Daryl Daley has kindly invited me to write some prefatory words, autobiographical in character, to the collection of articles which follows. I am very happy to do this, but first let me express my appreciation of the efforts which Daryl, Peter Thomson, and the contributors have put into the making of this volume. It is a wonderful tribute, which I greatly appreciate. May I also take the opportunity to acknowledge the help of all of those colleagues, whether or not they have contributed directly to this volume, whose support, professional advice, knowledge, competence and companionship have made my years as statistician and tyro seismologist extremely rewarding. To all such, a big thank you.

Several stochastic models have been proposed to describe the kinetic theory of reversible chemical reactions. However, in macroscopic systems the effects of stochastic variability are often outweighed by mean effects. In the present paper we show that some observed phenomena can be explained quite adequately by a stochastic model in which the stochastic variability is not negligible in comparison with mean effects. Our argument involves approximations to a stochastic model for competing chemical reactions.

We derive closed-form solutions to some discounted optimal stopping problems related to the perpetual American cancellable dividend-paying put and call option pricing problems in an extension of the Black–Merton–Scholes model. The cancellation times are assumed to occur when the underlying risky asset price process hits some unobservable random thresholds. The optimal stopping times are shown to be the first times at which the asset price reaches stochastic boundaries depending on the current values of its running maximum and minimum processes. The proof is based on the reduction of the original optimal stopping problems to the associated free-boundary problems and the solution of the latter problems by means of the smooth-fit and modified normal-reflection conditions. We show that the optimal stopping boundaries are characterised as the maximal and minimal solutions of certain first-order nonlinear ordinary differential equations.

Let {X n , n = 0, 1,…} be the sequence of the lower records for an arbitrary underlying distribution μ on [0, ∞). We show that is equal in distribution to where {τ i , i = 1, 2,…} is a Poisson flow of unit intensity and g is a right-continuous and nonincreasing function defined by μ. This observation allows us to extend results of Bose et al. and simplify their proofs.

A generalization of the Bernoulli–Laplace diffusion model is proposed. We consider the case where the number of balls exchanged is greater than one. We show that the stationary distribution is the same as in the classical scheme and we give the mean and the variance of the process. In a second stage, we study the asymptotic approximation based on the diffusion process. A solution of transition density is given using Legendre polynomials.

We consider an infinite server queueing system. An examination of sample path dynamics allows a straightforward development of integral equations having solutions that give time-dependent occupancy (number of customers) and backlog (unfinished work) distributions (conditioned on the time of the first arrival) for the GI/G/∞ queue. These integral equations are amenable to numerical evaluation and can be generalized to characterize GIX/G/∞ queue. Two examples are given to illustrate the results.

The mean and the variance of the time S(t) spent by a system below a random threshold until t are obtained when the system level is modelled by the current value of a sequence of independent and identically distributed random variables appearing at the epochs of a nonhomogeneous Poisson process. In the case of the homogeneous Poisson process, the asymptotic distribution of S(t)/t as t → ∞ is derived.

In this paper, the optimal control problem of system described by stochastic McShane differential equations is considered. It is shown that this problem can be reduced to an equivalent optimal control problem of distributed parameter systems of parabolic type with controls appearing in the coefficients of the differential operator. Further, to this reduced problem, necessary conditions for optimality and an existence theorem for optimal controls are given.

Conditions are derived for the components of the normed limit of a multi-type branching process with varying environments, to be continuous on (0, ∞). The main tool is an inequality for the concentration function of sums of independent random variables, due originally to Petrov. Using this, we show that if there is a discontinuity present, then a particular linear combination of the population types must converge to a non-random constant (Equation (1)). Ensuring this can not happen provides the desired continuity conditions.

We consider the uncovered set (i.e. the complement of the union of growing random intervals) in the one-dimensional Johnson-Mehl model. Let S(z,L) be the number of components of this set at time z > 0 which intersect (0, L]. An explicit bound is known for the total variation distance between the distribution of S(z,L) and a Poisson distribution, but due to clumping of the components the bound can be rather large. We here give a bound for the total variation distance between the distribution of S(z,L) and a simple compound Poisson distribution (a Pólya-Aeppli distribution). The bound is derived by interpreting S(z,L) as the number of visits to a ‘rare’ set by a Markov chain, and applying results on compound Poisson approximation for Markov chains by Erhardsson. It is shown that under a mild condition, if z→∞ and L→∞ in a proper fashion, then both the Pólya-Aeppli and the Poisson approximation error bounds converge to 0, but the convergence of the former is much faster.

