The Voronoi tessellation generated by a Gibbs point process is considered. Using the algebraic formalism of polymer expansion, the limit theorem and the large deviation principle for the number of Voronoi vertices are proved.

In the present paper a spatially homogeneous distance measure of Edwards and Cavalli-Sforza type is derived for multiple allele random genetic drift with a (possibly vanishing) symmetric mutation field, using the technique of random time substitution. Since the mutation field is of gradient type in gene frequency space, the transformed process is proved to be Brownian motion relative to a new Riemannian geometry and a new time measure. The new geometry is conformally related to spherical geometry of the original process but is not of constant curvature, generally. A formula relating the stationary density of the old and new process is derived and the edge length formula for the new geometry on the n-simplex frequency space is given and analysed.

A particular queue, the general arrival, general service-time, infinite-server queue (GA/G/∞), is introduced and certain of its properties studied. Motivated by a life situation in which the interarrival times for service converge to 0, a different sort of regularity condition (involving a tail property of random measures) is imposed on the arrival process to prove various limit theorems. There are similarities to heavy-traffic theory.

First it is shown that any generalized inverse of the infinitesimal generator of an irreducible Markov chain can be used to compute the exact stationary distribution and all the expected first-passage times of the chain. In the special case of a single-server queue this allows all computations to be done with upper-triangular matrices.

Next it is shown that the effect of a perturbation of the infinitesimal generator on the stationary distribution and expected first-passage times can also be computed using generalized inverses. These results extend and generalize Schweitzer's [9] original work using fundamental matrices. It is then shown that any perturbation can be broken up into a series of perturbations each involving a single state.

Let Un be the time between the nth and (n + 1)th arrivals to a single-server queuing system, and Vn the nth arrival's service time. There are quite a few models in which {Un, Vn, n ≥ 1} is a regenerative sequence. In this paper, some light and heavy traffic limit theorems are proved solely under this assumption; some of the light traffic results, and all the heavy traffic results, are new for two such models treated earlier by the author; and all the results are new for the semi-Markov queuing model.

In the last three sections, the results are applied to a single-server queue whose input is the output of a G/G/1 queue functioning in light traffic.

Assuming that the forward rates ftu are semimartingales, we give conditions on their components under which the discounted bond prices are martingales. To achieve this, we give sufficient conditions for the integrated processes ftu=∫0uftvdv to be semimartingales, and identify their various components. We recover the no-arbitrage conditions in models well known in the literature and, finally, we formulate a new random field model for interest rates and give its equivalent martingale measure (no-arbitrage) condition.

We show that load-sharing models (a very special class of multivariate probability models for nonnegative random variables) can be used to obtain basic results about a multivariate extension of stochastic precedence and related paradoxes. Such results can be applied in several different fields. In particular, applications of them can be developed in the context of paradoxes which arise in voting theory. Also, an application to the notion of probability signature may be of interest, in the field of systems reliability.

A stack is a structural unit in an RNA structure that is formed by pairs of hydrogen bonded nucleotides. Paired nucleotides are scored according to their ability to hydrogen bond. We consider stack/hairpin-loop structures for a sequence of independent and identically distributed random variables with values in a finite alphabet, and we show how to obtain an asymptotic Poisson distribution of the number of stack/hairpin-loop structures with a score exceeding a high threshold, given that we count in a proper, declumped way. From this result we obtain an asymptotic Gumbel distribution of the maximal stack score. We also provide examples focusing on the computation of constants that enter in the asymptotic distributions. Finally, we discuss the close relation to existing results for local alignment.

For a one-locus haploid infinite population with discrete generations, the celebrated model of Kingman describes the evolution of fitness distributions under the competition of selection and mutation, with a constant mutation probability. This paper generalises Kingman’s model by using independent and identically distributed random mutation probabilities, to reflect the influence of a random environment. The weak convergence of fitness distributions to the globally stable equilibrium is proved. Condensation occurs when almost surely a positive proportion of the population travels to and condenses at the largest fitness value. Condensation may occur when selection is favoured over mutation. A criterion for the occurrence of condensation is given.

