We consider some generalizations of the germ-grain growing model studied by Daley, Mallows and Shepp (2000). In this model, a realization of a Poisson process on a line with points Xi is fixed. At time zero, simultaneously at each Xi, a circle (grain) starts growing at the same speed. It grows until it touches another grain, and then it stops. The question is whether the point zero is eventually covered by some circle. In our note we expand this model in the following three directions. We study: a one-sided growth model with a fixed number of circles; a grain-growth model on a regular tree; and a grain-growth model on a line with non-Poisson distributed centres of the circles.

We consider the problem of estimating the boundary of a compact set S ⊂ ℝd from a random sample of points taken from S. We use the Devroye-Wise estimator which is a union of balls centred at the sample points with a common radius (the smoothing parameter in this problem). A universal consistency result, with respect to the Hausdorff metric, is proved and convergence rates are also obtained under broad intuitive conditions of a geometrical character. In particular, a shape condition on S, which we call expandability, plays an important role in our results. The simple structure of the considered estimator presents some practical advantages (for example, the computational identification of the boundary is very easy) and makes this problem quite close to some basic issues in stochastic geometry.

In this paper, we consider a model of social learning in a population of myopic, memoryless agents. The agents are placed at integer points on an infinite line. Each time period, they perform experiments with one of two technologies, then each observes the outcomes and technology choices of the two adjacent agents as well as his own outcome. Two learning rules are considered; it is shown that under the first, where an agent changes his technology only if he has had a failure (a bad outcome), the society converges with probability 1 to the better technology. In the other, where agents switch on the basis of the neighbourhood averages, convergence occurs if the better technology is sufficiently better. The results provide a surprisingly optimistic conclusion about the diffusion of the better technology through imitation, even under the assumption of extremely boundedly rational agents.

It is shown that the time to ruin and the recovery time in a risk process have the same distribution as the busy period in a certain queueing system. Similarly, the deficit at the time of ruin is distributed as the idle period in a single-server queueing system. These duality results are exploited to derive upper bounds for the expected time to ruin and the expected recovery time as defined by Egídio dos Reis (2000). When the claim size is generally distributed, Lorden's inequality is applied to derive the bounds. When the claim-size distribution is of phase type, tighter upper bounds are derived.

The full-information best-choice problem, as posed by Gilbert and Mosteller in 1966, asks us to find a stopping rule which maximizes the probability of selecting the largest of a sequence of n i.i.d. standard uniform random variables. Porosiński, in 1987, replaced a fixed n by a random N, uniform on {1,2,…,n} and independent of the observations. A partial-information problem, imbedded in a 1980 paper of Petruccelli, keeps n fixed but allows us to observe only the sequence of ranges (max - min), as well as whether or not the current observation is largest so far. Recently, Porosiński compared the solutions to his and Petruccelli's problems and found that the two problems have identical optimal rules as well as risks that are asymptotically equal. His discovery prompts the question: why? This paper gives a good explanation of the equivalence of the optimal rules. But even under the lens of a planar Poisson process model, it leaves the equivalence of the asymptotic risks as somewhat of a mystery. Meanwhile, two other problems have been shown to have the same limiting risks: the full-information problem with the (suboptimal) Porosiński-Petruccelli stopping rule, and the full-information ‘duration of holding the best’ problem of Ferguson, Hardwick and Tamaki, which turns out to be nothing but the Porosiński problem in disguise.

Stephens and Donnelly (2000) constructed an efficient sequential importance-sampling proposal distribution on coalescent histories of a sample of genes for computing the likelihood of a type configuration of genes in the sample. In the current paper a characterization of their importance-sampling proposal distribution is given in terms of the diffusion-process generator describing the distribution of the population gene frequencies. This characterization leads to a new technique for constructing importance-sampling algorithms in a much more general framework when the distribution of population gene frequencies follows a diffusion process, by approximating the generator of the process.

De Iorio and Griffiths (2004) developed a new method of constructing sequential importance-sampling proposal distributions on coalescent histories of a sample of genes for computing the likelihood of a type configuration of genes in the sample by simulation. The method is based on approximating the diffusion-process generator describing the distribution of population gene frequencies, leading to an approximate sample distribution and finally to importance-sampling proposal distributions. This paper applies that method to construct an importance-sampling algorithm for computing the likelihood of samples of genes in subdivided population models. The importance-sampling technique of Stephens and Donnelly (2000) is thus extended to models with a Markov chain mutation mechanism between gene types and migration of genes between subpopulations. An algorithm for computing the likelihood of a sample configuration of genes from a subdivided population in an infinitely-many-alleles model of mutation is derived, extending Ewens's (1972) sampling formula in a single population. Likelihood calculation and ancestral inference in gene trees constructed from DNA sequences under the infinitely-many-sites model are also studied. The Griffiths-Tavaré method of likelihood calculation in gene trees of Bahlo and Griffiths (2000) is improved for subdivided populations.

