While the general principles involved in the formulation of random walk and Brownian motion equations (whether the random changes are directly on the position of a particle or individual, or on the velocity) are well-known, there are various situations considered in the literature involving the assumption of a constant speed (in magnitude). Thus the derivation by Goldstein (1951) of a one-dimensional wave-like equation involved the tacit assumption Ut = ± a, where Ut is the vector velocity dRt/dt,Rt being the (column) position vector (Bartlett (1957)). Biological models may involve the assumption of individuals moving at constant speed (cf. Kendall (1974)). Finally, the derivation of Schrodinger-type equations from Brownian motion models has sometimes involved the assumption U′tUt = c2, where c is the velocity of light (Cane (1967), (1975)).

The annual premium for a life assurance contract is obtained by equating the expected discounted value of the sum payable on death to the expected discounted value of the series of premiums, and loading the resultant net premium for expenses and contingencies. The approach is essentially deterministic. An adequate stochastic model of mortality is readily available, but it is of limited practical value. A life office writes a large number of policies on independent lives and, apart from the effects of a few very large policies, the overall mortality behaviour of its portfolio is effectively deterministic.

Let X(t) be a real-valued stochastic process with sample functions which are almost surely continuously differentiable. We say that X(t) has an upcrossing at level u at time t if X(t) = u, X′(t) > 0.

Suppose where S is a compact set (of say). Let Φ (mapping S into S) and Ψ be continuous on S and such that Ψ(xk) is monotone (non-decreasing, say) as k increases, and xk+1 = Φ(xk). Put Ψ∗ = lim (k → ∞)Ψ(xk); if {xk(i)}∞i=0 is a convergent subsequence of {xk}, with a limit-point α, then Ψ∗ = Ψ(α), and and

Consider a genetic population evolving in time. In particular, suppose that the population is very large, that there is a large number of genetic types (alleles) possible for the genes making up the population, and that the numbers of the various alleles change stochastically in time, subject to Mendel's law of inheritance. Assume that there are no selective differences among the genotypes, but that mutation may occur among the alleles.

Many species of birds have a characteristic clutch size which is either fixed at k or is of the form k or k + 1 for some appropriate integer k. For example, the ground-living pigeons lay two eggs, the white stork four and the sulphur-crested cockatoo two and rarely three. The spotted nightjar lays one egg and the white-tailed nightjar lays two. The (European) swift lays two or three eggs, the (Australian) wood swallows lay three or four eggs and a number of (Australian) finches lay five or six eggs. These clutch sizes should be the result of evolutionary forces and it could be expected that the clutch size, for a particular species and environment, is the one which results in maximization of the probability of survival of the species to each time point t. It is consequently of interest to see whether it is possible to prescribe a population process which provides a reasonable description of a bird population and which exhibits this maximization behaviour when the clutch size is either k or is k or k + 1. Such a model has been suggested by Heyde and Schuh (1978) and it is the object of the present note to indicate how this model may be extended while still retaining the desired characteristic features.

Quantum mechanics is usually described in the terminology of probability theory even though the properties of the probability spaces associated with it are fundamentally different from the standard ones of probability theory. For example, Kolmogorov's axioms are not general enough to encompass the non-commutative situations that arise in quantum theory. There have been many attempts to generalise these axioms to meet the needs of quantum mechanics. The focus of these attempts has been the observation, first made by Birkhoff and von Neumann (1936), that the propositions associated with a quantum-mechanical system do not form a Boolean σ-algebra. There is almost universal agreement that the probability space associated with a quantum-mechanical system is given by the set of subspaces of a separable Hilbert space, but there is disagreement over the algebraic structure that this set represents. In the most popular model for the probability space of quantum mechanics the propositions are assumed to form an orthocomplemented lattice (Mackey (1963), Jauch (1968)). The fundamental concept here is that of a partial order, that is a binary relation that is reflexive and transitive but not symmetric. The partial order is interpreted as embodying the logical concept of implication in the set of propositions associated with the physical system. Although this model provides an acceptable mathematical expression of the probabilistic structure of quantum mechanics in that the subspaces of a separable Hilbert space give a representation of an ortho-complemented lattice, it has several deficiencies which will be discussed later.

The theoretical analyses of the extensive air showers developing from the cosmic radiation has its origins in the work of Carlson and Oppenheimer (1937) and Bhabha and Heitler (1937), at a time when it was thought that such showers were initiated by electrons. The realization that protons and other nuclei were the primary particles led to a reformulation of the theory by Heitler and Janossy (1949), Messel and Green (1952) and others, in which the production of energetic pions and the three-dimensional development of air showers were accounted for. But as the soft (electromagnetic) component of the cosmic radiation is the most prominent feature of air showers at sea level, there has been a sustained interest in the theory of this component. Most of the more recent work, such as that by Butcher and Messel (1960) and Thielheim and Zöllner (1972) has relied on computer simulation; but this method has disadvantages in terms of accuracy and presentation of results, especially where a simultaneous analysis of the development of air showers in terms of several physical variables is required. This is so for instance when the time of arrival is one of the variables. Moyal (1956) played an important part in the analytical formulation of a stochastic theory of cosmic ray showers, with time as an explicit variable, and it is essentially this approach which will be adopted in the following. The actual distribution of arrival times is cosmic ray showers, for which results are obtained, is of current experimental interest (McDonald, Clay and Prescott (1977)).

There is an extensive literature on estimating a probability density (or some other appropriate curve) f using statistics of the form Here X1, X2, · · ·, Xn is a sample from the population, the weight function w is constrained by suitable regularity conditions, and the sequence {bn} of band-widths satisfies bn → 0, nbn → ∞ as n → ∞. Rosenblatt (1971) presents details of this method of estimation and provides an extensive list of references.

Consider, initially, a time series regression model of the simplest kind, namely Assume that x(t) is second-order stationary with zero mean and absolutely continuous spectrum with density f(ω) so that The y(t) are taken to be part of a sequence generated entirely independently of x(t) and will be treated as constants. Let βN be the least squares estimate of β and Call the numerator and denominator of b(N), respectively, c(N), d(N). We shall use K for a positive finite constant, not always the same one. We have the following result, which is Menchoff's inequality [3].

The general age-dependent branching model of Crump, Mode and Jagers will be generalized towards generation-dependent varying lifespan and reproduction distributions. A system of integral and renewal equations is established for the generating functions and the first two moments of Zi(t) (the number of individuals alive at time t), if the population was initiated at time 0 by one ancestor of age 0 from generation i. Convergence in quadratic mean of Zi(t)/EZi(t) as t tends to infinity is obtained if the generation-dependent reproduction functions converge to a supercritical one. In particular, if this convergence is slow enough tγ exp (αt) is the asymptotic behavior of EZi(t) for t tending to infinity, where γ is a positive real number and α the Malthusian parameter of growth of the limiting reproduction function.

At times n = 0, 1, 2, · · · a Poisson number of particles enter each state of a countable state space. The particles then move independently according to the transition law of a Markov chain, until their death which occurs at a random time. Several limit theorems are then proved for various functionals of this infinite particle system. In particular, laws of large numbers and central limit theorems are proved.

It is shown that every measure of expected information which has the branching property is of the form where J is a given information measure which is compositive under a regular binary operation and the Ψn are antisymmetric, bi-additive functions. In a probability space, such measures (entropies) take the form

The nature and historical development of both stochastic and deterministic models for binary graphs are discussed. Here the focus of applications is sociological and emphasizes representations of networks of interpersonal relations as directed graphs. Models from the natural sciences and from the social sciences are examined and suggestions for future research are given.

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.

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.

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.

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.

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.