Symmetric diffusions on manifolds with boundary are studied. Symmetric diffusions are nicer to work with than non-symmetric diffusions because (1) it is easier to tell if an equilibrium density exists, and (2) it is easier to find the equilibrium density when it does exist. If an equilibrium density exists, a symmetric diffusion is time reversible. On the line essentially all diffusions are symmetric.

Using symmetric diffusions, it is shown that a large family of densities can be realized as equilibrium densities of time-reversible diffusions. Some examples are given.

In a supercritical branching random walk on Rp, a Galton–Watson process with the additional feature that people have positions, let be the set of positions of the nth-generation people, scaled by the factor n–1. It is shown that when the process survives looks like a convex set for large n. An analogous result is established for an age-dependent branching process in which people also have positions. In certain cases an explicit formula for the asymptotic shape is given.

A generalized semi-Markov process with speeds describes the fluctuation, in time, of the state of a certain general system involving, at any given time, one or more living components, whose residual lifetimes are being reduced at state-dependent speeds. Conditions are given for the stationary state distribution, when it exists, to depend only on the means of some of the lifetime distributions, not their exact shapes. This generalizes results of König and Jansen, particularly to the infinite-state case.

Markovian contact processes (mcp's) include some of the commoner models for the spatial spread of population processes, such as the ‘simple’ and ‘general’ epidemics, percolation processes, and birth, death and migration processes. They are here set in a framework which relates them to each other, and to a particular basic model, the contact birth process (cbp); indeed they are defined as modifications of the cbp. The immediate advantage of this is that bounds for the velocity of the cbp, which can be obtained from the linear equation describing its expected numbers (provided the contact distribution has exponentially bounded tail) apply also to general mcp's. Existence of moments (and for some processes their time-derivatives) of St(θ), the distance to the furthest individual in an arbitrary direction θ, can also be deduced whenever the corresponding moment of the contact distribution exists.

An important subclass consists of population-monotone mcp's, for which the distributions of St can be shown to be subconvolutive, so that the work of Hammersley (1974) can be applied to obtain convergence theorems for St/t; and these can be extended to convergence of the convex hull of the set of inhabited points. These results are particularly valuable because they apply to some non-linear processes, e.g. the simple epidemic. Some special results on the cbp (Section 6) emphasize the differences between linear and non-linear spatial stochastic processes.

Although the paper is written throughout in terms of continuous-time processes, the results on subconvolutive distributions are actually more easily applied in the discrete-time case. (Conversely, the approach used in the accompanying paper by Biggins (pp. 62–84), which is written in terms of discrete-time processes, can be extended to deal also with continuous-time processes.)

A plantation is modelled in terms of discs of random size centred on the points of a two-dimensional lattice, together with rules for partitioning the union of these discs. The resulting sets are functions of random sequences, and some properties of these functions are deduced. Results are given which relate various properties of measurements on a plantation to measurements of individual plants and of plants in isolation. These include limit theorems for complete plantations and for samples. They also lead to a rigorous demonstration of certain well-known empirical relations between typical measurements and survival frequencies and to some new relations which are amenable to test. The model is a formulation of computer simulations which have had success in describing competition in plantations, but are too costly for routine forest management.

Let be a family of random walks with For ε↓0 under certain conditions the random walk U(∊)n converges to an oscillating random walk. The ladder point distributions and expectations converge correspondingly. Let M∊ = max {U(∊)n, n ≧ 0}, v0 = min {n : U(∊)n = M∊}, v1 = max {n : U(∊)n = M∊}. The joint limiting distribution of ∊2σ∊–2v0 and ∊σ∊–2M∊ is determined. It is the same as for ∊2σ∊–2v1 and ∊σ–2∊M∊. The marginal ∊σ–2∊M∊ gives Kingman's heavy traffic theorem. Also lim ∊–1P(M∊ = 0) and lim ∊–1P(M∊ < x) are determined. Proofs are by direct comparison of corresponding probabilities for U(∊)n and for a special family of random walks related to MI/M/1 queues, using the central limit theorem.

We consider a Markov chain on the d-dimensional (d-allelî) non-negative lattice points with the sum of components being N, for which one-step transition consists of two stages—independent reproduction and random sampling. Convergence to a degenerate diffusion process when N → ∞ is proved. We show how difference among alleles in means and variances of offspring numbers affects the limit diffusion, giving a rigorous multi-allelic version of a result of Gillespie.

This paper considers the problem of estimating the growth rate ρ of a p-type Galton–Watson process {Zn}. To this end, a general approach of possible independent interest to central limit theorems for discrete-time branching processes is developed. The idea is to adapt martingale central limit theory to martingale difference triangular arrays indexed by the set of all individuals ever alive. Iterated logarithm laws are derived by similar methods. Asymptotic distribution results and the a.s. asymptotic behaviour are derived for a maximum likelihood estimator based upon all parent–offspring combinations in a given number N of generations, and for the estimator which depends on the total generation sizes only.

