This second meeting In the ORSA/TIMS Applied Probability series was attended by 90 participants, more than a quarter of them from Australia, Belgium, Britain, Canada, Denmark, Finland, Hungary, Italy, Israel, Japan, The Netherlands and New Zealand. The remaining participants were from the USA, among them several graduate students.

This paper deals with continuous-time branching processes which allow a temporally-decreasing immigration whenever the population size is 0. In the critical case the asymptotic behaviour of the probability of non-extinction and of the first two moments is investigated and different types of limit theorems are also proved.

A single-type general branching population develops by individuals reproducing according to i.i.d. point processes on R+, interpreted as the individuals' ages. Such a population can be measured or counted in many different ways: those born, those alive or in some sub-phase of life, for example. Special choices of reproduction point process and counting yield the classical Galton–Watson or Bellman–Harris process. This reasonably self-contained survey paper discusses the exponential growth of such populations, in the supercritical case, and the asymptotic stability of composition according to very general ways of counting. The outcome is a strict definition of a stable population in exponential growth, as a probability space, a margin of which is the well-known stable age distribution.

After some introductory general remarks on recent investigations involving population models, two broad classes of stochastic model are discussed further, viz. spatial nearest-neighbour lattice models and doubly stochastic models.

In Section 1 of the paper, the first type of model is considered primarily for its relevance to recent work by the author and others affecting the practical design and analysis of replicated field experiments.

In Section 2, doubly stochastic processes are discussed more theoretically, particularly models investigated recently by the author involving infinitesimal transition operators in continuous time linear in the (variable) parameters.

Some new numerical results on extinction and other ‘absorption' probabilities are presented; these are intended to throw further light on the extent to which the assumption of ‘white-noise' variability of the parameters can be a useful approximation to more realistic models.

For the haploid genetic model of Moran, the joint distribution of the numbers of distinct ancestors of a collection of nested subsamples is derived. These results are shown to apply to the diffusion approximations of a wide variety of other genetic models, including the Wright–Fisher process. The results allow us to relate the ancestries of populations sampled at different times. Analogous results for a line-of-descent process that incorporates the effect of mutation are given. Some results about the ages of alleles in an infinite-alleles model are described.

After some introductory general remarks on recent investigations involving population models, two broad classes of stochastic model are discussed, further, viz., spatial nearest-neighbour lattice models, and doubly stochastic models.

We consider a diploid model of random selection. Assuming that the selective fitness is of the SAS–CFF type introduced by Gillespie, we obtain a diffusion approximation. We also discuss the existence of stationary distributions and recurrent properties.

An efficient computational approach to the analysis of finite birth-and-death models in a Markovian environment is given. The emphasis is upon obtaining numerical methods for evaluating stationary distributions and moments of first-passage times.

In this paper we obtain a limit theorem for discrete-parameter random evolutions. This theorem is then used to obtain diffusion approximations to the Wright-Fisher model in a Markovian environment and to sequences of stochastic difference equations.

We consider d-dimensional lattice systems of bounded real-valued spins with ferromagnetic random interaction between nearest neighbours. We establish an outer and, in two dimensions, an inner bound of the parameter region where spontaneous magnetization occurs. These bounds provide an estimate of the singularity of the critical temperature at the threshold for percolation through active bonds. We derive a relationship between the ferromagnetic region and the percolative region for a correlated site–bond percolation problem, and we investigate the latter. Bounds for the level sets of the expected magnetization are also obtained.

This paper is in three parts. The first deals with the algebraic and topological structure of spaces of rational transfer function linear systems—ARMAX systems, as they have been called. This structure theory is dominated by the concept of a space of systems of order, or McMillan degree, n, because of the fact that this space, M(n), can be realised as a kind of high-dimensional algebraic surface of dimension n(2s + m) where s and m are the numbers of outputs and inputs. In principle, therefore, the fitting of a rational transfer model to data can be considered as the problem of determining n and then the appropriate element of M(n). However, the fact that M(n) appears to need a large number of coordinate neighbourhoods to cover it complicates the task. The problems associated with this program, as well as theory necessary for the analysis of algorithms to carry out aspects of the program, are also discussed in this first part of the paper, Sections 1 and 2.

