The relation is investigated between the distributions of the total time until failure and the time of exposure to risk until failure, for individuals who are at risk only intermittently during their lifetimes. A specific example is considered in the case of computer failures.

In this paper we obtain an upper bound for the maximum of random fields of the form , where CP denotes circles of fixed radius and dW(P′) is a plane white noise field.

The results presented are obtained by means of successive steps involving Slepian's lemma for random fields, inequalities on Brownian fields and planar stochastic integrals.

Let {an} be an increasing sequence of positive integers and let be the family of all functions from the positive integers into the positive integers, which satisfy Assume that are random functions with probabilities and for all n > 1 and 0 elsewhere, i.e., all permissible values of f, given the past, are equally likely.

This note shows how the initial density of the volume of a random simplex in a body, K, depends upon the fourth moment of sectional area and so rejects the conjecture that this initial density is independent of K.

We study Bose gases in $d \ge 2$ dimensions with short-range repulsive pair interactions at positive temperature, in the canonical ensemble and in the thermodynamic limit. We assume the presence of hard Poissonian obstacles and focus on the non-percolation regime. For sufficiently strong interparticle interactions, we show that almost surely there cannot be Bose–Einstein condensation into a sufficiently localized, normalized one-particle state. The results apply to the canonical eigenstates of the underlying one-particle Hamiltonian.

The classical condition for regularity of a Markov chain is extended to include semi-Markov chains. In addition, for any given semi-Markov chain, we find Markov chains which exhibit identical regularity properties. This is done either (i) by transforming the state space or, alternatively, (ii) by imposing conditions on the holding-time distributions. Brief consideration is given to the problem of extending the results to processes other than semi-Markov chains.

The (conditional or unconditional) distribution of the continuous scan statistic in a one-dimensional Poisson process may be approximated by that of a discrete analogue via time discretization (to be referred to as the discrete approximation). Using a change of measure argument, we derive the first-order term of the discrete approximation which involves some functionals of the Poisson process. Richardson's extrapolation is then applied to yield a corrected (second-order) approximation. Numerical results are presented to compare various approximations.

We provide constructions of age-structured branching processes without or with immigration as pathwise-unique solutions to stochastic integral equations. A necessary and sufficient condition for the ergodicity of the model with immigration is also given.

We show analogs of the classical arcsine theorem for the occupation time of a random walk in (−∞,0) in the case of a small positive drift. To study the asymptotic behavior of the total time spent in (−∞,0) we consider parametrized classes of random walks, where the convergence of the parameter to 0 implies the convergence of the drift to 0. We begin with shift families, generated by a centered random walk by adding to each step a shift constant a>0 and then letting a tend to 0. Then we study families of associated distributions. In all cases we arrive at the same limiting distribution, which is the distribution of the time spent below 0 of a standard Brownian motion with drift 1. For shift families this is explained by a functional limit theorem. Using fluctuation-theoretic formulae we derive the generating function of the occupation time in closed form, which provides an alternative approach. We also present a new form of the first arcsine law for the Brownian motion with drift.

We analyse several aspects of a class of simple counting processes that can emerge in some fields of applications where a change point occurs. In particular, under simple conditions we prove a significant inequality for the stochastic intensity.

The motion of electrons through a gas of particles is modelled as a two-state process: an electron is alternatingly moving and being held captive by a non-moving particle, during exponentially distributed periods.

The model contains a parameter p, the probability that collision of an electron with a gas particle does not lead to its capture but produces an extra electron.

The quantity of interest is the current C(t) produced by the moving electrons, as a function of time.

We generalize a theorem due to Keilson on the approximate exponentiality of waiting times for rare events in regenerative processes. We use the result to investigate the limit distribution for a family of first entrance times in a sequence of Ehrenfest urn models. As a second application, we consider approximate pattern matching, a problem arising in molecular biology and other areas.

The thinking behind the original Black-Scholes formula is criticised on the grounds that it holds out the quite unrealistic prospect of risk-free operation, that it can sacrifice asset maximisation to exact meeting of the contract, and that it restricts investment to those stocks on which an option is being sold. An alternative approach is given, based on a model of risk-averse asset maximisation, which, while meeting these criticisms, gives the option price in the familiar form of a discounted and weighted conditional expectation of the seller's liability at maturity. This evaluation is extended to a completely general stochastic model of stock price evolution; consideration is also given to the possibility of seller's ruin.

The paper considers an extension to the reversible counters system of Lampard [1]. In Lampard's model the input processes are two independent Poisson processes; this results in a gamma Markov sequence for the time-interval between successive output pulses and a negative binomial Markov sequence for the counts at the times of out-put pulses. We consider the input process to be a bivariate Poisson process and show that the first out-put process given above is not affected, while the second out-put-process becomes of a type studied in the theory of branching processes.

Falin (1984) examined the quasi-input process (the flow of service starting times) in the M/G/1/∞ queue and raised the question as to whether this process is a renewal process. We show that, except in the trivial case of instantaneous service, the quasi-input process is never renewal.

Looking at a large branching population we determine along which path the population that started at 1 at time 0 ended up in B at time N. The result describes the density process, that is, population numbers divided by the initial number K (where K is assumed to be large). The model considered is that of a Galton-Watson process. It is found that in some cases population paths exhibit the strange feature that population numbers go down and then increase. This phenomenon requires further investigation. The technique uses large deviations, and the rate function based on Cramer's theorem is given. It also involves analysis of existence of solutions of a certain algebraic equation.

The standard coalescent is widely used in evolutionary biology and population genetics to model the ancestral history of a sample of molecular sequences as a rooted and ranked binary tree. In this paper we present a representation of the space of ranked trees as a space of constrained ordered matched pairs. We use this representation to define ergodic Markov chains on labeled and unlabeled ranked tree shapes analogously to transposition chains on the space of permutations. We show that an adjacent-swap chain on labeled and unlabeled ranked tree shapes has a mixing time at least of order $n^3$ , and at most of order $n^{4}$ . Bayesian inference methods rely on Markov chain Monte Carlo methods on the space of trees. Thus it is important to define good Markov chains which are easy to simulate and for which rates of convergence can be studied.

In 1968 Lampard determined some limit distributions for a counter process in which the input and output are independent Poisson processes. In 1974 Phatarfod dealt with the generalizations of Lampard's formulas for the case when the input and output form a bivariate Poisson process; however, his reasoning is erroneous. The object of this paper is to determine the correct limit distributions for the generalized process.

Two types of limit theorems are proved for processes of randomly displaced regular events. Firstly, as the displacements tend to infinity, the counting process is shown to converge weakly to a Poisson process and secondly, as the interval between events tends to zero, convergence of the finite-dimensional distributions of the associated storage process to a diffusion is proved.

We consider a multivariate Lévy process where the first coordinate is a Lévy process with no negative jumps which is not a subordinator and the others are non-decreasing. We determine the Laplace–Stieltjes transform of the steady-state buffer content vector of an associated system of parallel queues. The special structure of this transform allows us to rewrite it as a product of joint Laplace–Stieltjes transforms. We are thus able to interpret the buffer content vector as a sum of independent random vectors.