A simple convergence theorem for sequences of Markov chains is presented in order to derive new ‘convergence-to-the-coalescent’ results for diploid neutral population models.

For the so-called diploid Wright-Fisher model with selfing probability s and mutation rate θ, it is shown that the ancestral structure of n sampled genes can be treated in the framework of an n-coalescent with mutation rate ̃θ := θ(1-s/2), if the population size N is large and if the time is measured in units of (2-s)N generations.

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 problem of estimating an unknown compact convex set K in the plane, from a sample (X1,···,Xn) of points independently and uniformly distributed over K, is considered. Let Kn be the convex hull of the sample, Δ be the Hausdorff distance, and Δn := Δ (K, Kn). Under mild conditions, limit laws for Δn are obtained. We find sequences (an), (bn) such that

(Δn - bn)/an → Λ (n → ∞), where Λ is the Gumbel (double-exponential) law from extreme-value theory. As expected, the directions of maximum curvature play a decisive role. Our results apply, for instance, to discs and to the interiors of ellipses, although for eccentricity e < 1 the first case cannot be obtained from the second by continuity. The polygonal case is also considered.

We study the expected delay in a cyclic polling model with mixtures of exhaustive and gated service in heavy traffic. We obtain closed-form expressions for the mean delay under standard heavy-traffic scalings, providing new insights into the behaviour of polling systems in heavy traffic. The results lead to excellent approximations of the expected waiting times in practical heavy-load scenarios and moreover, lead to new results for optimizing the system performance with respect to the service disciplines.

We consider Brownian motion with a negative drift conditioned to stay positive. We give a sufficient condition for an initial measure to be in the domain of attraction of a quasi-stationary distribution. We construct a counter-example that strongly suggests that this condition is optimal.

‘Convergence-to-the-coalescent’ theorems for two-sex neutral population models are presented. For the two-sex Wright-Fisher model the ancestry of n sampled genes behaves like the usual n-coalescent, if the population size N is large and if the time is measured in units of 4N generations. Generalisations to a larger class of two-sex models are discussed.

The influence of bivariate extremal dependence on the limiting behaviour of the concomitant of the largest order statistic is examined. Our approach is to fix the marginal distributions and derive a general tail characterisation of the joint survivor function. From this, we identify the normalisation required to obtain the limiting distribution of the concomitant of the largest order statistic, obtain its tail form, and investigate the limiting probability that the vector of componentwise maxima occurs as an observation of the bivariate process. The results are illustrated for a range of extremal dependence forms.

In this paper we study the so-called random coeffiecient autoregressive models (RCA models) and (generalized) autoregressive models with conditional heteroscedasticity (ARCH/GARCH models). Both models can be represented as random systems with complete connections. Within this framework we are led (under certain conditions) to CL-regular Markov processes and we will give conditions under which (i) asymptotic stationarity, (ii) a law of large numbers and (iii) a central limit theorem can be shown for the corresponding models.

A new class of Gibbsian models with potentials associated with the connected components or homogeneous parts of images is introduced. For these models the neighbourhood of a pixel is not fixed as for Markov random fields, but is given by the components which are adjacent to the pixel. The relationship to Markov random fields and marked point processes is explored and spatial Markov properties are established. Extensions to infinite lattices are also studied, and statistical inference problems including geostatistical applications and statistical image analysis are discussed. Finally, simulation studies are presented which show that the models may be appropriate for a variety of interesting patterns, including images exhibiting intermediate degrees of spatial continuity and images of objects against background.

We consider a problem similar in many respects to a finite horizon Markov decision process, except that the reward to the individual is a strictly concave functional of the distribution of the state of the individual at final time T. Reward structures such as these are of interest to biologists studying the fitness of different strategies in a fluctuating environment. The problem fails to satisfy the usual optimality equation and cannot be solved directly by dynamic programming. We establish equations characterising the optimal final distribution and an optimal policy π*. We show that in general π* will be a Markov randomised policy (or equivalently a mixture of Markov deterministic policies) and we develop an iterative, policy improvement based algorithm which converges to π*. We also consider an infinite population version of the problem, and show that the population cannot do better using a coordinated policy than by each individual independently following the individual optimal policy π*.