Wilfrid S. Kendall

Bias in measurement of fission-track length distributions

Abstract Problems in the measurement of fission-track length distributions include biases in various methods of sampling and the amount of information about the true lenght distribution that can be recovered. It is concluded that all length measurements are biased and that this bias must be corrected before meaningful geological interpretations can be made. It is recommended that track-length measurements in minerals be restricted to horizontal confined fission tracks, because the length bias is then simple and easy to correct. Projected track-length measurements are not recommended because of complicated bias and insensitivity to important features of the true length distribution. These points are illustrated by length measurements on two Australian apatite samples.

Perfect simulation using dominating processes on ordered spaces, with application to locally stable point processes

In this paper we investigate the application of perfect simulation, in particular Coupling from the Past (CFTP), to the simulation of random point processes. We give a general formulation of the method of dominated CFTP and apply it to the problem of perfect simulation of general locally stable point processes as equilibrium distributions of spatial birth-and-death processes. We then investigate discrete-time Metropolis-Hastings samplers for point processes, and show how a variant which samples systematically from cells can be converted into a perfect version. An application is given to the Strauss point process.

Networks and Chaos - Statistical and Probabilistic Aspects

Mathematical methods of neurocomputing.- Statistical aspects of neural networks.- Statistical aspects of chaos: a review.- Chaotic dynamical systems with a View towards statistics: a review.- A tutorial on queuing networks.- River networks: a brief guide to the literature for statisticians and probabilists.- Random graphical networks.

Perfect Simulation for the Area-Interaction Point Process

Because so many random processes arising in stochastic geometry are quite intractable to analysis, simulation is an important part of the stochastic geometry toolkit. Typically, a Markov point process such as the area-interaction point process is simulated (approximately) as the long-run equilibrium distribution of a (usually reversible) Markov chain such as a spatial birth-and-death process. This is a useful method, but it can be very hard to be precise about the length of simulation required to ensure that the long-run approximation is good. The splendid idea of Propp and Wilson [17] suggests a way forward: they propose a coupling method which delivers exact simulation of equilibrium distributions of (finite-state-space) Markov chains. In this paper their idea is extended to deal with perfect simulation of attractive area-interaction point processes in bounded windows. A simple modification of the basic algorithm is described which provides perfect simulation of the repulsive case as well (which being nonmonotonic might have been thought out of reach). Results from simulations using a C computer program are reported; these confirm the practicality of this approach in both attractive and repulsive cases. The paper concludes by mentioning other point processes which can be simulated perfectly in this way, and by speculating on useful future directions of research. Clearly workers in stochastic geometry should now seek wherever possible to incorporate the Propp and Wilson idea in their simulation algorithms.

Nonnegative ricci curvature and the brownian coupling property

This paper shows that if M is a complete Riemannian manifold with Ricci curvatures all nonnegative than M has the Brownian coupling property. From this one may immediately draw deductions concerning the nonexistence of certain harmonic maps

Perfect simulation in stochastic geometry

Simulation plays an important role in stochastic geometry and related fields, because all but the simplest random set models tend to be intractable to analysis. Many simulation algorithms deliver (approximate) samples of such random set models, for example by simulating the equilibrium distribution of a Markov chain such as a spatial birth-and-death process. The samples usually fail to be exact because the algorithm simulates the Markov chain for a long but finite time, and thus convergence to equilibrium is only approximate. The seminal work by Propp and Wilson made an important contribution to simulation by proposing a coupling method, coupling from the past (CFTP), which delivers perfect, that is to say exact, simulations of Markov chains. In this paper we introduce this new idea of perfect simulation and illustrate it using two common models in stochastic geometry: the dead leaves model and a Boolean model conditioned to cover a finite set of points.

SIMULATION OF CLUSTER POINT PROCESSES WITHOUT EDGE EFFECTS

The usual direct method of simulation for cluster processes requires the generation of the parent point process over a region larger than the actual observation window, since we have to allow for all possible parents giving rise to observed daughter points, and some of these parents may fall outwith the observation window. When there is no a priori bound on the distance between parent and child then we have to take care to control approximations arising from edge effects. In this paper, we present a simulation method which requires simulation only of those parent points actually giving rise to observed daughter points, thus avoiding edge effect approximation. The idea is to replace the cluster distribution by one which is conditioned to plant at least one daughter point in the observation window, and to modify the parent process to have an inhomogeneous intensity exactly balancing the effect of the conditioning. We furthermore show how the method extends to cases involving infinitely many potential parents, for example gamma-Poisson processes and shot-noise G-Cox processes, allowing us to avoid approximation due to truncation of the parent process.

Contours of Brownian processes with several-dimensional times

SummaryTwo generalisations of Brownian motion to several-dimensional time are considered and the topology of their level sets is analysed. It is shown that for these maps non-trivial contours are quite rare — their union has Lebesgue measure zero. The boundedness of all contours is established for the generalisation due to Lévy. For the other, the Brownian sheet, a partial result concerning the behaviour of the zero contour near the boundary is established.

Short-length routes in low-cost networks via Poisson line patterns

In designing a network to link n points in a square of area n, we might be guided by the following two desiderata. First, the total network length should not be much greater than the length of the shortest network connecting all points. Second, the average route length (taken over source-destination pairs) should not be much greater than the average straight-line distance. How small can we make these two excesses? Speaking loosely, for a nondegenerate configuration, the total network length must be at least of order n and the average straight-line distance must be at least of order n 1/2, so it seems implausible that a single network might exist in which the excess over the first minimum is o(n) and the excess over the second minimum is o(n 1/2). But in fact we can do better: for an arbitrary configuration, we can construct a network where the first excess is o(n) and the second excess is almost as small as O(log n). The construction is conceptually simple and uses stochastic methods: over the minimum-length connected network (Steiner tree) superimpose a sparse stationary and isotropic Poisson line process. Together with a few additions (required for technical reasons), the mean values of the excess for the resulting random network satisfy the above asymptotics; hence, a standard application of the probabilistic method guarantees the existence of deterministic networks as required (speaking constructively, such networks can be constructed using simple rejection sampling). The key ingredient is a new result about the Poisson line process. Consider two points a distance r apart, and delete from the line process all lines which separate these two points. The resulting pattern of lines partitions the plane into cells; the cell containing the two points has mean boundary length approximately equal to 2r + constant(log r). Turning to lower bounds, consider a sequence of networks in satisfying a weak equidistribution assumption. We show that if the first excess is O(n) then the second excess cannot be

New Directions in Dirichlet Forms

Nonlinear Dirichlet forms by J. Jost From stochastic parallel transport to harmonic maps by W. S. Kendall Dirichlet forms and self-similarity by U. Mosco Stochastic analysis on configuration spaces: Basic ideas and recent results by M. Rockner The geometric aspect of Dirichlet forms by K.-T. Sturm.