Recently Propp and Wilson [14] have proposed an algorithm, called coupling from the past (CFTP), which allows not only an approximate but perfect (i.e. exact) simulation of the stationary distribution of certain finite state space Markov chains. Perfect sampling using CFTP has been successfully extended to the context of point processes by, amongst other authors, Häggström et al. [5]. In [5] Gibbs sampling is applied to a bivariate point process, the penetrable spheres mixture model [19]. However, in general the running time of CFTP in terms of number of transitions is not independent of the state sampled. Thus an impatient user who aborts long runs may introduce a subtle bias, the user impatience bias. Fill [3] introduced an exact sampling algorithm for finite state space Markov chains which, in contrast to CFTP, is unbiased for user impatience. Fill's algorithm is a form of rejection sampling and similarly to CFTP requires sufficient monotonicity properties of the transition kernel used. We show how Fill's version of rejection sampling can be extended to an infinite state space context to produce an exact sample of the penetrable spheres mixture process and related models. Following [5] we use Gibbs sampling and make use of the partial order of the mixture model state space. Thus we construct an algorithm which protects against bias caused by user impatience and which delivers samples not only of the mixture model but also of the attractive area-interaction and the continuum random-cluster process.

In keeping with the intersection density of a stationary Poisson process of r-flats in Euclidean d-space, where r ≥ d/2, we introduce a notion of closeness, called proximity, for such processes if r < d/2. It is shown that the two notions are connected by a duality: the proximity of a stationary Poisson r-flat process is, up to a constant factor, the intersection density of a certain unique stationary Poisson (d − r)-flat process.

The no-aging property and the ℓ1-isotropic model it implies have been introduced to overcome certain shortcomings of the exponential model. However, its definition is abstract and not very useful for practitioners. This paper presents several additional characterizations of the no-aging property. Included are (1) characterizations that appropriately generalize the memoryless property and the constant-failure-rate property of the exponential, (2) behavioral characterizations based on fair bets, and (3) geometric characterizations of the survival and density function and differential-geometric characterizations based on tensor methods.

In design stereology, and in the context of geometric sampling in general, the problem often arises of estimating the integral of a bounded non-random function over a bounded manifold D ⊂ ℝn by systematic sampling with geometric probes. Variance predictors, often based on Matheron's theory of regionalized variables, are available when the relevant function is sampled at the points of a grid intersecting D, but not when the dimension of the probes is greater than zero. For instance, the volume of a bounded object may be estimated using parallel systematic planes, which amounts to sampling on ℝ1 with systematic points, or using parallel systematic slabs of thickness t > 0, which amounts to sampling on ℝ1 with non-overlapping systematic segments of length t > 0. Useful variance predictors exist for the former case, but not for the latter. In this paper we set out a general scheme to predict estimation variances when the dimension of either D, or of the probes, is n. We make some progress when both dimensions are equal to n, and obtain explicit results for n = 1 (e.g. for systematic slice sampling). We check and illustrate our results for the volume estimators of ellipsoids and of rat lung.

We study one-dimensional continuous loss networks with length distribution G and cable capacity C. We prove that the unique stationary distribution ηL of the network for which the restriction on the number of calls to be less than C is imposed only in the segment [−L,L] is the same as the distribution of a stationary M/G/∞ queue conditioned to be less than C in the time interval [−L,L]. For distributions G which are of phase type (= absorbing times of finite state Markov processes) we show that the limit as L → ∞ of ηL exists and is unique. The limiting distribution turns out to be invariant for the infinite loss network. This was conjectured by Kelly (1991).

An unbiased stereological estimator for surface area density is derived for gradient surface processes which form a particular class of non-stationary spatial surface processes. Vertical planar sections are used for the estimation. The variance of the estimator is studied and found to be infinite for certain types of surface processes. A modification of the estimator is presented which exhibits finite variance.

Mixings of stationary Poisson hyperplane tessellations in d-dimensional Euclidean space are considered. The intention of the paper is to show that the 0-cell of a mixed stationary Poisson hyperplane tessellation Y is in some sense larger than that of stationary Poisson hyperplane tessellations Y' with the same intensity and directional distribution as Y. Related results concerning the moments for the volume of the 0-cell are derived. In special cases, similar statements with respect to the typical cell are proved.

We give formulae for different types of contact distribution functions for stationary (not necessarily Poisson) Voronoi tessellations in ℝd in terms of the Palm void probabilities of the generating point process. Moreover, using the well-known relationship between the linear contact distribution and the chord length distribution we derive a closed form expression for the mean chord length in terms of the two-point Palm distribution and the pair correlation function of the generating point process. The results obtained are specified for Voronoi tessellations generated by Poisson cluster and Gibbsian processes, respectively.

In a real n-1 dimensional affine space E, consider a tetrahedron T 0, i.e. the convex hull of n points α1, α2, …, αn of E. Choose n independent points β1, β2, …, βn randomly and uniformly in T 0, thus obtaining a new tetrahedron T 1 contained in T 0. Repeat the operation with T 1 instead of T 0, obtaining T 2, and so on. The sequence of the T k shrinks to a point Y of T 0 and this note computes the distribution of the barycentric coordinates of Y with respect to (α1, α2, …, αn ) (Corollary 2.3). We also obtain the explicit distribution of Y in more general cases. The technique used is to reduce the problem to the study of a random walk on the semigroup of stochastic (n,n) matrices, and this note is a geometrical application of a former result of Chamayou and Letac (1994).

