An asymptotic expansion for the expected number, μ(t), of particles of an age-dependent branching process is obtained with a general submultiplicative estimate for the remainder term. The influence of the roots of the characteristic equation on the asymptotic behaviour of μ(t) is taken into account.

The paper considers the superposition of modified Omori functions as a conditional intensity function for a point process model used in the exploratory analysis of earthquake clusters. For the examples discussed, the maximum likelihood estimates converge well starting from appropriate initial values even though the number of parameters estimated can be large (though never larger than the number of observations). Three datasets are subjected to different analyses, showing the use of the model to discover and study individual clustering features.

A given number of bullets will be fired sequentially in an attempt to destroy as many targets as possible from a fixed number of targets. The probability of destroying a target at each shot is known. After each shot, there is a report on the state for the target; destroyed or intact. The reports are subject to the usual two types of errors and the probabilities of making these errors are also known. This paper shows that the myopic decision strategy that picks the next target to be the one with the highest intact posterior probability is the optimal strategy.

If τ is the lifetime of a coherent system, then the signature of the system is the vector of probabilities that the lifetime coincides with the ith order statistic of the component lifetimes. The signature can be useful in comparing different systems. In this treatment we give a characterization of the signature of a system with independent identically distributed components in terms of the number of path sets in the system as well as in terms of the number of what we call ordered cut sets. We consider, in particular, the signatures of indirect majority systems and compare them with the signatures of simple majority systems of the same size. We note that the signature of an indirect majority system of size r × s = n is symmetric around , and use this to show that the expected lifetime of an r × s = n indirect majority system exceeds that of a simple (direct) majority system of size n when the components are exponentially distributed with the same parameter.

The contact process is an interacting particle system which models a spatially restricted infection. In the basic contact process the infection can only spread to an uninfected neighbour, but the diffusive contact process allows an infected individual to move to an uninfected site. If the infection rate is too low, the process will die out. If the individual can move (or diffuse), the disease can spread with a lower infection rate. An idea of the relationship between these rates is obtained by obtaining rigorous lower bounds for the critical infection rate for various values of the diffusion rate. In this paper we also improve the lower bound for the critical infection rate for the basic contact process from 1.539 to 1.5517.

We consider a reflected superposition of a Brownian motion and a compound Poisson process as a model for the workload process of a queueing system with two types of customers under heavy traffic. The distributions of the duration of a busy cycle and the maximum workload during a cycle are determined in closed form.

This paper uses the epidemic-type aftershock sequence (ETAS) point process model to study certain seismicity features of the Jiashi swarm of certain earthquakes, investigating in particular whether there is relative quiescence prior to the quite big events within the Jiashi sequence. The seven earthquake sequences studied occurred in the region of Jiashi, south of Tianshan Mountain, Xinjiang, China. The particular ETAS model that is developed is consistent with the reality of seismic activity. The various features of Jiashi swarm activity can be described as focusing in different stages. There is obvious precursory quiescence prior to most big events with Ms ≥ 6.0 within the Jiashi swarm. Thus, checking for relative quiescence can be use for earthquake prediction.

The annual earthquake predictions of the China Seismological Bureau (CSB) are evaluated by means of an R score (an R score is approximately 0 for completely random guesses, and approximately 1 for completely successful predictions). The average R score of the annual predictions in China in the period 1990–1998 is about 0.184, significantly larger than 0.0. However, background seismicity is higher in seismically active regions. If a ‘random guess' prediction is chosen to be proportional to the background seismicity, the expected R score is 0.123, and the nine-year mean R score of 0.184 as observed is only marginally higher than this background value. Monte Carlo tests indicate that the probability of attaining an R score of actual prediction by background seismicity based on random guess is about . It is concluded that earthquake prediction in China is still in a very preliminary stage, barely above a pure chance level.

We consider the problem of estimating the rate of convergence to stationarity of a continuous-time, finite-state Markov chain. This is done via an estimator of the second-largest eigenvalue of the transition matrix, which in turn is based on conventional inference in a parametric model. We obtain a limiting distribution for the eigenvalue estimator. As an example we treat an M/M/c/c queue, and show that the method allows us to estimate the time to stationarity τ within a time comparable to τ.

A point process procedure can be used to study reservoir-induced seismicity (RIS), in which the intensity function representing earthquake hazard is a combination of three terms: a constant background term, an ETAS (epidemic-type aftershock sequence) term for aftershocks, and a time function derived from observation of water levels of a reservoir. This paper presents the results of such a study of the seismicity in the vicinity of the Tarbela reservoir in Pakistan. Making allowance for changes in detection capability and the background seismicity related to tectonic activity, earthquakes of magnitude ≥ 2.0, occurring between May 1978 and January 1982 and whose epicentres were within 100 km of the reservoir, were used in this analysis. Several different intensities were compared via their Akaike information criterion (AIC) values relative to those of a Poisson process. The results demonstrate that the seismicity within 20 km of the reservoir correlates with water levels of the reservoir, namely, active periods occur about 250 days after the appearance of low water levels. This suggests that unloading the reservoir activates the seismicity beneath it. Seasonal variations of the seismicity in an area up to 100 km from the reservoir were also found, but these could not be adequately interpreted by an appropriate RIS mechanism.

An explicit and computable criterion for strong ergodicity of single-birth processes is presented. As an application, some sufficient conditions are obtained for strong ergodicity of an extended class of continuous-time branching processes and multi-dimensional Q-processes by comparison methods respectively. Consequently strong ergodicity of the Q-process corresponding to the finite-dimensional Schlögl model is proven.

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.