The mean number of edges of a randomly chosen neighbouring cell of the typical cell in a planar stationary tessellation, under the condition that it has n edges, has been studied by physicists for more than 20 years. Experiments and simulation studies led empirically to the so-called Aboav's law. This law now plays a central role in Rivier's (1993) maximum entropy theory of statistical crystallography. Using Mecke's (1980) Palm method, an exact form of Aboav's law is derived. Results in higher-dimensional cases are also discussed.

We solve a problem proposed by V. Klee (1969). He asked for a calculation of κ, the expected value of V, the volume of a daughter tetrahedron whose vertices are chosen at random (i.e. independently and uniformly) in the interior of a parent tetrahedron of unit volume. We discover:

We also calculate the second, fourth and sixth moments of V.

The horizon ξ T(x) of a random field ζ (x, y) of right circular cones on a plane is investigated. It is supposed that bases of cones are centered at points sn = (xn, yn), n = 1, 2, ···, on the (X, Y)-plane, constituting a Poisson point process S with intensity λ0 > 0 in a strip ΠT = {(x, y): – ∞< x <∞, 0 ≦ y ≦ T}, while altitudes of the cones h1, h2, · ·· are of the form hn = hn* + f(yn), n = 1, 2, ···, where f(y) is an increasing continuous function on [0,∞), f(0) = 0, and h1*, h2*, · ·· is a sequence of i.i.d. positive random variables which are independent of the Poisson process S and have a distribution function F(h) with density p(h).

Let denote the expected mean number of local maxima of the process ξ T(x) per unit length of the X-axis. We obtain an exact formula for under an arbitrary trend function f(y). Conditions sufficient for the limit being infinite are obtained in two cases: (a) h1* has the uniform distribution in [0, H], f(y) = kyγ; (b) h1* has the Rayleigh distribution, f(y) = k[log(y + 1)]γ. (In both cases γ 0 and 0 < k∞.) The corresponding sufficient conditions are: 0 < γ< 1 in case (a), 0 < γ< 1/2 in case (b).

In this paper a simple approximation scheme is proposed for the problem of generating and computing expectations of functionals of a wide class of random variables with values in a compact Lie group. The algorithm is suggested by the time-discretization of an ergodic diffusion leaving invariant the distribution of interest. It is shown to converge as the discretization step goes to zero with the iterations in a natural way.

This paper is concerned with a model for the spread of an epidemic in a closed, homogeneously mixing population in which new infections occur at rate f(x, y) and removals occur at rate g(x, y), where x and y are the numbers of susceptible and infective individuals, respectively, and f and g are arbitrary but specified positive real-valued functions. Sequences of such epidemics, indexed by the initial number of susceptibles n, are considered and conditions are derived under which the epidemic processes converge almost surely to a birth and death process as n tends to infinity. Thus a threshold theorem for such an epidemic model is obtained. The results are extended to models which incorporate immigration and emigration of susceptibles. The theory is illustrated by several examples of models taken from the epidemic literature. Generalizations to multipopulation epidemics are discussed briefly.

We describe some asymptotic properties of a general S–I–R epidemic process in a large heterogeneous population. We assume that the infectives behave independently, that each infective has a generally distributed random number of contacts with the others in the population, and that among the initial susceptibles there is an arbitrary initial distribution of susceptibility. For the case of a large number of initial infectives, we demonstrate the asymptotic normality of the final size distribution as well as convergence of the final distribution of susceptibility as the population size approaches infinity. The relationship between the mean of the limiting final size distribution and the initial heterogeneity of susceptibility is explored, for a parametric example.

This paper describes the early and final properties of a general S–I–R epidemic process in which the infectives behave independently, each infective has a random number of contacts with the others in the population, and individuals vary in their susceptibility to infection. For the case of a large initial number of susceptibles and a small (finite) initial number of infectives, we derive the threshold behavior and the limiting distribution for the final state of the epidemic. Also, we show strong convergence of the epidemic process over any finite time interval to a birth and death process, extending the results of Ball (1983). These complement some results due to Butler (1994), who considers the case of a large initial number of infectives.

By imbedding the multitype version of the standard epidemic model in a multiparameter process, we derive a functional limit theorem for the total cost of the epidemics.

Let {Zn} be a supercritical Galton–Watson process in varying environments, and W be the limit of the non-negative martingale {Zn/EZn}. Under a condition which ensures that W is not identically equal to zero we give an upper bound on the possible rates of growth of the process on the set {W = 0}, and find a sufficient condition for the process to have only one rate of growth. We also give an example of a process whose offspring distributions have bounded pth moments, for some p > 1, and which has an infinite number of rates of growth.

Two urns initially contain r red balls and n – r black balls respectively. At each time epoch a ball is chosen randomly from each urn and the balls are switched. Effectively the same process arises in many other contexts, notably for a symmetric exclusion process and random walk on the Johnson graph. If Y(·) counts the number of black balls in the first urn then we give a direct asymptotic analysis of its transition probabilities to show that (when run at rate (n – r)/n in continuous time) for as n →∞, where π n denotes the equilibrium distribution of Y(·) and γ α = 1 – α /β (1 – β). Thus for large n the transient probabilities approach their equilibrium values at time log n + log|γ α | (≦log n) in a particularly sharp manner. The same is true of the separation distance between the transient distribution and the equilibrium distribution. This is an explicit analysis of the so-called cut-off phenomenon associated with a wide variety of Markov chains.

We consider an absorbing semi-Markov chain for which each time absorption occurs there is a resetting of the chain according to some initial (replacement) distribution. The new process is a semi-Markov replacement chain and we study its properties in terms of those of the imbedded Markov replacement chain. A time-dependent version of the model is also defined and analysed asymptotically for two types of environmental behaviour, i.e. either convergent or cyclic. The results contribute to the control theory of semi-Markov chains and extend in a natural manner a wide variety of applied probability models. An application to the modelling of populations with semi-Markovian replacements is also presented.

A generalization of the notion of a stationary Markov chain in more than one dimension is proposed, and is found to be a special class of homogeneous Markov random fields. Stationary Markov chains in many dimensions are shown to possess a maximum entropy property, analogous to the corresponding property for Markov chains in one dimension. In addition, a representation of Markov chains in many dimensions is provided, together with a method for their generation that converges to their stationary distribution.

Let Φ = {Φ n} be an aperiodic, positive recurrent Markov chain on a general state space, π its invariant probability measure and f ≧ 1. We consider the rate of (uniform) convergence of Ex[g(Φ n)] to the stationary limit π (g) for |g| ≦ f: specifically, we find conditions under which

as n →∞, for suitable subgeometric rate functions r. We give sufficient conditions for this convergence to hold in terms of

(i) the existence of suitably regular sets, i.e. sets on which (f, r)-modulated hitting time moments are bounded, and

(ii) the existence of (f, r)-modulated drift conditions (Foster–Lyapunov conditions).

The results are illustrated for random walks and for more general state space models.

Learning from Matheron's representation (1973), and using the increment vector (PIV) methodology introduced by Cressie (1988) and developed by Chen and Anderson (1994), this paper presents a theory for the representation and decomposition of integrated stationary time series and gives some applications.

Using the electric network approach, we give closed-form formulas for the expected hitting times in the Ehrenfest urn model.

We make some comments on recent papers involving the integrated Cauchy functional equation or specialized versions of it, and reveal in particular that these papers give an inaccurate picture of the current state of the literature on the topic.