Fixed-cycle traffic light queues have been investigated by probabilistic methods by many authors. Beckmann, McGuire and Winsten (1956) considered a discrete time queueing model with binomial arrivals and regular departure headways and derived a relation between the stationary mean delay per vehicle and the stationary mean queue-length at the beginning of a red period of the traffic light. Haight (1959) and Buckley and Wheeler (1964) considered models with Poisson arrivals and regular departure headways and investigated certain properties of the queue-length. Newell (1960) dealt with the model proposed by the first authors and obtained the probability generating function of the stationary queue-length distribution. Darroch (1964) discussed a more general discrete time model with stationary, independent arrivals and regular departure headways and derived a necessary and sufficient condition for the stationary queue-length distribution to exist and obtained its probability generating function. The above two authors, Little (1961), Miller (1963), Newell (1965), McNeil (1968), Siskind (1970) and others gave approximate expressions for the stationary mean delay per vehicle for fixed-cycle traffic light queues of various types. All of the authors mentioned above dealt with the queue-length.

We introduce a class of stochastic processes in discrete time with finite state space by means of a simple matrix product. We show that this class coincides with that of the hidden Markov chains and provides a compact framework for it. We study a measure obtained by a projection on the real line of the uniform measure on the Sierpinski gasket, finding that the dimension of this measure fits with the Shannon entropy of an associated hidden Markov chain.

Cars entering and leaving a parking place in a forest are registered automatically in connection with an investigation of forest recreational use. By subtraction the number of cars parked is computed. Due to erroneous and missing counts, errors occur and may for instance indicate a negative number. The theory of incompletely observed Markov chains is used to obtain an estimate of the number of cars parked.

We consider an insurance model, where the underlying point process is a Cox process. Using a martingale approach applied to diffusion processes, finite-time Lundberg inequalities are obtained. By change-of-measure techniques, Cramér–Lundberg approximations are derived.

The object of this paper is to present a probabilistic model for analyzing changes through time in a binary dyadic relation on a finite set of points. The total relation on the set takes the form of an aggregate of directed binary dyadic relations between ordered pairs of points belonging to the set; equivalently, the total relation on the set can be represented by means of a digraph or an incidence matrix isomorphic with the total relation. Such a relation, changing in its structure as time proceeds, is a reasonable mathematical model, for example, for the evolution of inter-relationships of the members of a social or any other group. A group of this kind is organized for a specific activity involving some sort of “communication” from one member to the other, and may be observed at successive discrete points in time generating statistics on the evolutionary process. (For a detailed treatment, see [3].) As a matter of fact, under suitable assumptions, the model presented here has potentialities for application in those situations which can be represented mathematically in terms of a finite set of points and an all-or-none relationship between ordered pairs of these points. Some of the other examples are communication networks, ecology, animal sociology, and management sciences (see [5]).

The aim of this paper is to provide the conditions necessary to reduce the complexity of state filtering for finite stochastic systems (FSSs). A concept of lumpability for FSSs is introduced. In this paper we assert that the unnormalised filter for a lumped FSS has linear dynamics. Two sufficient conditions for such a lumpability property to hold are discussed. We show that the first condition is also necessary for the lumped FSS to have linear dynamics. Next, we prove that the second condition allows the filter of the original FSS to be obtained directly from the filter for the lumped FSS. Finally, we generalise an earlier published result for the approximation of a general FSS by a lumpable FSS.

A stationary Gaussian process is exhibited with the following property: the covariance function of the process is not differentiable at the origin and yet almost all the sample paths of the process are differentiable in a set of points of the power of the continuum. The process provides a counter example to a statement of Slepian.

We propose an estimator for the number of sources, N, which transmit signals according to a Poisson process with a common rate μ. This estimation problem was originated by Starr and pursued by Vardi. Our procedure is a ‘double sampling’ one and has the advantage that it can be used when μ is unknown.

We consider the asymmetric simple exclusion process which starts from a product measure such that all the sites to the left of zero (including zero) are occupied and the right of 0 (excluding 0) are empty. We label the particle initially at 0 as the leading particle. We study the long-term behaviour of this process near large sites when the leading particle's holding time is different from that of the other particles. In particular, we assume that the leading particle moves at a slower rate than the other particles. We call this modified asymmetric simple exclusion process the road-hog process. Coupling and stochastic ordering techniques are used to derive the density profile of this process. Road-hog processes are useful in modelling series of exponential queues with Poisson and non-Poisson input process. The density profiles dramatically illustrate the flow of customers through the queues.

Independent observations X0, X1…, XN+1 are drawn from each of N populations whose distribution functions F(x – θi) have means θ i, 0 ≦ i < N, and we wish to calculate the probability Pk;N that X0 is the k th largest observation. For normal populations an approximation is given for PK;N based on a Taylor series expansion in the θ 's. If F(x) has an increasing failure rate, as is the case for the normal, an upper bound can be given for the ‘win' probability P1;N Some moment relations for normal order statistics are also given.