A body E is completely embedded within a convex body G. A line segment is generated by a measure depending only on E or on both E and G. This line segment is then projected to the surface of G in one or both directions. The distributions of selected line segments of this type are derived.

A detailed study of the asymptotic behavior of the first-passage-time p.d.f. and its moments is carried out for an unrestricted conditional Ornstein-Uhlenbeck process and for a constant boundary. Explicit expressions are determined which include those earlier discussed by Sato [15] and by Nobile et al. [9]. In particular, it is shown that the first-passage-time p.d.f. can be expressed as the sum of exponential functions with negative exponents and that the latter reduces to a single exponential density as time increases, irrespective of the chosen boundary. The explicit expressions obtained for the first-passage-time moments of any order appear to be particularly suitable for computation purposes.

The effects of computer technology on various aspects of mathematical printing are discussed. A brief account is given of the use of a microcomputer and commercial software in a publishing office environment.

Let X(t) be a temporally homogeneous irreducible Markov chain in continuous time defined on . For k < i < j, let H = {k + 1, ···, j − 1} and let kTij (jTik) be the upward (downward) conditional first-passage time of X(t) from i to j(k) given no visit to . These conditional passage times are studied through first-passage times of a modified chain HX(t) constructed by making the set of states absorbing. It will be shown that the densities of kTij and jTik for any birth-death process are unimodal and the modes kmij (jmik) of the unimodal densities are non-increasing (non-decreasing) with respect to i. Some distribution properties of kTij and jTik for a time-reversible Markov chain are presented. Symmetry among kTij, jTik, and is also discussed, where , and are conditional passage times of the reversed process of X(t).

Let Tx be the time it takes for a pure jump process, which starts at 0, to cross a threshold x > 0. Sufficient conditions on the parameters of this process under which Tx has increasing failure rate average (IFRA), increasing failure rate (IFR) or logconcave density (PF2) are identified. The conditions for IFRA are weaker than those of Drosen (1986). Sufficient conditions on the parameter of a pure jump process for Tx to the IFR or PF2 are not available in the literature.

An attempt has been made to generalise the concept of exponential mixture to the multivariate case through the random environment model. Some important properties of this class of mixture distributions have been studied, along with characterising properties. A further generalisation for a type of dependent exponential distribution has also been made.

For the simple two-dimensional random walk on the vertices of a rectangular lattice, the asymptotic forms of several properties are well known, but their forms can be insufficiently accurate to describe the transient process. Inequalities with the correct asymptotic form are derived for six such properties. The rates of approach to the asymptotic form are derived. The accuracy of the bounds and some practical implications of the results are discussed.

For a plane domain the circle transform assigns to each circle its length of intersection with the domain. The problem is to determine the geometry of the domain given the mean of these intersection lengths as the circles' centers vary and given how that mean varies with circle radius. We also consider the mean square and higher moments of the intersection lengths — as a function of the circle radius. The discussion includes an identification of the geometric content of the mean when centers are inside of a disk and of the second moment when the centers are arbitrary points in the plane. The latter is equivalent to the distribution of distance between pairs of points of the domain.

An autoregressive process ARP(1) with Pareto-distributed inputs, analogous to those of Lawrance and Lewis (1977), (1980), is defined and its properties developed. It is shown that the stationary distributions are Pareto. Further, the maximum and minimum processes are asymptotically Weibull, and the ARP(1) process is shown to be closed under maximization or minimization when the number of terms is geometrically distributed. The ARP(1) process leads naturally to an extremal process in the sense of Lamperti (1964). Statistical inference for the ARP(1) process is developed. An absolutely continuous variant of the Pareto process is described.

A survey of the development of statistical techniques of field experimentation refers to the unified approach of R. A. Fisher, in which analysis and design were closely linked. Randomization of treatment positions bypassed the need for a precise stochastic model, but sometimes at the cost of diminished accuracy.

Alternative covariance methods associated with the name of J. S. Papadakis, taking note of the yields of neighbouring plots, have stimulated much research in recent years. The results to date suggest that, while improved accuracy can often be achieved, the assessment of accuracy is more dependent on the stochastic model. Robust methods for such assessment still need to be finalized; with these alternative methods a closer link between analysis and design is also advocated.

