The convex hull of a random sample in
- 1 January 1987
- journal article
- research article
- Published by Taylor & Francis in Communications in Statistics. Stochastic Models
- Vol. 3 (1) , 1-27
- https://doi.org/10.1080/15326348708807044
Abstract
Let X1.......Xn be an iid sequence of random vectors in with common distribution function which satisfies a multivariate regular o variation condition. In the metric space of compact convex sets of with metric given by the Hausdorff distance we show that the sample convex hull converges in distribution to the convex hull of the points of a two-dimensional Poisson process. We also give necessary and sufficient conditions for the number of vertices of the limiting convex hull to be finite with probability one. Finally, we discuss weak convergence of the number of vertices of the sample convex hull as well as asymptotic behavior of the expected number of vertices.Keywords
This publication has 23 references indexed in Scilit:
- Point processes, regular variation and weak convergenceAdvances in Applied Probability, 1986
- On the Estimation of a Convex SetThe Annals of Statistics, 1984
- The convex hull of a spherically symmetric sampleAdvances in Applied Probability, 1981
- The distribution of the convex hull of a Gaussian sampleJournal of Applied Probability, 1980
- The probability that two samples in the plane will have disjoint convex hullsJournal of Applied Probability, 1978
- Finding the edge of a Poisson forestJournal of Applied Probability, 1977
- Sur L'enveloppe convexe des nuages de points aleatoires dans Rn. IJournal of Applied Probability, 1970
- ZufÄllige konvexe Polygone in einem RinggebietProbability Theory and Related Fields, 1968
- Geometrical Probability and Random Points on a HypersphereThe Annals of Mathematical Statistics, 1967
- The convex hull of a random set of pointsBiometrika, 1965