Given a two-dimensional Poisson process, X, with intensity λ, we are interested in the largest number of points, L, contained in a translate of a fixed scanning set, C, restricted to lie inside a rectangular area.

The distribution of L is accurately approximated for rectangular scanning sets, using a technique that can be extended to higher dimensions. Reasonable approximations for non-rectangular scanning sets are also obtained using a simple correction of the rectangular result.

The empty space function of a stationary point process in ℝd is the function that assigns to each r, r > 0, the probability that there is no point within distance r of O. In a recent paper Van Lieshout and Baddeley study the so-called J-function, which is defined as the ratio of the empty space function of a stationary point process and that of its corresponding reduced Palm process. They advocate the use of the J-function as a characterization of the type of spatial interaction.

Therefore it is natural to ask whether J ≡ 1 implies that the point process is Poisson. We restrict our analysis to the one-dimensional case and show that a classical construction by Szász provides an immediate counterexample. In this example the interpoint distances are still exponentially distributed. This raises the question whether it is possible to have J ≡ 1 but non-exponentially distributed interpoint distances. We construct a point process with J ≡ 1 but where the interpoint distances are bounded.

The shape of a rectangular prism in (d + 1)-dimensions is defined as Y = (Y1, Y2, · ··, Yd), Yn = Ln/Ln+1 where the Ln are the prism's edge lengths, in ascending order. We investigate shape distributions that are invariant when the prism is cut into two, also rectangular, prisms, with one prism retained for measurement and the other discarded. Interesting new distributions on [0, 1]d arise.

We derive a variety of estimates for the survival probability of a branching process in a random environment. There are three cases of interest, the critical, weakly and strongly subcritical. The large deviation result, first obtained by Dekking for the class of finite state space i.i.d. environments, is shown to hold in more general environments. We also obtain some finer convergence results.

Consider a system of interacting finite Markov chains in continuous time, where each subsystem is aggregated by a common partitioning of the state space. The interaction is assumed to arise from dependence of some of the transition rates for a given subsystem at a specified time on the states of the other subsystems at that time. With two subsystem classes, labelled 0 and 1, the superposition process arising from a system counts the number of subsystems in the latter class. Key structure and results from the theory of aggregated Markov processes are summarized. These are then applied also to superposition processes. In particular, we consider invariant distributions for the level m entry process, marginal and joint distributions for sojourn-times of the superposition process at its various levels, and moments and correlation functions associated with these distributions. The distributions are obtained mainly by using matrix methods, though an approach based on point process methods and conditional probability arguments is outlined. Conditions under which an interacting aggregated Markov chain is reversible are established. The ideas are illustrated with simple examples for which numerical results are obtained using Matlab. Motivation for this study has come from stochastic modelling of the behaviour of ion channels; another application is in reliability modelling.

The gating mechanism of a single ion channel is usually modelled by a continuous-time Markov chain with a finite state space. The state space is partitioned into two classes, termed ‘open’ and ‘closed’, and it is possible to observe only which class the process is in. In many experiments channel openings occur in bursts. This can be modelled by partitioning the closed states further into ‘short-lived’ and ‘long-lived’ closed states, and defining a burst of openings to be a succession of open sojourns separated by closed sojourns that are entirely within the short-lived closed states. There is also evidence that bursts of openings are themselves grouped together into clusters. This clustering of bursts can be described by the ratio of the variance Var (N(t)) to the mean [N(t)] of the number of bursts of openings commencing in (0, t]. In this paper two methods of determining Var (N(t))/[N(t)] and limt→∝ Var (N(t))/[N(t)] are developed, the first via an embedded Markov renewal process and the second via an augmented continuous-time Markov chain. The theory is illustrated by a numerical study of a molecular stochastic model of the nicotinic acetylcholine receptor. Extensions to semi-Markov models of ion channel gating and the incorporation of time interval omission are briefly discussed.

This paper studies the expected average cost control problem for discrete-time Markov decision processes with denumerably infinite state spaces. A sequence of finite state space truncations is defined such that the average costs and average optimal policies in the sequence converge to the optimal average cost and an optimal policy in the original process. The theory is illustrated with several examples from the control of discrete-time queueing systems. Numerical results are discussed.

