Brownian limits, local limits and variance asymptotics for convex hulls in the ball

Abstract

Schreiber and Yukich [Ann. Probab. 36 (2008) 363-396] establish an asymptotic representation for random convex polytope geometry in the unit ball ${\mathbb{B}}^d,d\ge 2$, in terms of the general theory of stabilizing functionals of Poisson point processes as well as in terms of generalized paraboloid growth processes. This paper further exploits this connection, introducing also a dual object termed the paraboloid hull process. Via these growth processes we establish local functional limit theorems for the properly scaled radius-vector and support functions of convex polytopes generated by high-density Poisson samples. We show that direct methods lead to explicit asymptotic expressions for the fidis of the properly scaled radius-vector and support functions. Generalized paraboloid growth processes, coupled with general techniques of stabilization theory, yield Brownian sheet limits for the defect volume and mean width functionals. Finally we provide explicit variance asymptotics and central limit theorems for the k-face and intrinsic volume functionals.

Published in at http://dx.doi.org/10.1214/11-AOP707 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (8)

• Normal approximation under local dependence
• Gaussian limits for random measures in geometric probability
• Laws of large numbers in stochastic geometry with statistical applications
• Variance asymptotics and central limit theorems for generalized growth processes with applications to convex hulls and maximal points
• Poisson polytopes
• Central limit theorems for random polygons in an arbitrary convex set
• New Monte Carlo method for planar Poisson-Voronoi cells
• Intrinsic volumes of inscribed random polytopes in smooth convex bodies

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• Fluctuations of lattice zonotopes and polygons
• The $\beta$-Delaunay tessellation III: Kendall's problem and limit theorems in high dimensions
• The Widom-Rowlinson model: Mesoscopic fluctuations for the critical droplet
• Universal Scaling Limits for Generalized Gamma Polytopes
• Random polytopes: central limit theorems for intrinsic volumes
• Convex hulls of perturbed random point sets