We consider an extension of the Poisson hail model where the service speed is either 0 or ∞ at each point of the Euclidean space. We use and develop tools pertaining to sub-additive ergodic theory in order to establish shape theorems for the growth of the ice-heap under light tail assumptions on the hailstone characteristics. The asymptotic shape depends on the statistics of the hailstones, the intensity of the underlying Poisson point process, and on the geometrical properties of the zero speed set.

We consider a queue where the server is the Euclidean space, and the customers are random closed sets (RACSs) of the Euclidean space. These RACSs arrive according to a Poisson rain and each of them has a random service time (in the case of hail falling on the Euclidean plane, this is the height of the hailstone, whereas the RACS is its footprint). The Euclidean space serves customers at speed 1. The service discipline is a hard exclusion rule: no two intersecting RACSs can be served simultaneously and service is in the first-in–first-out order, i.e. only the hailstones in contact with the ground melt at speed 1, whereas the others are queued. A tagged RACS waits until all RACSs that arrived before it and intersecting it have fully melted before starting its own melting. We give the evolution equations for this queue. We prove that it is stable for a sufficiently small arrival intensity, provided that the typical diameter of the RACS and the typical service time have finite exponential moments. We also discuss the percolation properties of the stationary regime of the RACS in the queue.

Motivated by the well-posedness of birth-and-growth processes, a stochastic geometric differential equation and, hence, a stochastic geometric dynamical system are proposed. In fact, a birth-and-growth process can be rigorously modeled as a suitable combination, involving the Minkowski sum and the Aumann integral, of two very general set-valued processes representing nucleation and growth dynamics, respectively. The simplicity of the proposed geometric approach allows to avoid problems of boundary regularities arising from an analytical definition of the front growth. In this framework, growth is generally anisotropic and, according to a mesoscale point of view, is non local, i.e. at a fixed time instant, growth is the same at each point of the space.

We study a flexible class of finite-disc process models with interaction between the discs. We let 𝒰 denote the random set given by the union of discs, and use for the disc process an exponential family density with the canonical sufficient statistic depending only on geometric properties of 𝒰 such as the area, perimeter, Euler-Poincaré characteristic, and the number of holes. This includes the quermass-interaction process and the continuum random-cluster model as special cases. Viewing our model as a connected component Markov point process, and thereby establishing local and spatial Markov properties, becomes useful for handling the problem of edge effects when only 𝒰 is observed within a bounded observation window. The power tessellation and its dual graph become major tools when establishing inclusion-exclusion formulae, formulae for computing geometric characteristics of 𝒰, and stability properties of the underlying disc process density. Algorithms for constructing the power tessellation of 𝒰 and for simulating the disc process are discussed, and the software is made public available.

Conditions are derived for the asymptotic normality of a general class of vector-valued functionals of stationary Boolean models in the d-dimensional Euclidean space, where a Lindeberg-type central limit theorem for m-dependent random fields, m ∈ N, is applied. These functionals can be used to construct joint estimators for the vector of specific intrinsic volumes of the underlying Boolean model. Extensions to functionals of more general germ–grain models satisfying some mixing and integrability conditions are also discussed.

The covariance C(r), r ≥ 0, of a stationary isotropic random closed set Ξ is typically complicated to evaluate. This is the reason that an exponential approximation formula for C(r) has been widely used in the literature, which matches C(0) and C(1)(0), and in many cases also limr→∞C(r). However, for 0 < r < ∞, the accuracy of this approximation is not very high in general. In the present paper, we derive representation formulae for the covariance C(r) and its derivative C(1)(r) using Palm calculus, where r ≥ 0 is arbitrary. As a consequence, an explicit expression is obtained for the second derivative C(2)(0). These results are then used to get a refined exponential approximation for C(r), which additionally matches the second derivative C(2)(0).

We define and analyse a random coverage process of the d-dimensional Euclidean space which allows us to describe a continuous spectrum that ranges from the Boolean model to the Poisson–Voronoi tessellation to the Johnson–Mehl model. As for the Boolean model, the minimal stochastic setting consists of a Poisson point process on this Euclidean space and a sequence of real valued random variables considered as marks of this point process. In this coverage process, the cell attached to a point is defined as the region of the space where the effect of the mark of this point exceeds an affine function of the cumulative effect of all marks. This cumulative effect is defined as the shot-noise process associated with the marked point process. In addition to analysing and visualizing this spectrum, we study various basic properties of the coverage process such as the probability that a point or a pair of points be covered by a typical cell. We also determine the distribution of the number of cells which cover a given point, and show how to provide deterministic bounds on this number. Finally, we also analyse convergence properties of the coverage process using the framework of closed sets, and its differentiability properties using perturbation analysis. Our results require a pathwise continuity property for the shot-noise process for which we provide sufficient conditions. The model in question stems from wireless communications where several antennas share the same (or different but interfering) channel(s). In this case, the area where the signal of a given antenna can be received is the area where the signal to interference ratio is large enough. We describe this class of problems in detail in the paper. The results obtained allow us to compute quantities of practical interest within this setting: for instance the outage probability is obtained as the complement of the volume fraction; the law of the number of cells covering a point allows us to characterize handover strategies, and so on.

For a random closed set X and a compact observation window W the mean coverage fraction of W can be estimated by measuring the area of W covered by X. Jensen and Gundersen, and Baddeley and Cruz-Orive described cases where a point counting estimator is more efficient than area measurement. We give two other examples, where at first glance unnatural estimators are not only better than the area measurement but by Grenander's Theorem have minimal variance. Whittle's Theorem is used to show that the point counting estimator in the original Jensen-Gundersen paradox is optimal for large randomly translated discs.

Non-parametric estimators of the distribution of the grain of the Boolean model are considered. The technique is based on the study of point processes of tangent points in different directions related to the Boolean model. Their second- and higher-order characteristics are used to estimate the mean body and the distribution of the typical grain. Central limit theorems for the improved estimator of the intensity and surface measures of the Boolean model are also proved.

In this paper, we develop the probabilistic foundations of the dynamic continuous choice problem. The underlying choice set is a compact metric space E such as the unit interval or the unit square. At each time point t, utilities for alternatives are given by a random function . To achieve a model of dynamic continuous choice, the theory of classical vector-valued extremal processes is extended to super-extremal processesY= {Yt, t > 0}. At any t > 0, Yt is a random upper semicontinuous function on a locally compact, separable, metric space E. General path properties of Y are discussed and it is shown that Y is Markov with state-space US(E). For each t > 0, Yt is associated.

For a compact metric E, we consider the argmax process M = {Mt, t > 0}, where . In the dynamic continuous choice application, the argmax process M represents the evolution of the set of random utility maximizing alternatives. M is a closed set-valued random process, and its path properties are investigated.

Random closed sets (in Matheron's sense) which are a.s. compact convex and contain the origin are considered. The totality of such random closed sets are closed under the Minkowski addition and we can define the concept of infinite divisibility with respect to Minkowski addition of random compact convex sets. Using a generalized notion of Laplace transformations we get Lévy-type canonical representations of infinitely divisible random compact convex sets. Isotropic and stable cases are also considered. Finally we get several mean formulas of Minkowski functionals of infinitely divisible random compact convex sets in terms of their Lévy spectral measures.