Parker's model (or the Scotch Auction) for a contest between two competitors has been studied by Rose (1978). He considers a form of the model in which every pure strategy is playable, and shows that there is no evolutionarily stable strategy (ess). In this paper, in order to discover more about the behaviour of strategies under the model, we shall assume that there are only a finite number of playable pure strategies I1, I2, ···, In where Ij is the strategy ‘play value mj′ and m1 < m2 < ··· < mn. The payoff matrix A for the contest is then given by where V is the reward for winning the contest, C is a constant added to ensure that each entry in A is non-negative (see Bishop and Cannings (1978)), and E[Ii, Ij] is the expected payoff for playing Ii against Ij. We also assume that A is regular (Taylor and Jonker (1978)) i.e. that all its rows are independent.

The paper discussed the value of the ‘Optimum Model” in describing the inter-relationship between metric characters and natural fitness. This model represents one extreme of a whole spectrum of models. It can be used to predict changes in the mean of the character under natural selection, but the basic premises underlying such use are not well understood.

We consider two distinct models of particle systems. In the first we have an infinite collection of identical Markov processes starting at random throughout Euclidean space. In the second a random sign is associated with each process. An interaction mechanism is introduced in each case via intersection local times, and the fluctuation theory of the systems studied as the processes become dense in space. In the first case the fluctuation theory always turns out to be Gaussian, regardless of the order of the intersections taken to introduce the interaction mechanism. In the second case, an interaction mechanism based on kth order intersections leads to a fluctuation theory akin to a :φ k: Euclidean quantum field theory. We consider the consequences of these results and relate them to different models previously studied in the literature.

We study the impact of a hard selective sweep on the genealogy of partially linked neutral loci in the vicinity of the positively selected allele. We consider a sexual population of stochastically varying size and, focusing on two neighboring loci, derive an approximate formula for the neutral genealogy of a sample of individuals taken at the end of the sweep. Individuals are characterized by ecological parameters depending on their genetic type, and governing their growth rate and interactions with other individuals (competition). As a consequence, the `fitness' of an individual depends on the population state and is not an intrinsic characteristic of individuals. We provide a deep insight into the dynamics of the mutant and wild-type populations during the different stages of a selective sweep.

The main purpose of this paper is to define and characterize random fields of bounded variation, that is, random fields with sample paths in the space of functions of bounded variation, and to study their mean total variation. Simple formulas are obtained for the mean total directional variation of random fields, based on known formulas for the directional variation of deterministic functions. It is also shown that the mean variation of random fields with stationary increments is proportional to the Lebesgue measure, and an expression of the constant of proportionality, called the variation intensity, is established. This expression shows, in particular, that the variation intensity depends only on the family of two-dimensional distributions of the stationary increment random field. When restricting to random sets, the obtained results give generalizations of well-known formulas from stochastic geometry and mathematical morphology. The interest of these general results is illustrated by computing the variation intensities of several classical stationary random field and random set models, namely Gaussian random fields and excursion sets, Poisson shot noises, Boolean models, dead leaves models, and random tessellations.

Partial differential equations are powerful tools for used to characterizing various physical systems. In practice, measurement errors are often present and probability models are employed to account for such uncertainties. In this paper we present a Monte Carlo scheme that yields unbiased estimators for expectations of random elliptic partial differential equations. This algorithm combines a multilevel Monte Carlo method (Giles (2008)) and a randomization scheme proposed by Rhee and Glynn (2012), (2013). Furthermore, to obtain an estimator with both finite variance and finite expected computational cost, we employ higher-order approximations.

We consider an extension of the Poisson hail model where the service speed is either 0 or ∞ at each point of the Euclidean space. We use and develop tools pertaining to sub-additive ergodic theory in order to establish shape theorems for the growth of the ice-heap under light tail assumptions on the hailstone characteristics. The asymptotic shape depends on the statistics of the hailstones, the intensity of the underlying Poisson point process, and on the geometrical properties of the zero speed set.

In this article we study the parabolic system of equations which is closely related to a multitype branching Brownian motion. Particular attention is paid to the monotone traveling wave solutions of this system. Provided with some moment conditions, we show the existence, uniqueness, and asymptotic behaviors of such waves with speed greater than or equal to a critical value c̲ and nonexistence of such waves with speed smaller than c̲.

We consider homogeneous STIT tessellations Y in the ℓ-dimensional Euclidean space ℝℓ and show the triviality of the tail σ-algebra. This is a sharpening of the mixing result by Lachièze-Rey (2001).