We consider extreme value theory for a class of stationary Markov chains with values in ℝd. The asymptotic distribution of Mn, the vector of componentwise maxima, is determined under mild dependence restrictions and suitable assumptions on the marginal distribution and the transition probabilities of the chain. This is achieved through computation of a multivariate extremal index of the sequence, extending results of Smith [26] and Perfekt [21] to a multivariate setting. As a by-product, we obtain results on extremes of higher-order, real-valued Markov chains. The results are applied to a frequently studied random difference equation.

In this paper we study a new kind of option, called hereinafter a Parisian barrier option. This option is the following variant of the so-called barrier option: a down-and-out barrier option becomes worthless as soon as a barrier is reached, whereas a down-and-out Parisian barrier option is lost by the owner if the underlying asset reaches a prespecified level and remains constantly below this level for a time interval longer than a fixed number, called the window. Properties of durations of Brownian excursions play an essential role. We also study another kind of option, called here a cumulative Parisian option, which becomes worthless if the total time spent below a certain level is too long.

In a recent paper [16], one of us identified all of the quasi-stationary distributions for a non-explosive, evanescent birth-death process for which absorption is certain, and established conditions for the existence of the corresponding limiting conditional distributions. Our purpose is to extend these results in a number of directions. We shall consider separately two cases depending on whether or not the process is evanescent. In the former case we shall relax the condition that absorption is certain. Furthermore, we shall allow for the possibility that the minimal process might be explosive, so that the transition rates alone will not necessarily determine the birth-death process uniquely. Although we shall be concerned mainly with the minimal process, our most general results hold for any birth-death process whose transition probabilities satisfy both the backward and the forward Kolmogorov differential equations.

A continuous-time threshold autoregressive process of order two (CTAR(2)) is constructed as the first component of the unique (in law) weak solution of a stochastic differential equation. The Cameron–Martin–Girsanov formula and a random time-change are used to overcome the difficulties associated with possible discontinuities and degeneracies in the coefficients of the stochastic differential equation. A sequence of approximating processes that are well-suited to numerical calculations is shown to converge in distribution to a solution of this equation, provided the initial state vector has finite second moments. The approximating sequence is used to fit a CTAR(2) model to percentage relative daily changes in the Australian All Ordinaries Index of share prices by maximization of the ‘Gaussian likelihood'. The advantages of non-linear relative to linear time series models are briefly discussed and illustrated by means of the forecasting performance of the model fitted to the All Ordinaries Index.

A criterion is given for the existence of a stationary and causal multivariate integer-valued autoregressive process, MGINAR(p). The autocovariance function of this process being identical to the autocovariance function of a standard Gaussian MAR(p), we deduce that the MGINAR(p) process is nothing but a MAR(p) process. Consequently, the spectral density is directly found and gives good insight into the stochastic structure of a MGINAR(p). The estimation of parameters of the model, as well as the forecasting of the series, is discussed.

The problem of estimating the integral of a stochastic process from observations at a finite number N of sampling points has been considered by various authors. Recently, Benhenni and Cambanis (1992) studied this problem for processes with mean 0 and Hölder index K + ½, K ; ℕ These results are here extended to processes with arbitrary Hölder index. The estimators built here are linear in the observations and do not require the a priori knowledge of the smoothness of the process. If the process satisfies a Hölder condition with index s in quadratic mean, we prove that the rate of convergence of the mean square error is N2s+1 as N goes to ∞, and build estimators that achieve this rate.

For sequences of i.i.d. random variables whose common tail 1 – F is regularly varying at infinity wtih an unknown index –α < 0, it is well known that the Hill estimator is consistent for α–1 and usually asymptotically normally distributed. However, because the Hill estimator is a function of k = k(n), the number of upper order statistics used and which is only subject to the conditions k →∞, k/n → 0, its use in practice is problematic since there are few reliable guidelines about how to choose k. The purpose of this paper is to make the use of the Hill estimator more reliable through an averaging technique which reduces the asymptotic variance. As a direct result the range in which the smoothed estimator varies as a function of k decreases and the successful use of the esimator is made less dependent on the choice of k. A tail empirical process approach is used to prove the weak convergence of a process closely related to the Hill estimator. The smoothed version of the Hill estimator is a functional of the tail empirical process.