The Annals of Probability

Variance asymptotics and central limit theorems for generalized growth processes with applications to convex hulls and maximal points

T. Schreiber and J. E. Yukich

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Abstract

We show that the random point measures induced by vertices in the convex hull of a Poisson sample on the unit ball, when properly scaled and centered, converge to those of a mean zero Gaussian field. We establish limiting variance and covariance asymptotics in terms of the density of the Poisson sample. Similar results hold for the point measures induced by the maximal points in a Poisson sample. The approach involves introducing a generalized spatial birth growth process allowing for cell overlap.

Article information

Source
Ann. Probab., Volume 36, Number 1 (2008), 363-396.

Dates
First available in Project Euclid: 28 November 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1196268683

Digital Object Identifier
doi:10.1214/009117907000000259

Mathematical Reviews number (MathSciNet)
MR2370608

Zentralblatt MATH identifier
1130.60031

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

Keywords
Convex hulls maximal points spatial birth growth processes Gaussian limits

Citation

Schreiber, T.; Yukich, J. E. Variance asymptotics and central limit theorems for generalized growth processes with applications to convex hulls and maximal points. Ann. Probab. 36 (2008), no. 1, 363--396. doi:10.1214/009117907000000259. https://projecteuclid.org/euclid.aop/1196268683


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