A correlated random walk in the plane is studied for which the direction of a step depends on the direction of the previous step. Both step direction and step length are continuous random variables. Such a random walk has been used to model the vegetative dispersal of certain plant populations. The analysis provides general conclusions about dependencies on parameters, an efficient scheme for generating numerical results, and testable predictions for plant populations.

Suppose that a certain population consists of N individuals. One member initially learns a rumour from an outside source, and starts telling it to other members, who continue spreading the information. A knower becomes inactive once he encounters somebody already informed. Daley and Kendall, who initiated the study of this model, conjectured that the number of eventual knowers is asymptotically normal with mean and variance linear in N. Our purpose is to confirm this conjecture.

For each n > 0, the Ewens sampling formula from population genetics is a measure on the set of all partitions of the integer n. To determine the limiting distributions for the part sizes of a partition with respect to the measures given by this formula, we associate to each partition a step function on [0, 1]. Each jump in the function equals the number of parts in the partition of a certain size. We normalize these functions and show that the induced measures on D[0, 1] converge to Wiener measure. This result complements Kingman's frequency limit theorem [10] for the Ewens partition structure.

Importance sampling is a Monte Carlo simulation technique in which the simulation distribution is different from the true underlying distribution. In order to obtain an unbiased Monte Carlo estimate of the desired parameter, simulated events are weighted to reflect their true relative frequency. In this paper, we consider the estimation via simulation of certain large deviations probabilities for time-homogeneous Markov chains. We first demonstrate that when the simulation distribution is also a homogeneous Markov chain, the estimator variance will vanish exponentially as the sample size n tends to∞. We then prove that the estimator variance is asymptotically minimized by the same exponentially twisted Markov chain which arises in large deviation theory, and furthermore, this optimization is unique among uniformly recurrent homogeneous Markov chain simulation distributions.

In this paper we show that the problem of existence and uniqueness of stationary distributions for piecewise-deterministic Markov processes (PDPs) is equivalent to the same problem for the associated Markov chain, so long as some mild conditions on the parameters of the PDP are satisfied. Our main result is the construction of an invertible mapping from the set of stationary distributions for the PDP to the set of stationary distributions for the Markov chain. Some sufficient conditions for existence are presented and an application to capacity expansion is given.

Products of the Laplace transforms of exponential distributions with different parameters are inverted to give a mixture of Erlang densities, i.e. an expression for the convolution of exponentials. The formula for these inversions is expressed both as an explicit sum and in terms of a recurrence relation which is better suited to numerical computation. The recurrence for the inversion of certain weighted sums of these transforms is then solved by converting it into a linear first-order partial differential equation. The result may be used to find the density function of passage times between states in a Markov process and it is applied to derive an expression for cycle time distribution in tree-structured Markovian queueing networks.

Let be i.i.d. non-negative random variables with d.f. F and Laplace transform L. Let N be integer valued and independent of In many applications one knows that for y → ∞ and a function φ where in turn τ is the solution of an equation On the basis of a sample of size n we derive an estimator τ n for τ by solving ψ (τ n, Ln(τ n), L′n(τ n), · ··) = 0 where Ln is the empirical version of L. This estimator is then used to derive the asymptotic behaviour of φ (y, τ n, Ln(τ n), L′n(τ n), · ··). We include five examples, some of which are taken from insurance mathematics.

The first-crossing-time problem through two time-dependent boundaries for one-dimensional diffusion processes is considered. The first-crossing p.d.f.'s from below and from above are proved to satisfy a new system of Volterra integral equations of the second kind involving two arbitrary continuous functions. By a suitable choice of such functions a system of continuous-kernel integral equations is obtained and an efficient algorithm for its solution is provided. Finally, conditions on the drift and infinitesimal variance of the diffusion process are given such that the system of integral equations reduces to a non-singular single integral equation for the first-crossing-time p.d.f.

Sufficient conditions under which two random vectors are ordered in the sense of uniform conditional stochastic order (Whitt (1980), (1982)) with respect to planar regions are given. A natural classification of distributions based on this notion of stochastic order is defined and studied. A negative dependence property of Block et al. (1985) is shown to hold for this class of distributions. An application of these results in statistics is also presented.

The length of the main channel in a river network is viewed as an extreme value statistic L on a randomly weighted binary rooted tree having M sources. Questions of concern for hydrologic applications are formulated as the construction of an extreme value theory for a dependence which poses an interesting contrast to the classical independent theory. Equivalently, the distribution of the extinction time for a binary branching process given a large number of progeny is sought. Our main result is that in the case of exponentially weighted trees, the conditional distribution of n–1/2L given M = n is asymptotically distributed as the maximum of a Brownian excursion. When taken with an earlier result of Kolchin (1978), this makes the maximum of the Brownian excursion a tree-dependent extreme value distribution whose domain of attraction includes both the exponentially distributed and almost surely constant weights. Moment computations are given for the Brownian excursion which are of independent interest.

