Matthias Reitzner

Central limit theorems for

A

U

U

-statistics of Poisson point processes

-statistic of a Poisson point process is defined as the sum

Random points on the boundary of smooth convex bodies

\sum f(x_1,\ldots,x_k)

Large Poisson-Voronoi cells and Crofton cells

over all (possibly infinitely many)

Gaussian polytopes: variances and limit theorems

k

Poisson–Voronoi approximation

-tuples of distinct points of the point process. Using the Malliavin calculus, the Wiener-It\^{o} chaos expansion of such a functional is computed and used to derive a formula for the variance. Central limit theorems for

Limit theory for the Gilbert graph

U

Set reconstruction by Voronoi cells

-statistics of Poisson point processes are shown, with explicit bounds for the Wasserstein distance to a Gaussian random variable. As applications, the intersection process of Poisson hyperplanes and the length of a random geometric graph are investigated.

Random points in halfspheres

The convex hull of n independent random points chosen on the boundary of a convex body K C R d according to a given density function is a random polytope. The expectation of its i-th intrinsic volume for i = 1,..., d is investigated. In the case that the boundary of K is sufficiently smooth, asymptotic expansions for these expected intrinsic volumes as n → oo are derived.