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

Pierre Calka (1) , Tomasz Schreiber (2) , J. E. Yukich (3)

1 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5
2 Faculty of Mathematics and Computer Science
3 Department of Mathematics [Bethlehem, USA]

The paper of Schreiber and Yukich [40] establishes an asymptotic representation for random convex polytope geometry in the unit ball $\B_d,\; d \geq 2,$ in terms of the general theory of stabilizing functionals of Poisson point processes as well as in terms of the so-called generalized paraboloid growth process. This paper further exploits this connection, introducing also a dual object termed the paraboloid hull process. Via these growth processes we establish local functional and measure-level limit theorems for the properly scaled radius-vector and support functions as well as for curvature measures and $k$-face empirical measures of convex polytopes generated by high density Poisson samples. We use general techniques of stabilization theory to establish Brownian sheet limits for the defect volume and mean width functionals, and we provide explicit variance asymptotics and central limit theorems for the $k$-face and intrinsic volume functionals. We establish extreme value theorems for radius-vector and support functions of random polytopes and we also establish versions of the afore-mentioned results for large isotropic cells of hyperplane tessellations, reducing the study of their asymptotic geometry to that of convex polytopes via inversion-based duality relations, as in Calka and Schreiber [14].

Domains: Probability [math.PR]

Pierre Calka, Tomasz Schreiber, J. E. Yukich. Brownian limits, local limits, extreme value and variance asymptotics for convex hulls in the ball. 2009. ⟨hal-00442653⟩