Iterative sampling procedures of a general type in a finite population are considered. They generalize the Reed-Frost process in that binomial sampling is replaced by an arbitrary symmetric sampling defined by a factorial series distribution. Threshold limit theorems are proved saying that the total number of sampled objects is either small with a certain limit distribution, or a finite fraction of the population with a Gaussian limit distribution as the size of the population gets large. These results extend earlier ones for the Reed-Frost process [1], and are proved in a more direct way than before.

A general exchangeable model is introduced to study gene survival in populations whose size changes without density dependence. Necessary and sufficient conditions for the occurrence of fixation (that is the proportion of one of the types tending to 1 with probability 1) are obtained. These are then applied to the Wright–Fisher model, the Moran model, and conditioned branching-process models. For the Wright–Fisher model it is shown that certain fixation is equivalent to certain extinction of one of the types, but that this is not the case for the Moran model.

Some classes of population-size-dependent Galton-Watson processes are considered where extinction occurs with probability 1. Results on the asymptotic behaviour of the probability of survival up to time t, mean population size and conditioned limit distributions are found to hold. They correspond to those obtained in the study of Galton-Watson processes with immigration stopped at 0.

A familiar relation links the densities that result for the intersection of a convex body and straight lines under uniform isotropic randomness with those that result under weighted randomness. An extension of this relation to the intersection of more general domains is utilized to obtain the variance of the n-dimensional measure of the intersection of two bodies under uniform isotropic randomness. The formula for the variance contains the point-pair-distance distributions for the two domains — or the closely related geometric reduction factors. The result is applied to derive the variance of the intersection of a Boolean scheme, i.e. a stationary, isotropic Poisson process of domains, with a fixed sampling region.

Point processes on the circle with circumference 1 are considered, which are related to the coverage problem of the circle by n randomly placed arcs of a fixed length. The anticlockwise endpoint of each arc is assumed to be uniformly distributed on the circle. We deal with a general limit result on the convergence of these point processes to a Poisson process on the circle. This result is then applied to several cases of the coverage problem, giving improved limit results in these cases. The proof uses a new convergence result of general point processes.

For each positive integer n, let Sn be the nth partial sum of a sequence of i.i.d. random variables which assume the values +1 and −1 with respective probabilities p and 1 – p, having mean μ= 2p − 1. The exact distribution of the random variable , where sup Ø= 0, is given for the case that λ > 0 and μ+ λ= k/(k + 2) for any non-negative integer k. Tables to the 99.99 percentile of some of these distributions, as well as a limiting distribution, are given for the special case of a symmetric simple random walk (p = 1/2).

The resolution of the optimal stopping problem for a partially observed Markov state process reduces to the computation of a function — the Snell envelope — defined on a measure space which is in general infinite-dimensional. To avoid these computational difficulties, we propose in this paper to approximate the optimal stopping time as the limit of times associated to similar problems for a sequence of processes converging towards the true state. We show on two examples that these approximating states can be chosen such that the Snell envelopes can be explicitly computed.

Consider a sequence U1, U2, · ·· of i.i.d. uniform (0, 1)-random variables. For fixed α ∈ (0, 1), let U(n, K) denote the [Kα]th order statistic of the subsample Un+1, · ··, Un+K, and set . Book and Truax (1976) proved the following analogue of the Erdös-Rényi (1970) strong law of large numbers: for α < u < 1 and C = C(α, u) such that −1/C = αlog(u/α)+ (1 – α)log((l – u)/(1 –α)), it holds almost surely that In view of the Deheuvels–Devroye (1983) improvements of the original Erdös-Rényi law, we determine the lim inf and lim sup of where K = [C log N]. This improves (∗), showing that it holds with a best-possible convergence rate of order O(log log N/log N). Using the quantile transformation the result can be extended to a general i.i.d. sequence X1, X2, · ·· with d.f. F satisfying a strict monotonicity condition.

Using a coupling technique, rates of convergence in total variation of certain k -dimensional diffusion processes to the stationary distribution are obtained. The results place assumptions only on the coefficients of the elliptic differential operator governing the diffusion.

In this paper a planar random motion governed by the two-dimensional telegraph equation is presented. It is proved that the particle performing motion is at any time t within a circle centred at the starting point and with radius . The explicit density of the particle position is obtained. Results concerning the trend of motion are also given.

