Vol. 307, No. 2, 2020

Download this article
Download this article For screen
For printing
Recent Issues
Vol. 307: 1  2
Vol. 306: 1  2
Vol. 305: 1  2
Vol. 304: 1  2
Vol. 303: 1  2
Vol. 302: 1  2
Vol. 301: 1  2
Vol. 300: 1  2
Online Archive
The Journal
Editorial Board
Special Issues
Submission Guidelines
Submission Form
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
Author Index
To Appear
Other MSP Journals
This article is available for purchase or by subscription. See below.
Strong negative type in spheres

Russell Lyons

Vol. 307 (2020), No. 2, 383–390

It is known that spheres have negative type, but only subsets with at most one pair of antipodal points have strict negative type. These are conditions on the (angular) distances within any finite subset of points. We show that subsets with at most one pair of antipodal points have strong negative type, a condition on every probability distribution of points. This implies that the function of expected distances to points determines uniquely the probability measure on such a set. It also implies that the distance covariance test for stochastic independence, introduced by Székely, Rizzo and Bakirov, is consistent against all alternatives in such sets. Similarly, it allows tests of goodness of fit, equality of distributions, and hierarchical clustering with angular distances. We prove this by showing an analogue of the Cramér–Wold theorem.

PDF Access Denied

However, your active subscription may be available on Project Euclid at

We have not been able to recognize your IP address as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

Please contact your institution's librarian suggesting a subscription, for example by using our journal-recom­mendation form. Or, visit our subscription page for instructions on purchasing a subscription.

You may also contact us at contact@msp.org
or by using our contact form.

Or, you may purchase this single article for USD 40.00:

Cramér–Wold, hemispheres, expected distances, distance covariance, equality of distributions, goodness of fit, hierarchical clustering.
Mathematical Subject Classification 2010
Primary: 44A12, 45Q05, 51K99, 51M10
Secondary: 62G20, 62H15, 62H20, 62H30
Received: 16 May 2019
Revised: 4 April 2020
Accepted: 19 May 2020
Published: 4 September 2020
Russell Lyons
Department of Mathematics
Indiana Univ, Bloomington
Bloomington, IN
United States