This paper is an edited version of a talk given in Sheffield as part of the celebration of fifty years of the Applied Probability Trust and its journals. I sought to sketch the background to the Trust's formation in the development of different applications of probability during the previous century, and to draw lessons for the future of the discipline and therefore of its journals.

Regenerative phenomena were introduced some forty years ago to address problems in the theory of continuous-time Markov processes. The early work in the theory left a number of difficult unsolved problems, in the classification of $p$-functions, oscillation and inequalities, the multiplicative theory, and the theory of unbounded semi-$p$-functions. Recent years have shown progress on all of these fronts, and this paper surveys these results, while drawing attention to significant problems that remain open.

This paper explores the possibility of a calculus of variations powerful enough to prove inequalities for the p-functions of regenerative phenomena such as that conjectured by Davidson and proved by Dai. It is shown that this is unlikely to be achieved by compactifying the space of standard p-functions, and a more promising approach is that of working in a compact subspace. The analysis leads to a class of candidate p-functions which contains all the maxima of general functionals.

A Markov chain model for a battle between two opposing forces is formulated, which is a stochastic version of one studied by F. W. Lanchester. Solutions of the backward equations for the final state yield martingales and stopping identities, but a more powerful technique is a time-reversal analogue of a known method for studying urn models. A general version of a remarkable result of Williams and McIlroy (1998) is proved.

In the model of the OK Corral formulated by Williams and McIlroy [2]: ‘Two lines of gunmen face each other, there being initially m on one side, n on the other. Each person involved is a hopeless shot, but keeps firing at the enemy until either he himself is killed or there is no one left on the other side.’ They are interested in the number S of survivors when the shooting ceases, and the surprising result, for which they give both numerical and heuristic evidence, is that when m=n,

formula here

as m→∞ (where E denotes expectation).

It is the occurrence of this curious power of m, rather than any application to real gunfights, which makes the Williams–McIlroy process of interest. The purpose of this paper is to give an essentially elementary proof of the fact that if m and n are not too different, then S/m3/4 has a simple asymptotic distribution that leads at once to (1.1).

For a time in the mid-1970s probabilists were tantalized by a seemingly simple problem posed by Araki and Kakutani[3]. An interval is repeatedly divided by points chosen successively at random, the nth point being uniformly distributed over the largest of the n intervals formed by the first n − 1 points. Is this sequence of points asymptotically uniformly distributed?

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.

In § 33 of [2], Professor Williams asks whether there are L1 random variables† X, Y, Z (not identically zero) such that

A new Markov chain is introduced which can be used to describe the family relationships among n individuals drawn from a particular generation of a large haploid population. The properties of this process can be studied, simultaneously for all n, by coupling techniques. Recent results in neutral mutation theory are seen as consequences of the genealogy described by the chain.