## Limit theorems for the convex hull of random points in higher dimensions

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- by Irene Hueter
- Trans. Amer. Math. Soc.
**351**(1999), 4337-4363 - DOI: https://doi.org/10.1090/S0002-9947-99-02499-X
- Published electronically: July 21, 1999
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## Abstract:

We give a central limit theorem for the number $N_n$ of vertices of the convex hull of $n$ independent and identically distributed random vectors, being sampled from a certain class of spherically symmetric distributions in $\mathbb {R}^d \; (d> 1),$ that includes the normal family. Furthermore, we prove that, among these distributions, the variance of $N_n$ exhibits the same order of magnitude as the expectation as $n \rightarrow \infty .$ The main tools are Poisson approximation of the point process of vertices of the convex hull and (sub/super)-martingales.## References

- David J. Aldous, Bert Fristedt, Philip S. Griffin, and William E. Pruitt,
*The number of extreme points in the convex hull of a random sample*, J. Appl. Probab.**28**(1991), no. 2, 287–304. MR**1104567**, DOI 10.2307/3214867 - H. Carnal,
*Die konvexe Hülle von $n$ rotationssymmetrisch verteilten Punkten*, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete**15**(1970), 168–176 (German, with English summary). MR**286153**, DOI 10.1007/BF00531885 - Rex A. Dwyer,
*Convex hulls of samples from spherically symmetric distributions*, Discrete Appl. Math.**31**(1991), no. 2, 113–132. First Canadian Conference on Computational Geometry (Montreal, PQ, 1989). MR**1106694**, DOI 10.1016/0166-218X(91)90064-4 - Bradley Efron,
*The convex hull of a random set of points*, Biometrika**52**(1965), 331–343. MR**207004**, DOI 10.1093/biomet/52.3-4.331 - William Feller,
*An introduction to probability theory and its applications. Vol. II.*, 2nd ed., John Wiley & Sons, Inc., New York-London-Sydney, 1971. MR**0270403** - Piet Groeneboom,
*Limit theorems for convex hulls*, Probab. Theory Related Fields**79**(1988), no. 3, 327–368. MR**959514**, DOI 10.1007/BF00342231 - Wassily Hoeffding,
*Probability inequalities for sums of bounded random variables*, J. Amer. Statist. Assoc.**58**(1963), 13–30. MR**144363** - Hueter, I. (1992).
*The convex hull of $n$ random points and its vertex process.*Doctoral Dissertation, University of Berne. - Irene Hueter,
*The convex hull of a normal sample*, Adv. in Appl. Probab.**26**(1994), no. 4, 855–875. MR**1303866**, DOI 10.2307/1427894 - I. A. Ibragimov and Yu. V. Linnik,
*Independent and stationary sequences of random variables*, Wolters-Noordhoff Publishing, Groningen, 1971. With a supplementary chapter by I. A. Ibragimov and V. V. Petrov; Translation from the Russian edited by J. F. C. Kingman. MR**0322926** - Olav Kallenberg,
*Random measures*, 3rd ed., Akademie-Verlag, Berlin; Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1983. MR**818219** - H. Raynaud,
*Sur l’enveloppe convexe des nuages de points aléatoires dans $R^{n}$. I*, J. Appl. Probability**7**(1970), 35–48 (French). MR**258089**, DOI 10.2307/3212146 - A. Rényi and R. Sulanke,
*Über die konvexe Hülle von $n$ zufällig gewählten Punkten*, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete**2**(1963), 75–84 (1963) (German). MR**156262**, DOI 10.1007/BF00535300 - Daniel W. Stroock,
*Diffusion processes associated with Lévy generators*, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete**32**(1975), no. 3, 209–244. MR**433614**, DOI 10.1007/BF00532614

## Bibliographic Information

**Irene Hueter**- Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611
- Email: hueter@math.ufl.edu
- Received by editor(s): December 2, 1998
- Received by editor(s) in revised form: January 22, 1999
- Published electronically: July 21, 1999
- © Copyright 1999 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**351**(1999), 4337-4363 - MSC (1991): Primary 52A22, 60D05
- DOI: https://doi.org/10.1090/S0002-9947-99-02499-X
- MathSciNet review: 1670156