The stationary characteristics of an n-component periodic review system which is subject to stochastic deterioration (but not to failure) are investigated. When the n-component vector which expresses the state of deterioration of the system pierces a certain surface the entire multicomponent system is replaced by items of identical cost structure at the time of the next review. In the absence of this situation nothing is replaced. It is assumed that there is a fixed cost associated with each replacement and that the operating cost of each item is a strictly increasing function of its state of deterioration. The conditions for minimizing the long-term cost of maintaining a system which operates under the stated policy were found through solution of a problem in variational calculus. Two examples are worked. A useful graph which aids in the solution of such problems is provided.

A random spatial series is a collection of random variables F(xi, · · ·, xn) depending on several spatial coordinates (x1, · · ·, xn). An attempt is made to construct a statistical second-order theory of such series when (x1, · · ·, xn) varies over a regular cartesian lattice. Using the properties of the linear (Hilbert) space associated with the series, the concepts of innovation and purely non-deterministic (p.n.d.) series are introduced. For a p.n.d. series F(x1, · · ·, xn) a unilateral representation is obtained in terms of a white innovations series Z(y1, · · ·, yn) where The representatation is specialized to the homogeneous case and we discuss spectral conditions for p.n.d. The familiar time-series condition ∫ log f(λ) dλ > –∞ on the spectral density f is necessary but not sufficient. A sufficient condition is stated. Motivated by the p.n.d. unilateral representation results we define unilateral arma (autoregressive-moving average) spatial series models. Stability and invertibility conditions are formulated in terms of the location of zero sets of polynomials relative to the unit polydisc in Cn, and a rigorous shift operator formalism is established. For autoregressive spatial series a Yule–Walker-type matrix equation is formulated and it is shown how this can be used to obtain estimates of the autoregressive parameters. It is demonstrated that under mild conditions the estimates are consistent and asymptotically normal.

In the present paper a spatially homogeneous distance measure of Edwards and Cavalli-Sforza type is derived for multiple allele random genetic drift with a (possibly vanishing) symmetric mutation field, using the technique of random time substitution. Since the mutation field is of gradient type in gene frequency space, the transformed process is proved to be Brownian motion relative to a new Riemannian geometry and a new time measure. The new geometry is conformally related to spherical geometry of the original process but is not of constant curvature, generally. A formula relating the stationary density of the old and new process is derived and the edge length formula for the new geometry on the n-simplex frequency space is given and analysed.

The principal results of this paper concern the asymptotic behavior of the number of arcs in the optimal routes of first-passage percolation processes on the square lattice. Assuming that the underlying distribution has an atom at zero less than λ–1, where λ is the connectivity constant, Lp and (in some cases) almost sure convergence theorems are proved for the normalized route length processes. The proofs involve the extension of much of the existing theory of first-passage percolation to the case where negative time coordinates are permitted.

Let qn(t) be the conditioned probability of finding a birth-and-death process in state n at time t, given that absorption into state 0 has not occurred by then. A family {q1(t), q2(t), · · ·} that is constant in time is a quasi-stationary distribution. If any exist, the quasi-stationary distributions comprise a one-parameter family related to quasi-stationary distributions of finite state-space approximations to the process.

Consider a pair of single server queues arranged in series. (This is the simplest example of a queuing network.) In an earlier paper [2], a limit theorem was proved to justify a heavy traffic approximation for the (two-dimensional) equilibrium waiting-time distribution. Specifically the waiting-time distribution was shown to be approximated by the limit distribution F of a certain vector stochastic process Z. The process Z was defined as an explicit, but relatively complicated, transformation of vector Brownian motion, and the general problem of determining F was left unsolved.

It is shown in this paper that Z is a diffusion process (continuous strong Markov process) whose state space S is the non-negative quadrant. On the interior of S, the process behaves as an ordinary vector Brownian motion, and it reflects instantaneously at each boundary surface (axis). At one axis, the reflection is normal, but at the other axis it has a tangential component as well. The generator of Z is calculated.

It is shown that the limit distribution F is the solution of a first-passage problem for a certain dual diffusion process Z∗. The generator of Z∗ is calculated, and the analytical theory of Markov processes is used to derive a partial differential equation (with boundary conditions) for the density f of F. Necessary and sufficient conditions are found for f to be separable (for the limit distribution to have independent components). This extends slightly the class of explicit solutions found previously in [2]. Another special case is solved explicitly, showing that the density is not in general separable.