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The distribution of the convex hull of a Gaussian sample

Published online by Cambridge University Press:  14 July 2016

William F. Eddy
William F. Eddy*
Affiliation:
Carnegie-Mellon University
*
∗Postal address: Department of Statistics, Carnegie-Mellon University, Schenley Park, Pittsburgh, PA 15213, U.S.A.
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Abstract

The distribution of the convex hull of a random sample of d-dimensional variables is described by embedding the collection of convex sets into the space of continuous functions on the unit sphere. Weak convergence of the normalized convex hull of a circular Gaussian sample to a process with extreme-value marginal distributions is demonstrated. The proof shows that an underlying sequence of point processes converges to a Poisson point process and then applies the continuous mapping theorem. Several properties of the limit process are determined.

Keywords

POISSON RANDOM MEASURECONTINUOUS FUNCTIONSEXTREME VALUE PROCESSWEAK CONVERGENCEPOINTWISE MAXIMUM
Type
Research Papers
Information
Journal of Applied Probability , Volume 17 , Issue 3 , September 1980 , pp. 686 - 695
Copyright
Copyright © Applied Probability Trust 1980 

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Footnotes

Research supported in part by NSF Grant MCS 78–02422 to Carnegie-Mellon University

References

Barnett, V. (1976) The ordering of multivariate data. J. R. Statist. Soc. A 139, 318–354.Google Scholar
Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
Brown, B. M. and Resnick, S. I. (1977) Extreme values of independent stochastic processes. J. Appl. Prob. 14, 732–739.CrossRefGoogle Scholar
David, H. A. (1970) Order Statistics. Wiley, New York.Google Scholar
De Haan, L. (1970) On Regular Variation and its Application to the Weak Convergence of Sample Extremes. MC Tract 32, Mathematisch Centrum, Amsterdam.Google Scholar
Eddy, W. F. (1980) Optimum kernel estimators of the mode. Ann. Statist. 8.Google Scholar
Eddy, W. F. and Hartigan, J. A. (1977) Uniform convergence of the empirical distribution function over convex sets. Ann. Statist. 5, 370–374.CrossRefGoogle Scholar
Efron, B. (1965) The convex hull of a random set of points. Biometrika 52, 331–343.Google Scholar
Jagers, P. (1974) Aspects of random measure and point processes. In Advances in Probability 3, Dekker, New York.Google Scholar
Kendall, D. G. (1974) Foundations of a theory of random sets. In Stochastic Geometry , ed. Harding, E. F. and Kendall, D. G., Wiley, New York.Google Scholar
Matheron, G. (1975) Random Sets and Integral Geometry , Wiley, New York.Google Scholar
Rényi, A. and Sulanke, R. (1963), (1964) Über die konvexe Hülle von n zufällig gewählten Punkten, I, II. Z. Wahrscheinlichkeitsth. 2, 27–84; 3, 138–147.Google Scholar
Resnick, S. I. (1975) Weak convergence to extremal processes. Ann. Prob. 3, 951–960.Google Scholar
Sager, T. W. (1979) An iterative method for estimating a multivariate mode and isopleth. J. Amer. Statist. Assoc. 74, 329–339.Google Scholar
Sibuya, M. (1960) Bivariate extreme statistics. Ann. Inst. Statist. Math. 11, 195–210.Google Scholar
Valentine, F. A. (1964) Convex Sets. McGraw-Hill, New York.Google Scholar