Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-03T08:47:47.873Z Has data issue: false hasContentIssue false
  • English
  • Français

The volume of a random simplex in an n-ball is asymptotically normal

Published online by Cambridge University Press:  14 July 2016

Harold Ruben
Harold Ruben*
Affiliation:
McGill University, Montreal
Get access Rights & Permissions [Opens in a new window]

Abstract

A proof is given of a conjecture in the literature of geometrical probability that the r-content of the r-simplex whose r + 1 vertices are independent random points of which p are uniform in the interior and q uniform on the boundary of a unit n-ball (1 ≦ r ≦ n; 0 ≦ p, q ≦ r + 1, p + q = r + 1) is asymptotically normal (n →∞) with asymptotic mean and variance and , respectively.

Keywords

BLASCHKE-PETKANTSCHIN REPRESENTATIONCOMPLEX MOMENTS OF r-VOLUME OF RANDOM r-SIMPLEX IN n-BALLDIFFERENCE EQUATIONCHARACTERISTIC FUNCTION OF LOGARITHM OF VOLUMEASYMPTOTIC NORMALITY OF VOLUME
Type
Short Communications
Information
Journal of Applied Probability , Volume 14 , Issue 3 , September 1977 , pp. 647 - 653
Copyright
Copyright © Applied Probability Trust 1977 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Blaschke, W. (1935) Integralgeometrie 1. Ermittlung der Dichten für lineare Unterräume im En. Hermann, Paris (Act. Sci. Indust. No. 252) Google Scholar
[2] Erdelyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G. (1953) Higher Transcendental Functions , Vol. 1. McGraw-Hill, New York.Google Scholar
[3] Hammersley, J. ?. (1950) The distribution of distance in a hypersphere. Ann. Math. Statist. 21, 447–452.Google Scholar
[4] Miles, R. E. (1971) Isotropic random simplices. Adv. Appl. Prob. 3, 353–382.Google Scholar
[5] Petkantschin, B. (1936) Integralgeometrie 6. Zusammenhänge zwischen den Dichten der linearen Unterräume im n-dimensionalen Raum. Abh. Math. Seminar Hamburg 11, 249–310.Google Scholar
[6] Rao, C. R. (1973) Linear Statistical Inference and Its Applications , 2nd ed. Wiley, London.Google Scholar