Vector autoregressive processes of the first order are considered which are non-negative and optimize a linear objective function. These processes may be used in stochastic linear programming with a dynamic structure. By using Tweedie's results from the theory of Markov chains, conditions for geometric rates of convergence to stationarity (i.e. so-called geometric ergodicity) and for existence and geometric convergence of moments of these processes are obtained.

Bounds on the rate of convergence measured by the variation distance are obtained for the number of large spacings and for two occupancy problems connected with multinomial and Pólya sampling. The bounds are derived by imbedding techniques together with the elementary coupling inequality.

We discuss the role of stochastic processes in modelling the life-cycle of a biological cell and the growth of cell populations. Results for multiphase age-dependent branching processes have proved invaluable for the interpretation of many of the basic experimental studies of the life-cycle. Moreover problems from cell kinetics, in particular those related to diurnal rhythm in cell-growth, have stimulated research into ‘periodic' renewal theory, and the asymptotic behaviour of populations of cells with periodic death rate.

An iterated sequence of Bernoulli trials is carried out and the success probability estimated at each point on the sequence by the current success ratio. We find the probability P1 that this estimate always lies above some pre-selected rational fraction p′, and its complement P2, the probability that it will reach p′ or below at least once. In the region p′ ≧ p, P1 = 0. In the region p′ < p, P1 ≠ 0 and is furthermore a discontinuous function of p′ at every rational p′.

The spacings have been widely applied to many situations, such as for example the coverage problem of the circle or the line, under the hypothesis of uniformity. The paper examines the asymptotic behaviour of the maximal spacings under non-uniformity, assuming that the existing density f has a minimum point m with f(m) = 0. The results are applied to the coverage of the circle by randomly placed arcs.

Let Xi, i ≧ 1, be a sequence of independent N(0, 1) random variables and Sj,m = Xj + · ·· + Xj+m–1, the jth moving sum. Let τ m = inf{j ≧ 1 : Sj,m > a} + m – 1, the boundary crossing time. Approximation in the spirit of Glaz and Johnson (1984), (1986) and Samuel-Cahn (1983) are given for Pr(τm > n), E(τ m), and σ (τ m),the standard deviation of τm.

A simple Bayesian model for oil exploration is suggested to investigate strategies for drilling. A condition on the way successes and failures affect the prior distribution implies a certain form of the detection mechanism. It is shown that the problem of finding strategies for drilling reduces to an optimal stopping problem. Two new families of distributions are obtained with generating functions related to classical work on partitions of integers. By using such distributions and simple mixtures of them as priors, the stopping problem can be solved explicitly. This leads to the construction of simple strategies and their effectiveness is demonstrated by evaluating suitable operating characteristics.

It is known that several different alleles can be maintained at a locus by selection, but only when the various genotypic fitnesses satisfy very special conditions. It is shown in this paper that a population with many possible mutations will evolve in such a way that these conditions arise naturally for most fitness regimes. The first steps are taken towards the assessment of the likely size and shape of the resulting stable polymorphisms.

Analysis of time-series models has, in the past, concentrated mainly on second-order properties, i.e. the covariance structure. Recent interest in non-Gaussian and non-linear processes has necessitated exploration of more general properties, even for standard models. We demonstrate that the powerful Markov property which greatly simplifies the distributional structure of finite autoregressions has an analogue in the (non-Markovian) finite moving-average processes. In fact, all the joint distributions of samples of a qth-order moving average may be constructed from only the (q + 1)th-order distribution. The usefulness of this result is illustrated by references to three areas of application: time-reversibility; asymptotic behaviour; and sums and associated point and count processes. Generalizations of the result are also considered.

Consider N urns into which n balls are dropped independently with equal probability of hitting each urn and constant probability p of staying in the urn. We find the characteristic function of the joint distribution of KN and LN where KN is the number of urns that have not been hit, and LN is the number of urns where all balls have fallen through. Furthermore, we study the asymptotic distribution of (KN, LN) as n, N → ∞ at various rates.