/06/21

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 ℓ^p spaces and norms are of particular interest in this setting. In this paper, we establish a limit theorem for distributions on ℓ^p 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 ℓ^p balls in a high-dimensional Euclidean space.

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