J. D. Biggins

Sperm Precedence in the Domestic Fowl

The aim of this study was to examine last-male sperm precedence in the domestic fowl. We used sperm from two different genotypes to assign paternity, and in seven experiments females were artificially inseminated with either equal or unequal numbers of sperm at intervals of 4 or 24 h. We were unable to replicate the results of a previous study by Compton et al. (1978) in which a strong last-male precedence effect had been recorded when two equal sized inseminations were made 4 h apart. We observed no marked last-male sperm precedence and our results did not differ significantly from that predicted by the passive sperm loss model, in which a last-male effect is determined by the rate at which sperm are lost from the female tract and the interval between successive inseminations. The most likely explanation for the disparity between our result and Compton et al.s is a difference in the timing of inseminations. The implications of this for studies of sperm competition in birds is discussed.

The asymptotic shape of the branching random walk

In a supercritical branching random walk on R p , a Galton–Watson process with the additional feature that people have positions, let be the set of positions of the n th-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.

The supercritical Galton- Watson process in varying environments

Let {Zn} be a supercritical Galton-Watson process in varying environments. It is known that Zn when normed by its mean EZn converges almost surely to a finite random variable W. It is possible, however, for such a process to exhibit more than one rate of growth so that in particular {W> 0} need not coincide with {Zn --> [infinity]}. Here a natural sufficient condition is given which ensures that this cannot happen. Under a weaker condition it is shown that the possible rates of growth cannot differ very much in that {Zn/EZn}1/n --> 1 on {Zn --> [infinity]}.

Lindley-type equations in the branching random walk

An analogue of the Lindley equation for random walk is studied in the context of the branching random walk, taking up the studies of Karpelevich, Kelbert and Suhov [(1993a) In: Boccara, N., Goles, E., Martinez, S., Picco, P. (Eds.), Cellular Automata and Cooperative Behaviour. Kluwer, Dordrecht, pp. 323-342; (1994a) Stochast. Process. Appl. 53, 65-96]. The main results are: (i) close to necessary conditions for the equation to have a solution, (ii) mild conditions for there to be a one-parameter family of solutions and (iii) mild conditions for this family to be the only possible solutions.

Near-constancy phenomena in branching processes

The occurrence of certain ‘near-constancy phenomena’ in some aspects of the theory of (simple) branching processes forms the background for the work below. The problem arises out of work by Karlin and McGregor [8, 9]. A detailed study of the theoretical and numerical aspects of the Karlin–McGregor near-constancy phenomenon was given by Dubuc[7], and considered further by Bingham[4]. We give a new approach which simplifies and generalizes the results of these authors. The primary motivation for doing this was the recent work of Barlow and Perkins [3], who observed near-constancy in a framework not immediately covered by the results then known.

HOW FAST DOES A GENERAL BRANCHING RANDOM WALK SPREAD

New results on the speed of spread of the one-dimensional spatial branching process are described. Generalizations to the multitype case and to d dimensions are discussed. The relationship of the results with deterministic theory is also indicated. Finally the theory developed is used to re-prove smoothly (and improve slightly) results on certain data-storage algorithms arising in computer science.

A note on the growth of random trees

We provide a unification and generalization of several recent results on the asymptotics, as the number of nodes increases, of the heights of trees grown according to various rules.

Convergence results on multitype, multivariate branching random walks

We consider a multi-type branching random walk on d-dimensional Euclidian space. The~uniform convergence, as n goes to infinity, of a scaled version of the Laplace transform of the point process given by the nth generation particles of each type is obtained. Similar results in the one-type case, where the transform gives a martingale, were obtained in Biggins (1992) and Barral (2001). This uniform convergence of transforms is then used to obtain limit results for numbers in the underlying point processes. Supporting results, which are of interest in their own right, are obtained on (i) ‘Perron-Frobenius theory’ for matrices that are smooth functions of a variable λ∈L and are nonnegative when λ∈L −⊂L, where L is an open set in ℂ d , and (ii) saddlepoint approximations of multivariate distributions. The saddlepoint approximations developed are strong enough to give a refined large deviation theorem of Chaganty and Sethuraman (1993) as a by-product.

Multi-type branching in varying environment

This paper considers the asymptotic theory of the varying environment Galton-Watson process with a countable set of types. This paper examines the convergence in Lp and almost surely of the numbers of the various types when normalised by the corresponding expected number. The harmonic functions of the mean matrices play a central role in the analysis. Many previously studied models provide particular cases.

RANDOM WALK CONDITIONED TO STAY POSITIVE

A random walk that is certain to visit (0,∞) has associated with it, via a suitable htransform, a Markov chain called random walk conditioned to stay positive, which will be defined properly in a moment. In continuous time, if the random walk is replaced by Brownian motion then the analogous associated process is Bessel-3. Let φ(x) = log log x. The main result obtained here, which is stated formally in Theorem 1, is that, when the random walk has zero mean and finite variance, the total time that the random walk conditioned to stay positive is below x ultimately lies between Lx/φ(x) and Uxφ(x), for suitable (non-random) positive L and finite U , as x goes to infinity. For Bessel-3, the best L and U are identified.