A spatial epidemic process where the individuals are located at positions in the Euclidean space R2 is considered. The infective individuals, with an infection period that is exponentially distributed with parameter µ, move in R2 according to a Brownian motion with a diffusion coefficient σ2. The susceptible individuals may also move. But we shall use the approximation that they remain unchanged in numbers and therefore assume that the averaged ‘density' of susceptibles per unit area is the same throughout space and time. The transition probability rate of infection of a susceptible in the infinitesimal element of area dy by an infective in dx is assumed to be a function h(x – y |) of the distance | x – y | between x and y. Then our process can be considered as a two-dimensional birth and death Brownian motion. Let be the number of infective individuals in the set D at time t and . The almost everywhere convergence of the random variables to a limit random variable W(D) is established.

The stochastic version of the simple epidemic is considered in this paper. A method which has computational advantages is developed for computing the probability distribution of the process. The method is based on the Jordan canonical form of the matrix representing the system of differential-difference equations for the simple epidemic. It is also shown that the method can be applied to the general epidemic as well as any other right-shift process.

In this paper it is shown that a random time-scale transformation leads to a simple derivation of some asymptotic results describing the progress of a major outbreak in the standard epidemic model. These results find application in approximation of the size distribution and in estimation of the threshold parameter.

In addition to the concept of the evolutionarily stable strategy (ESS), developed specifically for considering intraspecific conflicts, concepts such as the Nash equilibrium from game theory and the attractor or sink from dynamical systems theory appear relevant to the problem of characterizing populations of stable composition. The three concepts mentioned are discussed for one simple standard population model. It is found that evolutionarily stable strategies of one type are necessarily Nash equilibrium strategies, although the converse is not true. The dynamical systems characterization is found to provide a model for populations susceptible to invasion by ‘co-operative' strategies, but capable of evolving back in average to the original equilibrium.

A general branching process is extended to allow life length and reproduction probabilities to depend on randomly changing environmental states. First and second moments of the population size with respect to time are derived assuming that the environmental process is semi-Markovian. The results are similar to Markov-renewal type equations which allow, under discrete time, iterative computation of both moments of the population through time.

By assembling the transition matrix of a finite discrete Markov chain from overlays of matrices which are defined only by the serial correlation coefficients and marginal distribution of the chain to be modelled, a considerable saving is made in the number of parameters required to define a multilag Markov chain. This parsimony is achieved without detriment to the marginal distribution or serial correlation structure of the modelled chain. Applications to daily precipitation sequences and reservoir reliability are outlined to demonstrate the model's versatility.

Let {W(t), 0≦t<∞} be the standard Wiener process. The computation schemes developed in the past are not computationally efficient for the absorption probabilities of the type P{sup0≦t≦TW(t) − f(t) ≧ 0} when either T is large or f(0) > 0 is small. This paper gives an efficient and accurate algorithm to compute such probabilities, and some applications to other Gaussian stochastic processes are discussed.

We consider the general Markov chain on the vertices of a regular dodecahedron D such that P[Xn+1 = j | Xn = i] depends only on the distance between i and j. We consider also a Markov chain on the oriented edges (i, j) of D for which the only non-zero transition probabilities are and fix a vertex A. This paper computes explicitly P[Xn = A | X0 = A] and P[In = A | I0 = A]. The methods used are applicable to other solids.

In this paper the global behaviour of percolation processes on the d-dimensional square lattice is studied. Using techniques of Richardson (1973) we prove, under weak moment assumptions on the time coordinate distribution, the following result. There exists a norm N(·) on Rd such that, for all 0 < ε < 1, we have that almost surely for all sufficiently large t the N-ball of radius (1 – ε)t is contained in η, (the set of all sites occupied by time t) and η, is contained in the N-ball of radius (1 + ε)t. Richardson (1973) derived the corresponding ‘in probability' result for a class of spread processes on Rd, satisfying certain conditions.

A random mapping of a countable set N into itself is studied. An explicit formula is found for the probability that a subset of N will not contain cycles of the mapping. This formula is applied then for a detailed asymptotic analysis of two special mappings.

In recent publications of Hadwiger, Bokowski, and Wills it has been shown that it is possible to evaluate the average distance between a fixed and a randomly distributed convex set in terms of the mean projection measures of these sets. In the present note it is pointed out that these and more general results can be obtained from a theorem concerning the relationship between integrals over arbitrary measure spaces and corresponding Lebesgue–Stieltjes integrals.

The cumulative amount of time that a regenerative or semi-stationary process exceeds a high level and other measures of these exceedances are considered as special cases of a non-decreasing stochastic process of partial sums. We present necessary and sufficient conditions for these exceedance processes to converge in distribution to Poisson processes or processes with stationary independent non-negative increments as the level goes to infinity. We apply our results to random walks, M/M/s queues, and thinnings of point processes.

A general formula is obtained for the variance of the maximum of partial sums of n-exchangeable random variables, derived from a result of Spitzer's. The formula is applied in particular to obtain the variance of the maximum of adjusted rescaled partial sums of normal summands. This is of direct relevance to the Hurst effect.

The ‘snake' theorem, i.e., the weak convergence of the complex quantogram, is studied. The moment assumption is weakened by a strong approximation result of the author.

Let X(t) be a continuous, homogeneous stochastic process with independent increments characterized by a, σ, M, N in the Lévy representation formula. In this note we obtain the Lévy canonical representation of the characteristic function of a stochastic integral (in the sense of convergence in probability) of the form (where υ(t) is a non-decreasing, non-negative and left-continuous function) in terms of υ(t), a, σ, M, N.

We consider fractional derivatives of characteristic functions. We use these fractional derivatives for the formulation of new conditions for the existence of non-integer moments. We also compare the known conditions for the existence of moments of arbitrary random variables with our new conditions. As a consequence many conditions can be written in a unified terminology.

We introduce a class of increasing processes with independent increments that reduce to gamma processes under the further condition of stationarity. Each such process can be reduced to a simple gamma process by a stochastic integral transformation. Applications to deformation laws of materials such as concrete are mentioned.

For a maintained reliability system, i.e., components may be repaired upon failures, we prove that the distribution of time to first system failure starting with all components new is NBU under the following conditions: (a) non-repairable components have NBU uptime distributions, (b) repairable components have exponential uptime and DFR downtime distributions. This is a slight generalization of a theorem of Barlow and Proschan (1976). However, our method of proof is different from theirs. We also show that the initial up-period of the system is stochastically larger than any other system up-period. Finally, a method of finding a lower bound for the expected time until first system failure is outlined using our main result.

In this paper a formula is derived for the probability density function of the lifetime for a system of identical units. The lifetime of the system is the sum of the lifetimes of the independent units when the units have lifetimes distributed independently by the law proposed by Birnbaum and Saunders (1969).

This paper discusses infinite-server queues with phase-type input. The problems of obtaining the transient and steady-state distributions and moments of the queue length are reduced to the solution of certain well-behaved systems of linear differential equations. Sample computations, provided with as many as ten phases, show that although (even the time-dependent) mean queue length is very insensitive to substantial random variability in the arrival process, the higher moments of the queue length are highly sensitive. These examples indicate that considerable caution should be exercised in using robustness results for such stochastic models.