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A class of spherical and elliptical distributions with Gaussian-like limit properties

Research output: Contribution to journalJournal article

Published

Article number467187
Journal publication date2012
JournalJournal of Probability and Statistics
Volume2012
Number of pages17
Original languageEnglish

Abstract

We present a class of spherically symmetric random variables defined by the property that as dimension increases to infinity the mass becomes concentrated in a hyperspherical shell, the width of which is negligible compared to its radius. We provide a sufficient condition for this property in terms of the functional form of the density and then show that the property carries through to equivalent elliptically symmetric distributions, provided that the contours are not too eccentric, in a sense which we make precise. Individual components of such distributions possess a number of appealing Gaussian-like limit properties, in particular that the limiting one-dimensional marginal distribution along any component is Gaussian.