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 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.

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.

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.

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.

When analyzing the equilibrium behavior of M/G/1 type Markov chains by transform methods, restrictive hypotheses are often made to avoid technical problems that arise in applying results from complex analysis and linear algebra. It is shown that such restrictive assumptions are unnecessary, and an analysis of these chains using generating functions is given under only the natural hypotheses that first moments (or second moments in the null recurrent case) exist. The key to the analysis is the identification of an important subspace of the space of bounded solutions of the system of homogeneous vector-valued Wiener–Hopf equations associated with the chain. In particular, the linear equations in the boundary probabilities obtained from the transform method are shown to correspond to a spectral basis of the shift operator on this subspace. Necessary and sufficient conditions under which the chain is ergodic, null recurrent or transient are derived in terms of properties of the matrix-valued generating functions determined by transitions of the Markov chain. In the transient case, the Martin exit boundary is identified and shown to be associated with certain eigenvalues and vectors of one of these generating functions. An equilibrium analysis of the class of G/M/1 type Markov chains by similar methods is also presented.

We consider the application of importance sampling in steady-state simulations of finite Markov chains. We show that, for a large class of performance measures, there is a choice of the alternative transition matrix for which the ratio of the variance of the importance sampling estimator to the variance of the naive simulation estimator converges to zero as the sample path length goes to infinity. Obtaining this ‘optimal’ transition matrix involves computing the performance measure of interest, so the optimal matrix cannot be computed in precisely those situations where simulation is required to estimate steady-state performance. However, our results show that alternative transition matrices of the form Q = P + E/T, where P is the original transition matrix and T is the sample path length, can be expected to provide good results. Moreover, we provide an iterative algorithm for obtaining alternative transition matrices of this form that converge to the optimal matrix as the number of iterations increases, and present an example that shows that spending some computer time iterating this algorithm and then conducting the simulation with the resulting alternative transition matrix may provide considerable variance reduction when compared to naive simulation.

A homogeneous Gaussian Markov lattice-process model has a regression coefficient that determines the extent to which a random variable of a vertex is dependent on those of the neighbors. In many studies, the absolute value of this parameter has been assumed to be less than the reciprocal of the number of neighbors. This condition is shown to be necessary and sufficient for the existence of the Gaussian process satisfying the model equations under some assumptions on lattices using the notion of dual processes. We also give examples of models that neither satisfy the condition imposed on the region for the parameter nor the assumptions on lattices. A formula for autocovariance functions of Gaussian Markov processes on general lattices is derived, and numerical procedures to calculate the autocovariance functions are proposed.

We consider a real-valued random walk S which drifts to –∞ and is such that E(exp θS1) < ∞ for some θ > 0, but for which Cramér's condition fails. We investigate the asymptotic tail behaviour of the distributions of the all time maximum, the upwards and downwards first passage times and the last passage times. As an application, we obtain new limit theorems for certain conditional laws.

A general model for the evolution of the frequency distribution of types in a population under mutation and selection is derived and investigated. The approach is sufficiently general to subsume classical models with a finite number of alleles, as well as models with a continuum of possible alleles as used in quantitative genetics. The dynamics of the corresponding probability distributions is governed by an integro-differential equation in the Banach space of Borel measures on a locally compact space. Existence and uniqueness of the solutions of the initial value problem is proved using basic semigroup theory. A complete characterization of the structure of stationary distributions is presented. Then, existence and uniqueness of stationary distributions is proved under mild conditions by applying operator theoretic generalizations of Perron–Frobenius theory. For an extension of Kingman's original house-of-cards model, a classification of possible stationary distributions is obtained.

We investigate the asymptotic behaviour of the distribution of the number of comparisons needed by a quicksort-style selection algorithm that finds the lth smallest in a set of n numbers. Letting n tend to infinity and considering the values l = 1, ···,n simultaneously we obtain a limiting stochastic process. This process admits various interpretations: it arises in connection with a representation of real numbers induced by nested random partitions and also in connection with expected path lengths of a random walk in a random environment on a binary tree.

The series expansion for the solution of the integral equation for the first-passage-time probability density function, obtained by resorting to the fixed point theorem, is used to achieve approximate evaluations for which error bounds are indicated. A different use of the fixed point theorem is then made to determine lower and upper bounds for asymptotic approximations, and to examine their range of validity.

A queueing network with arbitrary topology, state dependent routing and flow control is considered. Customers may enter the network at any queue and they are routed through it until they reach certain queues from which they may leave the system. The routing is based on local state information. The service rate of a server is controlled based on local state information as well. A distributed policy for routing and service rate control is identified that achieves maximum throughput. The policy can be implemented without knowledge of the arrival and service rates. The importance of flow control is demonstrated by showing that, in certain networks, if the servers cannot be forced to idle, then no maximum throughput policy exists when the arrival rates are not known. Also a model for exchange of state information among neighboring nodes is presented and the network is studied when the routing is based on delayed state information. A distributed policy is shown to achieve maximum throughput in the case of delayed state information. Finally, some implications for deterministic flow networks are discussed.

Batch departures arise in various applications of queues. In particular, such models have been studied recently in connection with production systems. For the most part, however, these models assume Poisson arrivals and exponential service times; little is known about them under more general settings. We consider how their stationary queue length distributions are affected by the distributions of interarrival times, service times and departing batch sizes of customers. Since this is not an easy problem even for single departure models, we first concentrate on single-node queues with a symmetric service discipline, which is known to have nice properties. We start with pre-emptive LIFO, a special case of the symmetric service discipline, and then consider symmetric queues with Poisson arrivals. Stability conditions and stationary queue length distributions are obtained for them, and several stochastic order relations are considered. For the symmetric queues and Poisson arrivals, we also discuss their network. Stability conditions and the stationary joint queue length distribution are obtained for this network.