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A conditional limit theorem for high-dimensional ℓᵖ-spheres

  • Steven S. Kim (a1) and Kavita Ramanan (a1)
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.

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Corresponding author
* Postal address: Brown University, 182 George Street, Box F, Providence, RI 02912, USA.
** Email address: steven_kim@alumni.brown.edu
*** Email address: kavita_rananan@brown.edu
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