The second part, Sections 3 and 4, deals with algorithms to carry out the fitting of a model and exhibits these algorithms through simulations and the analysis of real data.

The third part of the paper discusses the asymptotic properties of the algorithm. These properties depend on uniform rates of convergence being established for covariances up to some lag increasing indefinitely with the length of record, T. The necessary limit theorems and the analysis of the algorithms are given in Section 5. Many of these results are of interest independent of the algorithms being studied.

There are two modes by which gross deformation can occur in a solid. Slips may occur along crystal planes, or the atomic planes may be torn apart and create microcracks. Both are highly stochastic phenomena requiring attention from probabilists. The present work attempts to understand the creation and growth of microcracks and their effect on the macro-level deformations.

Let v(x, t) denote the displacement of an infinitely long, idealized string performing damped vibrations caused by white noise.

Upper and lower bounds for the distribution of maxsv(x, s) and maxxv(x, t) are presented. The results are obtained by adapting Lévy-type inequalities and exploiting a connection of v(x, t) with the Ornstein-Uhlenbeck process through Slepian's theorem.

The case of forced-damped vibrations is also analysed. Finally, a section is devoted to the case of a semi-infinite string performing damped vibrations.

Professor Neuts opened the discussion by suggesting that Applied Probability appeared to be in a precarious situation in the USA. Applied probabilists were widely spread in mathematics, statistics, electrical engineering, biomathematics and other departments; the fact that they defied classification was a drawback. They were perhaps not pure enough to be accepted by the mathematicians, but yet not applied enough for the practitioners. ‘Where’ he asked ‘do we fit in?’.

Random walks on N associated with orthogonal polynomials have properties similar to classical random walks on . In fact such processes have independent increments with respect to a hypergroup structure on with a convolution and a Fourier transform which is the basic tool for their study. We illustrate these ideas by giving a description of the asymptotic behaviour (CLT and ILL) of the random walks associated with Gegenbauer's polynomials. Moreover we can then use these random walks as a reference scale to deduce asymptotic properties of other Markov chains on via a comparison theorem which is of independent interest.

A local limit theorem for is obtained, where τ a is the first time a random walk Sn with positive drift exceeds a. Applications to large-deviation probabilities and to the crossing of a non-linear boundary are given.

Consider a general single-server queue where the customers arrive according to a Poisson process whose rate is modulated according to an independent Markov process. The authors have previously reported on limits for the average delay in light and heavy traffic. In this paper we review these results, extend them to multiserver queues, and describe some approximations obtained from them for general delays.

We consider the statistical estimation problem of potential functions of Gibbs models on the plane lattice. We assume that the area on which a random point pattern is observed is sufficiently large and take an asymptotic point of view. The main result is to show the locally asymptotic normality of the Gibbs model under certain conditions. From this result we can show the optimality of the maximum likelihood estimator employing known results about locally asymptotic normal families, though a practical computation of the maximum likelihood estimator presents difficulties. An estimation procedure based on the method of moments is also proposed.

This paper discusses perturbation bounds for the stationary distribution of a finite indecomposable Markov chain. Existing bounds are reviewed. New bounds are presented which more completely exploit the stochastic features of the perturbation and which also are easily computable. Examples illustrate the tightness of the bounds and their application to bounding the error in the Simon–Ando aggregation technique for approximating the stationary distribution of a nearly completely decomposable Markov chain.

For certain stationary random sets X, densities Dφ (X) of additive functionals φ are defined and formulas for are derived when K is a compact convex set in . In particular, for the quermassintegrals and motioninvariant X, these formulas are in analogy with classical integral geometric formulas. The case where X is the union set of a Poisson process Y of convex particles is considered separately. Here, formulas involving the intensity measure of Y are obtained.