Extensions of ESS theory to situations well outside the classical formulation often assume, as a convenience, that the population being modelled is, in some sense, monomorphic. While this assumption is in keeping with the original approach used in developing the theory, it is rendered less plausible by the observation that the original models do not preclude the possibility of polymorphism, a potentially serious omission. We consider a generalisation of the classical bilinear fitness function, and examine the circumstances that will tend to favour monomorphism.

We consider records from the sequence Yj = Xj, + cj, j ≧ 1 where c > 0 if upper records are of interest and c < 0 if lower records are studied. If {Xj} is stationary the record rate is asymptotically constant and if {Xj} is i.i.d. the record rate is, in addition, asymptotically normal. These results are illustrated by analysis of the times in the mile run.

The evolution of strategies in animal contests is examined where the dynamical equation takes into account population growth rates. This leads to a different definition of evolutionary stable strategy (ESS) from the usual one. Consequences for independent haploid species are then contrasted with the previous theory. Inheritance patterns for male–female contests with sex-dependent payoffs are considered. In particular, if males and females evolve independently to the same ESS, then so does the diploid species under random mating. Finally, the evolution of diploid populations where strategies are determined at a diallelic locus is investigated.

The paper is concerned with Markov renewal processes satisfying a certain non-singularity condition. The relation of this condition to irreducibility, Harris recurrence and regularity of the associated forward Markov process is studied. This enables one to prove limit theorems of a total variation type for Markov renewal processes and semi-regenerative processes by applying Orey's theorem to the forward process. The results are applied to a GI/G/1 queue and a growth-catastrophe population model.

We consider critical Bellman-Harris processes which admit an immigration component only in the state 0. The asymptotic behaviour of the probability of extinction and of the first two moments is investigated and a limit theorem is also proved.

We consider the model Zt = φ (0, k)+ φ(1, k)Zt–1 + at (k) whenever rk−1<Zt−1≦rk, 1≦k≦l, with r0 = –∞ and rl =∞. Here {φ (i, k); i = 0, 1; 1≦k≦l} is a sequence of real constants, not necessarily equal, and, for 1≦k≦l, {at(k), t≧1} is a sequence of i.i.d. random variables with mean 0 and with {at(k), t≧1} independent of {at(j), t≧1} for j ≠ k. Necessary and sufficient conditions on the constants {φ (i, k)} are given for the stationarity of the process. Least squares estimators of the model parameters are derived and, under mild regularity conditions, are shown to be strongly consistent and asymptotically normal.

The present paper deals with continuous-time Markov branching processes allowing immigration. The immigration rate is allowed to be random and time-dependent where randomness may stem from an external source or from state-dependence. Unlike the traditional approach, we base the analysis of these processes on the theory of multivariate point processes. Using the tools of this theory, asymptotic results on parametric inference are derived for the subcritical case. In particular, the limit distributions of some parametric estimators and of Pearson-type statistics for testing simple and composite hypotheses are established.

Sequence symmetries in DNA and RNA are being discovered at an increasing rate. Conjectures and hypotheses are being proposed for their possible structural and functional role in the nucleic acid. In this paper a probability model is studied which evaluates the probabilities of various repeats occurring by chance alone. Expressions are derived for the mean and variance of the statistics employed. The central limit theorem for dependent trials is used to obtain the asymptotic distributions. An indication is given of how to use the model to search for various gene amplification events in the evolutionary history of the sequences.

We consider first-passage percolation in an infinite horizontal strip of finite height. Using methods from the theory of Markov chains, we prove a central limit theorem for first-passage times, and compute the time constants for some special cases.

Some classes of population-size-dependent Galton–Watson processes {Z(t)}t=0,1, …, whose transition probability generating functions allow for certain upper or lower bounds, can be treated by means of functional iteration methods. Criteria for almost certain extinction are obtained as well as gammatype limit distributions for Z(t)/t as t → ∞ the results can be stated under conditions on moments of the reproduction distributions.

Equivalence of rates of convergence in the central limit theorem for the vector of maximum sums and the corresponding first-passage variables is established. A similar result for the vector of partial sums and the corresponding renewal variables is also given. The results extend to several dimensions the bivariate results of Ahmad (1981).