In this paper, we consider a failure point process related to the Markovian arrival process defined by Neuts. We show that it converges in distribution to a homogeneous Poisson process. This convergence takes place in the context of rare occurrences of failures. We also provide a convergence rate of the convergence in total variation of this point process using an approach developed by Kabanov, Liptser and Shiryaev for the doubly stochastic Poisson process driven by a finite Markov process.

In this paper we derive some of the main ergodicity properties of a class of Markov renewal processes and the associated marked point processes. This class represents a generic model of applied probability and is of importance in earthquake modeling, reliability theory and queueing.

This paper considers a Markovian bulk-arriving queue modified to allow both mass arrivals when the queue is idle and mass departures which allow for the possibility of removing the entire workload. Properties of queues which terminate when the server becomes idle are developed first, since these play a key role in later developments. Results for the case of mass arrivals, but no mass annihilation, are then constructed with specific attention being paid to recurrence properties, equilibrium queue-size structure, and waiting-time distribution. A closed-form expression for the expected queue size and its Laplace transform are also established. All of these results are then generalised to allow for the removal of the entire workload, with closed-form expressions being developed for the equilibrium size and waiting-time distributions.

The basic queueing system considered in this paper is the M/G/1 processor-sharing queue with or without impatience and with finite or infinite capacity. Under some mild assumptions, a criterion for the validity of the reduced-service-rate approximation is established when service times are heavy tailed. This result is applied to various models based on M/G/1 processor-sharing queues.

The Kesten-Stigum theorem for the one-type Galton-Watson process gives necessary and sufficient conditions for mean convergence of the martingale formed by the population size normed by its expectation. Here, the approach to this theorem pioneered by Lyons, Pemantle and Peres (1995) is extended to certain kinds of martingales defined for Galton-Watson processes with a general type space. Many examples satisfy stochastic domination conditions on the offspring distributions and suitable domination conditions combine nicely with general conditions for mean convergence to produce moment conditions, like the X log X condition of the Kesten-Stigum theorem. A general treatment of this phenomenon is given. The application of the approach to various branching processes is indicated. However, the main reason for developing the theory is to obtain martingale convergence results in a branching random walk that do not seem readily accessible with other techniques. These results, which are natural extensions of known results for martingales associated with binary branching Brownian motion, form the main application.

We consider a single-type supercritical or near-critical size-dependent branching process {Nn}n such that the offspring mean converges to a limit m ≥ 1 with a rate of convergence of order as the population size Nn grows to ∞ and the variance may vary at the rate where −1 ≤ β < 1. The offspring mean m(N) = m + μN-α + o(N-α) depends on an unknown parameter θ0 belonging either to the asymptotic model (θ0 = m) or to the transient model (θ0 = μ). We estimate θ0 on the nonextinction set from the observations {Nh,…,Nn} by using the conditional least-squares method weighted by (where γ ∈ ℝ) in the approximate model mθ,ν̂n(·), where ν̂n is any estimation of the parameter of the nuisance part (O(N-α) if θ0 = m and o(N-α) if θ0 = μ). We study the strong consistency of the estimator of θ0 as γ varies, with either h or n - h remaining constant as n → ∞. We use either a minimum-contrast method or a Taylor approximation of the first derivative of the contrast. The main condition for obtaining strong consistency concerns the asymptotic behavior of the process. We also give the asymptotic distribution of the estimator by using a central-limit theorem for random sums and we show that the best rate of convergence is attained when γ = 1 + β.

We propose a stochastic modelling of the PCR amplification process by a size-dependent branching process starting as a supercritical Bienaymé-Galton-Watson transient phase and then having a saturation near-critical size-dependent phase. This model allows us to estimate the probability of replication of a DNA molecule at each cycle of a single PCR trajectory with a very good accuracy.

We analyse the role of Euler summation in a numerical inversion algorithm for Laplace transforms due to Abate and Whitt called the EULER algorithm. Euler summation is shown to accelerate convergence of a slowly converging truncated Fourier series; an explicit bound for the approximation error is derived that generalizes a result given by O'Cinneide. An enhanced inversion algorithm called EULER-GPS is developed using a new variant of Euler summation. The algorithm EULER-GPS makes it possible to accurately invert transforms of functions with discontinuities at arbitrary locations. The effectiveness of the algorithm is verified through numerical experiments. Besides numerical transform inversion, the enhanced algorithm is applicable to a wide range of other problems where the goal is to recover point values of a piecewise-smooth function from the Fourier series.

Discrete time-series models are commonly used to represent economic and physical data. In decision making and system control, the first-passage time and level-crossing probabilities of these processes against certain threshold levels are important quantities. In this paper, we apply an integral-equation approach together with the state-space representations of time-series models to evaluate level-crossing probabilities for the AR(p) and ARMA(1,1) models and the mean first passage time for AR(p) processes. We also extend Novikov's martingale approach to ARMA(p,q) processes. Numerical schemes are used to solve the integral equations for specific examples.