A conditional limit theorem for high-dimensional ℓᵖ-spheres. (16th January 2019)

• Record Type: Journal Article
• Authors: Kim, Steven S.; Ramanan, Kavitar
• Is Part Of: Journal of applied probability. Volume 55:Number 4(2018)
• Page Start: 1060
• Page End: 1077
• Publication Date: 2019-01-16
• Subjects: Conditional limit theorem, -- ℓᵖ-sphere, -- large deviation, -- Gibbs conditioning principle, -- cone measure, -- surface measure, -- relative entropy, -- Wasserstein topology; Primary 60F10, -- 52A20, -- Secondary 52A23, -- 60D05; 519.2
• DOI: 10.1017/jpr.2018.71
• Languages: English
• ISSNs: 0021-9002

Abstract

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.