We consider two independent homogeneous Poisson processes Π0 and Π1 in the plane with intensities λ0 and λ1, respectively. We study additive functionals of the set of Π0-particles within a typical Voronoi Π1-cell. We find the first and the second moments of these variables as well as upper and lower bounds on their distribution functions, implying an exponential asymptotic behavior of their tails. Explicit formulae are given for the number and the sum of distances from Π0-particles to the nucleus within a typical Voronoi Π1-cell.

Grenander et al. (1991) proposed a conditional cyclic Gaussian Markov random field model for the edges of a closed outline in the plane. In this paper the model is recast as an improper cyclic Gaussian Markov random field for the vertices. The limiting behaviour of this model when the vertices become closely spaced is also described and in particular its relationship with the theory of ‘snakes' (Kass et al. 1987) is established. Applications are given in Grenander et al. (1991), Mardia et al. (1991) and Kent et al. (1992).

The problem addressed is to reverse the degradation which occurs when images are digitised: they are blurred, subjected to noise and rounding error, and sampled only at a lattice of points. Inference is considered for the fundamental case of binary scenes, binary data and isotropic blur. The inferential process is separable into two stages: first from the lattice points to a binary image in continuous space and then the reversal of thresholding and blur. Methods are motivated by, and illustrated using, an electron micrograph of an immunogold-labelled section of tulip virus.

The workshops on stochastic geometry, stereology and image analysis have been held every second year since 1981. The first workshops were small meetings, but with prominent speakers, who have attended most of the workshops since then. Over the years the workshops have increased in size and in impact too.

In design-based stereology, fixed parameters (such as volume, surface area, curve length, feature number, connectivity) of a non-random geometrical object are estimated by intersecting the object with randomly located and oriented geometrical probes (e.g. test slabs, planes, lines, points). Estimation accuracy may in principle be increased by increasing the number of probes, which are usually laid in a systematic pattern. An important prerequisite to increase accuracy, however, is that the relevant estimators are unbiased and consistent. The purpose of this paper is therefore to give sufficient conditions for the unbiasedness and strong consistency of design-based stereological estimators obtained by systematic sampling. Relevant mechanisms to increase sample size, compatible with stereological practice, are considered.

In machine vision, objects are observed subject to an unknown projective transformation, and it is usual to use projective invariants for either testing for a false alarm or for classifying an object. For four collinear points, the cross-ratio is the simplest statistic which is invariant under projective transformations. We obtain the distribution of the cross-ratio under the Gaussian error model with different means. The case of identical means, which has appeared previously in the literature, is derived as a particular case. Various alternative forms of the cross-ratio density are obtained, e.g. under the Casey arccos transformation, and under an arctan transformation from the real projective line of cross-ratios to the unit circle. The cross-ratio distributions are novel to the probability literature; surprisingly various types of Cauchy distribution appear. To gain some analytical insight into the distribution, a simple linear-ratio is also introduced. We also give some results for the projective invariants of five coplanar points. We discuss the general moment properties of the cross-ratio, and consider some inference problems, including maximum likelihood estimation of the parameters.

The object studied in this paper is a pair (Φ, Y), where Φ is a random surface in and Y a random vector field on . The pair is jointly stationary, i.e. its distribution is invariant under translations. The vector field Y is smooth outside Φ but may have discontinuities on Φ. Gauss' divergence theorem is applied to derive a flow conservation law for Y. For this specializes to a well-known rate conservation law for point processes. As an application, relationships for the linear contact distribution of Φ are derived.

In design-based stereology, fixed parameters (such as volume, surface area, curve length, feature number, connectivity) of a non-random geometrical object are estimated by intersecting the object with randomly located and oriented geometrical probes (e.g. test slabs, planes, lines, points), [4], [5], [8], [11], [12]. Estimation accuracy may in principle be increased by increasing the number of probes, which are usually laid in a systematic pattern, [1], [2], [3], [7], [9], [10]. An important prerequisite to increase accuracy, however, is that the relevant estimators are unbiased and consistent. Our purpose is therefore to give sufficient conditions for the unbiasedness and strong consistency of design-based stereological estimators obtained by systematic sampling.

A rectangular tessellation is a covering of the plane by non-overlapping rectangles. A basic theory for general homogeneous random rectangular tessellations is developed, and it is shown that many first-order mean values may be expressed in terms of just three basic quantities. Corresponding values for independent superpositions of two or more such tessellations are derived. The most interesting homogeneous rectangular tessellations are those with only T-vertices (i.e. no X-vertices). Gilbert's (1967) isotropic model adapted to this two-orthogonal-orientations case, although simply specified, appears theoretically intractable, due to a complex ‘blocking' effect. However, the approximating penetration model, also introduced by Gilbert, is found to be both tractable and informative about the true model. A multi-stage method for simulating the model is developed, and the distributions of important characteristics estimated.

Motivated by a statistical application, we consider continuum percolation in two or more dimensions, restricted to a large finite box, when above the critical point. We derive surface order large deviation estimates for the volume of the largest cluster and for its intersection with the boundary of the box. We also give some natural extensions to known, analogous results on lattice percolation.