This paper establishes the existence of an optimal stationary strategy in a leavable Markov decision process with countable state space and undiscounted total reward criterion.

Besides assumptions of boundedness and continuity, an assumption is imposed on the model which demands the continuity of the mean recurrence times on a subset of the stationary strategies, the so-called ‘good strategies'. For practical applications it is important that this assumption is implied by an assumption about the cost structure and the transition probabilities. In the last part we point out that our results in general cannot be deduced from related works on bias-optimality by Dekker and Hordijk, Wijngaard or Mann.

It is shown that if parameters occurring linearly in the transition intensity of a Markov process are replaced by their respective ‘Bayesian estimates' then the new process thus generated has an equilibrium distribution which is a mixture (over parameter values) of the original parametrised equilibrium distribution.

One effectively then has an extra state dependence in that one selects from a given class of transition rules those rules which are most consistent with the value of current state. The effect of this is thus to preserve the status quo, in that unlikely transitions are made even less likely. By this means one can construct processes which show several distinct and metastable modes of behaviour, and which can serve as models for memory devices.

Let {xt} be a discrete-time multivariate stationary process possessing an infinite autoregressive representation and let ΓB(k), ΓF(k) and Γ be the block Toeplitz covariance matrices of xB(k) = [x′–1, x′–2, · ··, x′–k]′, xF(k) = [x′1, x′2 · ·· x′k] and x = [·· ·x′–2, x′–1, x′0, x′1, x′2 · ··]′ respectively, where k ≧ 1, is finite or infinite. Also let φ m,n(j) and δm,n(u) be the coefficients of xt+ j and xt– u respectively in the linear least-squares interpolator of xt from xt+ 1, · ··, xt+ m; xt− 1, · ··, xt– n, where m, n ≧ 0, 0 ≦ j ≦ m, 0 ≦ u ≦ n are integers, zt(m, n) denote the interpolation error and τ2(m, n) = E[zt(m, n)zt(m, n)′]. A physical interpretation for the components of ΓB(k)–1, ΓF(k)–1 and Γ–1 is given by relating these components to the φm,n(j) δm,n(u) and τ2(m, n). A similar result is shown to hold also for the estimators of ΓB(k)–l and the interpolation parameters when these have been obtained from a realization of length T of {xt}. Some of the applications of the results are considered.

Non-negative time series which are first-order autoregressive with a mixed exponential innovation process are studied. Properties and approximate marginal distributions for such series are found. Modifications to include exact zeroes and to increase variability so that the time series is a more realistic model of rivers which are dry for part of the year are discussed.

A sampling system, MIL-STD-105D, used in quality control consists of three sampling plans with different acceptance probabilities used in turn for lot inspection. The decision to switch plan is based on the history of the lot acceptance records and a set of stopping rules. We derive the performance measure, average outgoing quality (AOQ), of this sampling system from a renewal process in which AOQ is expressed in terms of the moments of the stopping times. The renewal approach is simpler than that of the Markov chain generally used in computing AOQ for an infinite sequence of lots; it also provides a formula for AOQ for a finite sequence of lots.

The problem of how to compare repairable systems has long been of interest to those who are engaged in reliability analysis. In this article we have introduced a type of comparison of two counting processes. Various properties and applications of our stochastic comparison are also studied.

A relatively simple formula is presented for the probability distribution of the number K of components of a random function. This formula facilitates the (computer) calculation of the factorial moments of K and yields new expressions for the mean and variance of K.

Let pk, k ≧ 0, be a probability distribution having finite binomial moments Br, r ≧ 0, and the probability generating function U(z) with a radius of convergence α (≧ 1). In this note explicit and recursive formulae are derived allowing computation of the pk in terms of the Br if α > 1. Ch. Jordan's formula, which holds if α > 2, turns out to be a special case.

Existing procedures for strategy evaluation based on earlier work by the author have led (inter alia) to procedures for the assessment of heuristics for stochastic scheduling and to the development of an approach to sensitivity analysis. Recent theoretical work has now led to the development of more effective evaluation procedures. The key results are reported here.

In this paper we derive a new property of the exponential distribution, closely related to a single-server queueing system with unreliable server. We show that this new property is another characterization of the exponential law.