In this paper we show that the waiting and the sojourn times of a customer in a single-stage, multiple-server, G/G/c queueing system are increasing and starshaped with respect to the mean service time. Usefulness of this result in the design of the optimal service speed in the G/G/c queueing system is also demonstrated.

In Michel and Paroux (2003) the authors proposed a new proof of a well-known convergence result for the scaled elementary connected vacant component in the high intensity Boolean model towards the Crofton cell of the Poisson hyperplane process (see, e.g. Hall (1985)). In this paper we investigate the second-order term in this convergence when the two-dimensional Boolean model and the Poisson line process are coupled on the same probability space. We consider the particular case where the grains are discs with random radii. A precise coupling between the Boolean model and the Poisson line process is first established. A result of directional convergence in distribution for the difference of the two sets involved is then derived. Eventually, we show the convergence of the process, measuring the difference between the two random sets, once rescaled, as a function of the direction.

We propose a two-parameter family of conjugate prior distributions for the number of undiscovered objects in a class of Bayesian search models. The family contains the one-parameter Euler and Heine families as special cases. The two parameters may be interpreted respectively as an overall success rate and a rate of depletion of the source of objects. The new family gives enhanced flexibility in modelling.

The primary objective in the present paper is to gain fundamental understanding of the performance achievable in ATM networks as a function of the various system characteristics. We derive limit theorems that characterize the achievable performance in terms of the offered traffic, the admissible region, and the revenue measure. The insights obtained allow for substantial simplifications in the design of real-time connection admission control algorithms. In particular, we describe how the boundaries of admissible regions with convex complements may be linearized - thus reducing the admissible region - so as to obtain a convenient loss network representation. The asymptotic results for the achievable performance suggest that the potential reduction in revenue is immaterial in high-capacity networks. Numerical experiments confirm that the actual reduction is typically negligible, even in networks of moderate capacity.

The Holme‒Kim random graph process is a variant of the Barabási‒Álbert scale-free graph that was designed to exhibit clustering. In this paper we show that whether the model does indeed exhibit clustering depends on how we define the clustering coefficient. In fact, we find that the local clustering coefficient typically remains positive whereas global clustering tends to 0 at a slow rate. These and other results are proven via martingale techniques, such as Freedman's concentration inequality combined with a bootstrapping argument.

We consider a service system (QS) that operates according to the first-come-first-served (FCFS) discipline, and in which the service rate is an increasing function of the queue length. Customers arrive sequentially at the system, and decide whether or not to join using decision rules based upon the queue length on arrival. Each customer is interested in selecting a rule that meets a certain optimality criterion with regard to their expected sojourn time in the system; as a consequence, the decision rules of other customers must be taken into account. Within a particular class of decision rules for an associated infinite-player game, the structure of the Nash equilibrium routeing policies is characterized. We prove that, within this class, there exist a finite number of Nash equilibria, and that at least one of these is nonrandomized. Finally, with the aid of simulation experiments, we explore the extent to which the Nash equilibria are characteristic of customer joining behaviour under a learning rule based on system-wide data.

Quasi-stationary distributions (QSDs) arise from stochastic processes that exhibit transient equilibrium behaviour on the way to absorption. QSDs are often mathematically intractable and even drawing samples from them is not straightforward. In this paper the framework of sequential Monte Carlo samplers is utilised to simulate QSDs and several novel resampling techniques are proposed to accommodate models with reducible state spaces, with particular focus on preserving particle diversity on discrete spaces. Finally, an approach is considered to estimate eigenvalues associated with QSDs, such as the decay parameter.

Consider a bundle that has been constructed by placing n fibers in parallel between two clamps. The simplest assumption on fiber load sharing is that all non-failed fibers share the load applied to the bundle equally. Phoenix (1978b) shows that, as n tends to ∞, the time to failure of such a bundle is asymptotically normally distributed under certain assumptions on the stochastic nature of the fibers. In this paper we show that this asymptotic normality is preserved but with changed mean and variance if random slacks are introduced, as caused by random variations in the lengths of the fibers. A theorem on random linear combinations of order statistics, which is given in the appendix, is used to prove the result.

We obtain series expansions of the q-scale functions of arbitrary spectrally negative Lévy processes, including processes with infinite jump activity, and use these to derive various new examples of explicit q-scale functions. Moreover, we study smoothness properties of the q-scale functions of spectrally negative Lévy processes with infinite jump activity. This complements previous results of Chan et al. (Prob. Theory Relat. Fields 150, 2011) for spectrally negative Lévy processes with Gaussian component or bounded variation.