We study the contribution made by three or four points to certain areas associated with a typical polygon in a Voronoi tessellation of a planar Poisson process. We obtain some new results about moments and distributions and give simple proofs of some known results. We also use Robbins' formula to obtain the first three moments of the area of a typical polygon and hence the variance of the area of the polygon covering the origin.

Knowing the (geometric) covariogram of a convex body is equivalent to knowing, for each direction u, the distribution of the lengths of the chords of that body which are parallel to u. We prove that the covariogram determines convex polygons, among all convex bodies, up to translation and reflection. This gives a partial answer to a problem posed by Matheron.

The set covariance of a dead leaves model, constructed from hard spheres of constant diameter, is calculated analytically. The calculation is based on the covariance of a single sphere and on the pair correlation function of the centres of the spheres. There exist applications in the field of random sequential adsorption and in the interpretation of small-angle scattering experiments.

The main purpose of this work is to study and apply generalized contact distributions of (inhomogeneous) Boolean models Z with values in the extended convex ring. Given a convex body L ⊂ ℝd and a gauge body B ⊂ ℝd , such a generalized contact distribution is the conditional distribution of the random vector (d B (L,Z),u B (L,Z),p B (L,Z),l B (L,Z)) given that Z∩L = ∅, where Z is a Boolean model, d B (L,Z) is the distance of L from Z with respect to B, p B (L,Z) is the boundary point in L realizing this distance (if it exists uniquely), u B (L,Z) is the corresponding boundary point of B (if it exists uniquely) and l B (L,·) may be taken from a large class of locally defined functionals. In particular, we pursue the question of the extent to which the spatial density and the grain distribution underlying an inhomogeneous Boolean model Z are determined by the generalized contact distributions of Z.

Let U 1,U 2,… be a sequence of i.i.d. random vectors distributed uniformly in a compact plane region A of unit area. Sufficient conditions on the geometry of A are provided under which the Euclidean diameter D n of the first n of the points converges weakly upon suitable rescaling.

The estimation of a star-shaped set S from a random sample of points X 1,…,X n ∊ S is considered. We show that S can be consistently approximated (with respect to both the Hausdorff metric and the ‘distance in measure’ between sets) by an estimator ŝ n defined as a union of balls centered at the sample points with a common radius which can be chosen in such a way that ŝ n is also star-shaped. We also prove that, under some mild conditions, the topological boundary of the estimator ŝ n converges, in the Hausdorff sense, to that of S; this has a particular interest when the proposed estimation problem is considered from the point of view of statistical image analysis.

Siegmund and Worsley (1995) considered the problem of testing for signals with unknown location and scale in a Gaussian random field defined on ℝN . The test statistic was the maximum of a Gaussian random field in an N+1 dimensional ‘scale space’, N dimensions for location and 1 dimension for the scale of a smoothing filter. Scale space is identical to a continuous wavelet transform with a kernel smoother as the wavelet, though the emphasis here is on signal detection rather than image compression or enhancement. Two methods were used to derive an approximate null distribution for N=2 and N=3: one based on the method of volumes of tubes, the other based on the expected Euler characteristic of the excursion set. The purpose of this paper is two-fold: to show how the latter method can be extended to higher dimensions, and to apply this more general result to χ2 fields. The result of Siegmund and Worsley (1995) then follows as a special case. In this paper the results are applied to the problem of searching for activation in brain images obtained by functional magnetic resonance imaging (fMRI).

Seeds are randomly scattered in ℝd according to an m-dependent point process. Each seed has its own potential germination time. From each seed that succeeds in germinating, a spherical inhibited region grows to prohibit germination of any seed with later potential germination time. We show that under certain conditions on the distribution of the potential germination time, the number of germinated seeds in a large region has an asymptotic normal distribution.

Let n points be placed uniformly at random in a subset A of the plane. A point is said to be maximal in the configuration if no other point is larger in both coordinates. We show that, for large n and for many sets A, the number of maximal points is approximately normally distributed. The argument uses Stein's method, and is also applicable in higher dimensions.

A complete and sufficient statistic is found for various stationary Poisson processes of compact sets with known primary grain. In the particular case of a segment process, the uniformly best unbiased estimator for the length density is the number of segments hitting the sampling window divided by a certain constant and multiplied by the mean segment length.

