Open Access
January 2013 Brownian limits, local limits and variance asymptotics for convex hulls in the ball
Pierre Calka, Tomasz Schreiber, J. E. Yukich
Ann. Probab. 41(1): 50-108 (January 2013). DOI: 10.1214/11-AOP707


Schreiber and Yukich [Ann. Probab. 36 (2008) 363–396] establish an asymptotic representation for random convex polytope geometry in the unit ball $\mathbb{B}^{d}$, $d\geq 2$, 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.


Download Citation

Pierre Calka. Tomasz Schreiber. J. E. Yukich. "Brownian limits, local limits and variance asymptotics for convex hulls in the ball." Ann. Probab. 41 (1) 50 - 108, January 2013.


Published: January 2013
First available in Project Euclid: 23 January 2013

zbMATH: 1278.60020
MathSciNet: MR3059193
Digital Object Identifier: 10.1214/11-AOP707

Primary: 60F05
Secondary: 60D05

Keywords: Brownian sheets , Functionals of random convex hulls , paraboloid growth and hull processes , stabilization

Rights: Copyright © 2013 Institute of Mathematical Statistics

Vol.41 • No. 1 • January 2013
Back to Top