The transition p.d.f. for a one-dimensional Rayleigh process in the presence of an absorption condition or a zero-flux condition in the origin is obtained in closed form. The first-passage-time problem through an arbitrary constant boundary is then considered and the moment-generating function is determined. In some particular cases the first-passage-time p.d.f. is explicitly derived. Use of some of these results is finally made to obtain the transition p.d.f. of the affine drift-linear infinitesimal-variance diffusion process when the origin is an entrance or a regular boundary in the presence of a reflection condition.

This paper extends a result of Godambe's theory of parametric estimation for discrete-time stochastic processes to the continuous-time case. Let P ={P} be a family of probability measures such that (Ω, F, P) is complete, (Ft, t≧0) is a standard filtration, and X = (Xt Ft, t ≧ 0) is a semimartingale for every P ∈ P. For a parameter θ (Ρ), suppose Xt = Vt,θ + Ht,θ where the Vθ process is predictable and locally of bounded variation and the Hθ process is a local martingale. Consider estimating equations for θ of the form process is predictable. Under regularity conditions, an optimal form for α θ in the sense of Godambe (1960) is determined. When Vt,θ is linear in θ the optimal , corresponds to certain maximum likelihood or least squares estimates derived previously in special cases. Asymptotic properties of , are discussed.

In assessing the reliability of a system of components, it is usual to suppose the components to fail independently of each other. Often this is inappropriate because the common environment acting on all components induces correlation. For example, a harsh environment will encourage early failure of all components. A simple model that incorporates such dependencies is described, and several properties of this model investigated. Calculations are carried out for a parallel system of two components. Inequalities for multicomponent systems are suggested. The results generalize easily.

For a large class of replacement models for stochastically deteriorating systems the optimality criteria of total expected discounted cost and long-run (expected) average cost per unit time have a common structure. In the present paper a formal description of this structure is given and the optimal rule is determined. A so-called ‘λ -minimization technique' is applied. This method is discussed in general terms.

We give a unified way of obtaining approximation formulas for the steady-state distribution of the queue length in the M/GI/s queue. The approximations of Hokstad (1978) and Case A of Tijms et al. (1981) are derived again. The main interest of this paper is in considering the theoretical meaning of the assumptions given in the literature. Having done this, we derive new approximation formulas. Our discussion is based on one version of the steady-state equations, called the basic equations in this paper. The basic equations are derived for M/GI/s/k with finite and infinite k. Similar approximations are possible for M/GI/s/k (k < +∞).

Two inverse problems in queueing are studied, namely: (i) a restoration of system characteristics, and (ii) a restoration of model classes. These problems are discussed in the framework of a general concept of characterization problems given by Kalashnikov and Rachev (1985).

In this paper we study the equilibrium behavior of the queue lengths in a product-form closed queueing network when the service rates at a subset of stations (nodes) are increased. Univariate and multivariate likelihood ratio orderings as well as multivariate stochastic ordering of the queue lengths are established to indicate the effect of increasing the service rates. Relations among these orderings are also developed.

In this paper, we give good-quality approximations for multiserver queues with general service-time distributions. For the interarrival-time distribution we consider hyperexponential distributions and mixtures of Erlang distributions with the same scale parameters. We give an algorithm to compute various operating characteristics such as the delay probability and the mean queue length.

The approximations are obtained by making assumptions regarding the residual service times of services in progress at service completion epochs, whereas the arrival process is modelled exactly. As the operating characteristics are far more sensitive to changes in the parameters of the arrival process, the latter is very important.

The quality of the approximations is extensively tested by using the exact results of more than 2000 cases. In heavy traffic the average relative error is up to 1% and in moderate traffic about 4%.

The system provides each customer with a service made up of a random number of stages in sequence. Arrivals are Poisson, and the stage times are independent exponential random variables. The number of stages of a particular service depends on the customer's random demand and on the arrival process, in a way which is aimed at preventing the queue from growing fast while matching the customer's demand as well as possible.

The Wright–Fisher model with varying population size is examined in the case where the selective advantage varies from generation to generation. Models are considered where the selective advantage is not always in favour of a particular genotype. Sufficient conditions in terms of the selection coefficients and the population growth are given to ensure ultimate homozygosity.