This paper raises the following question: let {Φn (A), A ⊂ ℝd } be a Poisson process with intensity nf(x), x ∈ ℝd and let c(X i | Φn ) be a Voronoi tile with nucleus X i (a jump point of Φn ). Let μ(.) denote Lebesgue measure in ℝd . Is it true that, for any bounded measurable subset B of ℝd , ∑X i∈B μ(c(X i | Φn )) → μ(B) almost surely as n → ∞ only if f > 0 almost everywhere? This statement can be viewed as the strong law of large numbers for Voronoi tessellation. Though the positive answer may seem ‘obvious’, we could not find any such statement, especially for arbitrary measurable B and nonhomogeneous Poisson processes. For B with the boundary of Lebesgue measure 0 the proof is simple. We prove in this paper that the statement is true for ℝ1.

Products of independent identically distributed random stochastic 2 × 2 matrices are known to converge in distribution under a trivial condition. Rates for this convergence are estimated in terms of the minimal L p -metrics and the Kolmogoroff metric and applications to convergence rates of related interval splitting procedures are discussed.

A network is a system of segments or edges in ℝd which intersect only in the segment endpoints, which are called vertices. An example is the system of edges of a tessellation. It is possible to give formulas for the specific connectivity number of a random network; in the stationary case, the intensity of the 0-curvature measure is equal to the difference of the intensities of the point processes of vertices and edge centres.

The paper yields retrieval formulae of the directional distribution of a stationary k-flat process in ℝd if its rose of intersections with all r-flats is known. Cases k = d −1, 1 ≤ r ≤ d - 1 for arbitrary d and d = 4, k = 2, r = 2 are considered. Some generalizations to manifold processes in ℝd are made. The proofs use the methods of harmonic analysis on higher Grassmannians (spherical harmonics, integral transforms).

We use Jacobi field arguments and the contraction mapping theorem to locate Fréchet means of a class of probability measures on locally symmetric Riemannian manifolds with non-negative sectional curvatures. This leads, in particular, to a method for estimating Fréchet mean shapes, with respect to the distance function ρ determined by the induced Riemannian metric, of a class of probability measures on Kendall's shape spaces. We then combine this with the technique of ‘horizontally lifting’ to the pre-shape spheres to obtain an algorithm for finding Fréchet mean shapes, with respect to ρ, of a class of probability measures on Kendall's shape spaces in terms of the vertices of random shapes. This gives us, for example, an algorithm for finding Fréchet mean shapes of samples of configurations on the plane which is expressed directly in terms of the vertices.

Intrinsic volumes are key functionals in convex geometry and have also appeared in several stochastic settings. Here we relate them to questions of regularity in Gaussian processes with regard to Itô–Nisio oscillation and metrization of GB/GC indexing sets. Various bounds and estimates are presented, and questions for further investigation are suggested. From alternate points of view, much of the discussion can be interpreted in terms of (i) random sets and (ii) properties of (deterministic) infinite-dimensional convex bodies.

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.

We are interested in estimating the intensity parameter of a Boolean model of discs (the bombing model) from a single realization. To do so, we derive the conditional distribution of the points (germs) of the underlying Poisson process. We demonstrate how to apply coupling from the past to generate samples from this distribution, and use the samples thus obtained to approximate the maximum likelihood estimator of the intensity. We discuss and compare two methods: one based on a Monte Carlo approximation of the likelihood function, the other a stochastic version of the EM algorithm.

Two non-parametric methods for the estimation of the directional measure of stationary line and fibre processes in d-dimensional space are presented. The input data for both methods are intersection counts with finitely many test windows situated in hyperplanes. The first estimator is a measure valued maximum likelihood estimator, if applied to Poisson line processes. The second estimator uses an approximation of the associated zonoid (the Steiner compact) by zonotopes. Consistency of both estimators is proved (without use of the Poisson assumption). The estimation methods are compared empirically by simulation.

Consider the Boolean model in ℝ2, where the germs form a homogeneous Poisson point process with intensity λ and the grains are convex compact random sets. It is known (see, e.g., Cressie (1993, Section 9.5.3)) that Laslett's rule transforms the exposed tangent points of the Boolean model into a homogeneous Poisson process with the same intensity. In the present paper, we give a simple proof of this result, which is based on a martingale argument. We also consider the cumulative process of uncovered area in a vertical strip and show that a (linear) Poisson process with intensity λ can be embedded in it.