arXiv subjects

Günter Last

Publications and source records attributed to Günter Last.

43 records · Page 3Linked to original sources

arXiv2014

Normal approximation on Poisson spaces: Mehler's formula, second order Poincaré inequalities and stabilization

We prove a new class of inequalities, yielding bounds for the normal approximation in the Wasserstein and the Kolmogorov distance of functionals of a general Poisson process (Poisson random measure). Our approach is based on an iteration of the classical Poincaré inequality, as well as on the use of Malliavin operators, of Stein's method, and of an (integrated) Mehler's formula, providing a representation of the Ornstein-Uhlenbeck semigroup in terms of thinned Poisson processes. Our estimates only involve first and second order differential operators, and have consequently a clear geometric interpretation. In particular we will show that our results are perfectly tailored to deal with the normal approximation of geometric functionals displaying a weak form of stabilization, and with non-linear functionals of Poisson shot-noise processes. We discuss two examples of stabilizing functionals in great detail: (i) the edge length of the $k$-nearest neighbour graph, (ii) intrinsic volumes of $k$-faces of Voronoi tessellations. In all these examples we obtain rates of convergence (in the Kolmogorov and the Wasserstein distance) that one can reasonably conjecture to be optimal, thus significantly improving previous findings in the literature. As a necessary step in our analysis, we also derive new lower bounds for variances of Poisson functionals.

math.PR↗

arXiv2013

Percolation on stationary tessellations: models, mean values and second order structure

We consider a stationary face-to-face tessellation $X$ of $\mathbb{R}^d$ and introduce several percolation models by colouring some of the faces black in a consistent way. Our main model is cell percolation, where cells are declared black with probability $p$ and white otherwise. We are interested in geometric properties of the union $Z$ of black faces. Under natural integrability assumptions we first express asymptotic mean-values of intrinsic volumes in terms of Palm expectations associated with the faces. In the second part of the paper we study asymptotic covariances of intrinsic volumes of $Z\cap W$, where the observation window $W$ is assumed to be a polytope. Here we need to assume the existence of suitable asymptotic covariances of the face processes of $X$. We check these assumptions in the important special case of a Poisson Voronoi tessellation. In the case of cell percolation on a normal tessellation, especially in the plane, our formulae simplify considerably.

math.PR↗

arXiv2013

On a class of growth-maximal hard-core processes

Generalizing the well-known lilypond model we introduce a growth-maximal hard-core model based on a space-time point process of convex particles. Using a purely deterministic algorithm we prove under fairly general assumptions that the model exists and is uniquely determined by the point process. Under an additional stationarity assumption we show that the model does not percolate. Our model generalizes the lilypond model considerably even if all grains are born at the same time. In that case and under a Poisson assumption we prove a central limit theorem in a large volume scenario.

math.PR↗

arXiv2013

Statistics for Poisson models of overlapping spheres

The paper considers the stationary Poisson Boolean model with spherical grains and proposes a family of nonparametric estimators for the radius distribution. These estimators are based on observed distances and radii, weighted in an appropriate way. They are ratio-unbiased and asymptotically consistent for growing observation window. It is shown that the asymptotic variance exists and is given by a fairly explicit integral expression. Asymptotic normality is established under a suitable integrability assumption on the weight function. The paper also provides a short discussion of related estimators as well as a simulation study.

math.PR↗

arXiv2010

On the convex hull of symmetric stable processes

Let alpha \in (1, 2] and X be an R^d-valued alpha-stable process with independent and symmetric components starting in 0. We consider the closure S_t of the path described by X on the interval [0, t] and its convex hull Z_t. The first result of this paper provides a formula for certain mean mixed volumes of Z_t and in particular for the expected first intrinsic volume of Z_t. The second result deals with the asymptotics of the expected volume of the stable sausage Z_t+B (where B is an arbitrary convex body with interior points) as t \to 0.

math.PR↗

arXiv2009

Palm pairs and the general mass-transport principle

We consider a lcsc group G acting properly on a Borel space S and measurably on an underlying sigma-finite measure space. Our first main result is a transport formula connecting the Palm pairs of jointly stationary random measures on S. A key (and new) technical result is a measurable disintegration of the Haar measure on G along the orbits. The second main result is an intrinsic characterization of the Palm pairs of a G-invariant random measure. We then proceed with deriving a general version of the mass-transport principle for possibly non-transitive and non-unimodular group operations first in a deterministic and then in its full probabilistic form.

math.PR↗

arXiv2009

Invariant transports of stationary random measures and mass-stationarity

We introduce and study invariant (weighted) transport-kernels balancing stationary random measures on a locally compact Abelian group. The first main result is an associated fundamental invariance property of Palm measures, derived from a generalization of Neveu's exchange formula. The second main result is a simple sufficient and necessary criterion for the existence of balancing invariant transport-kernels. We then introduce (in a nonstationary setting) the concept of mass-stationarity with respect to a random measure, formalizing the intuitive idea that the origin is a typical location in the mass. The third main result of the paper is that a measure is a Palm measure if and only if it is mass-stationary.

math.PR↗