A theory for volume estimation from independent, unbounded plane probes is available. In practice, however, probes cannot be unbounded and independent at the same time, hence the interest of systematic sectioning. Previous empirical studies on real specimens showed that the mean square error of a volume ratio estimator obtained from m systematic sections behaved roughly as m–3 for small m. This drastic increase in efficiency with respect to independent sectioning prompted us to develop some theoretical models in this paper. We study specimens consisting of two ellipsoids in Rn. In virtue of an invariance result, an ellipsoid–ellipsoid model can be studied via a simple model consisting of two concentric n-balls. Explicit results are obtained for n = 1 and n = 3. In the latter case the bias and the mean square error of the relevant volume ratio estimator are both shown to be of O (m–4).

In this article we examine R&D projects where the project status changes according to a general dynamic stochastic equation. This allows for both continuous and jump behavior of the project status. The time parameter is continuous. The decision variable includes a non-stationary resource expenditure strategy and a stopping policy which determines when the project should be terminated. Characterization of stationary policies becomes straightforward in the present setting. A non-linear equation is determined for the expected discounted return from the project. This equation, which is of a very general nature, has been considered in certain special cases, where it becomes manageable. The examples include situations where the project status changes according to a compound Poisson process, a geometric Brownian motion, and a Brownian motion with drift. In those cases we demonstrate how the exact solution can be obtained and the optimal policy found.

The kin number problem in its simplest form is that of the relationship between sibship sizes and offspring numbers. The fact that the distributions are different, and the relationship between the two, is well known to demographers. It is important in such applications as estimating fertility from sibship rather than offspring counts. Further studies have been made, concerning relatives of other degrees of affinity than siblings, but these did not usually yield joint distributions. Recently this aspect of the problem has been studied in the framework of a Galton–Watson process (Waugh (1981), Joffe and Waugh (1982)).

In these studies the population is treated as monotype. Applications such as pedigree studies of diseases require a multitype approach (in the example, two types: victims and others). In this paper such a study is undertaken. The existence of more than one type complicates the time-reversal used in the previous studies, and raises questions about the way in which the focal individual (called ‘Ego') is sampled. These are dealt with, and joint distributions obtained, under a number of sampling schemes which might arise naturally.

We derive asymptotic forms for the distributions of k-point scan statistics as the interval L under study becomes infinite, while k and the window length are held fixed. In the Poisson case the intensity is also held fixed. In the uniform case the number of points N becomes infinite and N/L tends to a limit, representing a limiting intensity. These results are made explicit for k = 3, and in the Poisson case provide approximations which are typically accurate to six or seven figures, even for small L.

A methodology for an analytical derivation of visibility probabilities of n stationary target points in the plane is developed for the case when shadows are cast by a Poisson random field of obscuring elements. In addition, formulae for the moments of a measure of the total proportional visibility along a star-shaped curve are given.

Several different orderings for the comparison of point processes have been introduced and their relationships discussed in Whitt [9], Daley [2] and Deng [4]. It is of some interest to know whether these orderings, in general, are preserved under various operations on point processes. Some results concerning limit operations were given in Deng [4].

In the present paper, we first further introduce some convex and concave orderings for counting processes, and survey the relationships among all orderings mentioned in [9], [4] and this paper. Then we focus our attention on the study of the conditions for the preservation of orderings under the operations of superposition, thinning, shift, and random change of time.

An analogue of the Kesten–Stigum theorem, and sufficient conditions for the geometric rate of growth in the rth mean and almost surely, are obtained for population-size-dependent branching processes.

Let be i.i.d. uniform on (0,1) random variables and define Si,n = Ui,n–1Ui–1,n–1, i = 1, · ··, n where the Ui–n–1 are the order statistics from a sample of size n – 1 and U0,n–1 =0 and Un,n–1 = 1. The Si,n are called the spacings divided by U1,· ··,Un–1. For a fixed integer l, set . Exact and weak limit results are obtained for the Ml,n. Further we show that with probability 1 This extends results of Cheng.

We consider the problem of finding a betting strategy for an infinite sequence of wagers where the optimality criterion is the minimization of the expected exit time of wealth from an interval. We add the side constraint that the right boundary is hit first with at least some specified probability. The optimal strategy is derived for a diffusion approximation.