Let n points be placed independently in ν-dimensional space according to the standard ν-dimensional normal distribution. Let M n be the longest edge-length of the minimal spanning tree on these points; equivalently let M n be the infimum of those r such that the union of balls of radius r/2 centred at the points is connected. We show that the distribution of (2 log n)1/2M n - b n converges weakly to the Gumbel (double exponential) distribution, where b n are explicit constants with b n ~ (ν - 1)log log n. We also show the same result holds if M n is the longest edge-length for the nearest neighbour graph on the points.

We prove central limit theorems for certain geometrical characteristics of the convex polygons determined by a standard Poisson line process in the plane, such as: the angles at the vertices of the polygons, the empirical mean of the number of vertices and the empirical mean of the perimeter of the polygons.

A stochastic dynamical context is developed for Bookstein's shape theory. It is shown how Bookstein's shape space for planar triangles arises naturally when the landmarks are moved around by a special Brownian motion on the general linear group of invertible (2×2) real matrices. Asymptotics for the Brownian transition density are used to suggest an exponential family of distributions, which is analogous to the von Mises-Fisher spherical distribution and which has already been studied by J. K. Jensen. The computer algebra implementation Itovsn3 (W. S. Kendall) of stochastic calculus is used to perform the calculations (some of which actually date back to work by Dyson on eigenvalues of random matrices and by Dynkin on Brownian motion on ellipsoids). An interesting feature of these calculations is that they include the first application (to the author's knowledge) of the Gröbner basis algorithm in a stochastic calculus context.

Let us consider, in the Euclidean space E n , a fixed n-dimensional convex body K 0 of volume V 0 and a system K 1,…,K m of mn-dimensional convex bodies, congruent to a convex set K. Assume that the sets K i (i = 1,…,m) have random positions, being stochastically independent and uniformly distributed on a limited domain of E n and denote by V m the volume of the convex body K m = K 0 ∩ (K 1 ∩ … ∩ K m ). The aim of this paper is the evaluation of the second moment of the random variable V m .

An asymptotic expression for the expected area of the union of n random rectangles is derived by Mellin transforms, where their two diagonal corners are independently and uniformly distributed over (0,1)2. The general result for d-dimensional hyper-rectangles is also stated.

The problem of estimating an unknown compact convex set K in the plane, from a sample (X 1,···,X n ) of points independently and uniformly distributed over K, is considered. Let K n be the convex hull of the sample, Δ be the Hausdorff distance, and Δn := Δ (K, K n ). Under mild conditions, limit laws for Δn are obtained. We find sequences (a n ), (b n ) such that

(Δn - b n )/a n → Λ (n → ∞), where Λ is the Gumbel (double-exponential) law from extreme-value theory. As expected, the directions of maximum curvature play a decisive role. Our results apply, for instance, to discs and to the interiors of ellipses, although for eccentricity e < 1 the first case cannot be obtained from the second by continuity. The polygonal case is also considered.

A new class of Gibbsian models with potentials associated with the connected components or homogeneous parts of images is introduced. For these models the neighbourhood of a pixel is not fixed as for Markov random fields, but is given by the components which are adjacent to the pixel. The relationship to Markov random fields and marked point processes is explored and spatial Markov properties are established. Extensions to infinite lattices are also studied, and statistical inference problems including geostatistical applications and statistical image analysis are discussed. Finally, simulation studies are presented which show that the models may be appropriate for a variety of interesting patterns, including images exhibiting intermediate degrees of spatial continuity and images of objects against background.

We discuss the uniqueness of the Fréchet mean of a class of distributions on the shape space of k labelled points in ℝ2, the supports of which could be the entire space. From this it follows that the shape of the means is the unique Fréchet mean shape of the induced distribution with respect to an appropriate metric structure, provided the distribution of k labelled points in ℝ2 is isotropic and satisfies a further mild condition. This result implies that an increasing sequence of procrustean mean shapes defined in either of the two ways used in practice will tend almost surely to the shape of the means.

We consider a stationary germ-grain model Ξ with convex and compact grains and the distance r(x) from x ε ℝd to Ξ. For almost all points x ε ℝd there exists a unique point p(x) in the boundary of Ξ such that r(x) is the length of the vector x-p(x), which is called the spherical contact vector at x. In this paper we relate the distribution of the spherical contact vector to the times it takes a typical boundary point of Ξ to hit another grain if all grains start growing at the same time and at the same speed. The notion of a typical point is made precise by using the generalized curvature measures of Ξ. The result generalizes a well known formula for the Boolean model. Specific examples are discussed in detail.

For a stationary point process X of sets in the convex ring in ℝd , a relation is given between the mean particles of the section process X ∩ E (where E varies through the set of k-dimensional subspaces in ℝd ) and a mean particle of X. In particular, it is shown that the mean bodies of all planar sections of X determine the Blaschke body of X and hence the mean normal distribution of X.

Consider a continuum percolation model in which, at each point of a d-dimensional Poisson process of rate λ, a ball of radius 1 is centred. We show that, for any d ≧ 3, there exists a phase where both the regions, occupied and vacant, contain unbounded components. The proof uses the concept of enhancement for the Boolean model, and along the way we prove that the critical intensity of a Boolean model defined on a slab is strictly larger than the critical intensity of a Boolean model defined on the whole space.