In this paper we establish an extension of the method of approximating optimal discrete-time stopping problems by related limiting stopping problems for Poisson-type processes. This extension allows us to apply this method to a larger class of examples, such as those arising, for example, from point process convergence results in extreme value theory. Furthermore, we develop new classes of solutions of the differential equations which characterize optimal threshold functions. As a particular application, we give a fairly complete discussion of the approximative optimal stopping behavior of independent and identically distributed sequences with discount and observation costs.

We consider statistical inference for a parametric cooperative sequential adsorption model for spatial time series data, based on maximum likelihood. We establish asymptotic normality of the maximum likelihood estimator in the thermodynamic limit. We also perform and discuss some numerical simulations of the model, which illustrate the procedure for creating confidence intervals for large samples.

We describe the processes obtained by time reversal of a class of stationary jump diffusion processes that model the dynamics of genetic variation in populations subject to repeated bottlenecks. Assuming that only one lineage survives each bottleneck, the forward process is a diffusion on [0,1] that jumps to the boundary before diffusing back into the interior. We show that the behavior of the time-reversed process depends on whether the boundaries are accessible to the diffusive motion of the forward process. If a boundary point is inaccessible to the forward diffusion then time reversal leads to a jump diffusion that jumps immediately into the interior whenever it arrives at that point. If, instead, a boundary point is accessible then the jumps off of that point are governed by a weighted local time of the time-reversed process.

As a version of the secretary problem, Ferguson, Hardwick and Tamaki (1992) considered an optimal stopping problem called the duration problem. The basic duration problem is the classical duration problem, in which the objective is to maximize the time of possession of a relatively best object when a known number of rankable objects appear in random order. In this paper we generalize this classical problem in two directions by allowing the number N (of available objects) to be a random variable with a known upper bound n and also allowing the objects to appear in accordance with Bernoulli trials. Two models can be considered for our random horizon duration problem according to whether the planning horizon is N or n. Since the form of the optimal rule is in general complicated, our main concern is to give to each model a sufficient condition for the optimal rule to be simple. For N having a uniform, generalized uniform, or curtailed geometric distribution, the optimal rule is shown to be simple in the so-called secretary case. The asymptotic results, as n → ∞, will also be given for these priors.

A stochastic model of a dynamic marker array in which markers could disappear, duplicate, and move relative to its original position is constructed to reflect on the nature of long DNA sequences. The sequence changes of deletions, duplications, and displacements follow the stochastic rules: (i) the original distribution of the marker array {…, X−2, X−1, X0, X1, X2, …} is a Poisson process on the real line; (ii) each marker is replicated l times; replication or loss of marker points occur independently; (iii) each replicated point is independently and randomly displaced by an amount Y relative to its original position, with the Y displacements sampled from a continuous density g(y). Limiting distributions for the maximal and minimal statistics of the r-scan lengths (collection of distances between r + 1 successive markers) for the l-shift model are derived with the aid of the Chen-Stein method and properties of Poisson processes.

This paper demonstrates the occurrence of the feature called BRAVO (balancing reduces asymptotic variance of output) for the departure process of a finite-buffer Markovian many-server system in the QED (quality and efficiency-driven) heavy-traffic regime. The results are based on evaluating the limit of an equation for the asymptotic variance of death counts in finite birth-death processes.

In this paper we estimate the expectation of the size of the largest component in a supercritical random geometric graph; the expectation tends to a polynomial on a rate of exponential decay. We sharpen the expectation's asymptotic result using the central limit theorem. Similar results can be obtained for the size of the biggest open cluster, and for the number of open clusters of percolation on a box, and so on.

In the literature on optimal stopping, the problem of maximizing the expected discounted reward over all stopping times has been explicitly solved for some special reward functions (including (x+)ν, (ex − K)+, (K − e− x)+, x ∈ ℝ, ν ∈ (0, ∞), and K > 0) under general random walks in discrete time and Lévy processes in continuous time (subject to mild integrability conditions). All such reward functions are continuous, increasing, and logconcave while the corresponding optimal stopping times are of threshold type (i.e. the solutions are one-sided). In this paper we show that all optimal stopping problems with increasing, logconcave, and right-continuous reward functions admit one-sided solutions for general random walks and Lévy processes, thereby generalizing the aforementioned results. We also investigate in detail the principle of smooth fit for Lévy processes when the reward function is increasing and logconcave.