Journal of Probability and Statistics
Volume 2012 (2012), Article ID 467187, 17 pages
Research Article

A Class of Spherical and Elliptical Distributions with Gaussian-Like Limit Properties

Department of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF, UK

Received 4 May 2011; Accepted 29 November 2011

Academic Editor: Mohammad Fraiwan Al-Saleh

Copyright © 2012 Chris Sherlock and Daniel Elton. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.