Families of Poisson processes defined on general state spaces and with the intensity measure scaled by a positive parameter are investigated. In particular, mean value relations with respect to the scale parameter are established and used to derive various Gamma-type results for certain geometric characteristics determined by finite subprocesses. In particular, we deduce Miles' complementary theorem. Applications of the results within stochastic geometry and particularly for random tessellations are discussed.

We study the dispersion of a collection of particles carried by an isotropic Brownian flow in Of particular interest are the center of mass and the centered spatial second moments. Their asymptotic behavior depends strongly on the spatial dimension and the largest Lyapunov exponent of the flow. We use estimates for the pair separation process to give a fairly complete picture of this behavior as t → ∞. In particular, for incompressible flows in two dimensions, we show that the variance of the center of mass grows sublinearly, while dispersion relative to the center of mass grows linearly.

Let (Φ1 Φ2) be a bivariate point process. Let be the probability that (Φ1 Φ2) - (0, s) (the process without the 0 and s points) verify U when we have a point of <1>1 in the origin and a point of Φ2 in s. This is the reduced cross Palm distribution. Some correlation measures for bivariate point processes based on this reduced cross Palm distribution are proposed. Their estimators and expressions under the independence and the random labelling hypothesis are considered. The differences and improvements with respect to the cross intensity function and its integrated version, the cross function (Stoyan et al. 1987), are studied. Some Monte Carlo tests for testing the independence and the random labelling hypothesis are proposed. They are applied to real bivariate point patterns: positions of hickories and maples in the Lansing Woods (Diggle 1983) and cases and controls of childhood leukaemia and lymphoma in North Humberside (Cuzick and Edwards 1990).

Let θ (a) be the first time when the range (Rn; n ≧ 0) is equal to a, Rn being equal to the difference of the maximum and the minimum, taken at time n, of a simple random walk on ℤ. We compute the g.f. of θ (a); this allows us to compute the distributions of θ (a) and Rn. We also investigate the asymptotic behaviour of θ (n), n going to infinity.

Denote by A(x) = {a: |aτx| ≦ h} a circle zone on the three-dimensional sphere surface for each given h > 0. For a given integer m, we investigate how many zones chosen randomly are needed to contain at least one of the points on the sphere surface m times. As an application, the lifetime of a sphere roller is investigated. We present empirical formulas for the mean, standard deviation and distribution of the lifetime of the sphere roller. Furthermore, some limit behaviors of the above stopping time are obtained, such as the limit distribution, the law of the iterated logarithm, and the upper and lower bounds of the tail probability with the same convergent order.

The study of networks with random conductivities or resistances is of considerable interest in physics, biology, geology and mathematics. The talk concentrated on a rather flexible class of parametric models for trees with random conductivities, based on properties of the inverse Gaussian distribution. For finite trees some exact distributional results were presented and extensions of the models to infinite trees were discussed. Joint work with Gerard Letac and Angelo E. Koudou (Toulouse), extending Barndorff-Nielsen (1994).

We investigate the ‘clumping versus local finiteness' behavior in the infinite backward tree for a class of branching particle systems in ℝd with symmetric stable migration and critical ‘genuine multitype' branching. Under mild assumptions on the branching we establish, by analysing certain ergodic properties of the individual ancestral process, a critical dimension dc such that the (measure-valued) tree-top is almost surely locally finite if and only if d > dc. This result is used to obtain L1-norm asymptotics of a corresponding class of systems of non-linear partial differential equations.

In this paper we investigate operators that map one or more probability distributions on the positive real line into another via their Laplace–Stieltjes transforms. Our goal is to make it easier to construct new transforms by manipulating known transforms. We envision the results here assisting modelling in conjunction with numerical transform inversion software. We primarily focus on operators related to infinitely divisible distributions and Lévy processes, drawing upon Feller (1971). We give many concrete examples of infinitely divisible distributions. We consider a cumulant-moment-transfer operator that allows us to relate the cumulants of one distribution to the moments of another. We consider a power-mixture operator corresponding to an independently stopped Lévy process. The special case of exponential power mixtures is a continuous analog of geometric random sums. We introduce a further special case which is remarkably tractable, exponential mixtures of inverse Gaussian distributions (EMIGs). EMIGs arise naturally as approximations for busy periods in queues. We show that the steady-state waiting time in an M/G/1 queue is the difference of two EMIGs when the service-time distribution is an EMIG. We consider several transforms related to first-passage times, e.g. for the M/M/1 queue, reflected Brownian motion and Lévy processes. Some of the associated probability density functions involve Bessel functions and theta functions. We describe properties of the operators, including how they transform moments.

We establish a necessary condition for any importance sampling scheme to give bounded relative error when estimating a performance measure of a highly reliable Markovian system. Also, a class of importance sampling methods is defined for which we prove a necessary and sufficient condition for bounded relative error for the performance measure estimator. This class of probability measures includes all of the currently existing failure biasing methods in the literature. Similar conditions for derivative estimators are established.