We analyze a class of linear birth and death processes X(t) with killing. The generator is of the form λ i = bi + θ, µi = ai, γ i = ci, where γ i is the killing rate. Then P{killed in (t, t + h) | X(t) = i} = γ ih + o(h), h ↓ 0. A variety of explicit results are found, and an example from population genetics is given.

The invariant probability distribution is found for a class of birth-and-death processes on the integers with phases and one or two boundaries. The invariant vector has a matrix geometric form and is found by solving a non-linear matrix equation and then finding an invariant probability distribution on the boundary states. Levy's concept of watching a Markov process in a subset is used to naturally decouple the computation of distributions on the boundary and interior states.

In each of a large number N of independent cells a breakdown mechanism is under way and proceeds until the first of the cells actually fails. At such a time, in each cell, the situation reverts to some initial state and the mechanism restarts. In this paper we consider those mechanisms for which breakdown may be modelled as the explosion of a pure birth process. Of interest is the distribution of time between failures and the possibility of estimating N and/or model parameters by observing a sequence of failure times. Saddlepoint approximation methods are used in the relevant extreme-value theory analysis for two important cases.

Results connecting Yaglom's limit and Z1 are obtained. For example, if α > 1, E[Zα1] and converge or diverge together.

Random digits are collected one at a time until a given k -digit sequence is obtained, or, more generally, until one of several k -digit sequences is obtained. In the former case, a recursive formula is given, which determines the distribution of the waiting time until the sequence is obtained and leads to an expression for the probability generating function. In the latter case, the mean waiting time is given until one of the given sequences is obtained, or, more generally, until a fixed number of sequences have been obtained, either different sequences or not necessarily different ones. Several results are known before, but the methods of proof seem to be new.

The optimal control of dynamic models which are not completely known to the controller often requires some kind of estimation of the unknown parameters. We present conditions under which a minimum contrast estimator will be strongly consistent independently of the control used. This kind of estimator is appropriate for the adaptive or ‘estimation and control' approach in dynamic programming under uncertainty. We consider a countable-state Markov renewal model and we impose bounding and recurrence conditions of the so-called Liapunov type.

Consider the placement of a finite number of arcs on the circle of circumference 2π where the midpoint and length of each arc follows an arbitrary bivariate distribution. In the case where each arc has lengthπ, the probability that the circle is completely covered is equal to the probability that the convex hull of a finite random sample of points, chosen according to a certain bivariate distribution in the plane contains the origin. In general, we show that evaluating the probability that the random convex hull contains a fixed disc is equivalent to solving the general coverage problem where the midpoint and length of each arc follows an arbitrary bivariate distribution. Exact formulae for the above probabilities are obtained and some examples are considered.

The coverage problem on the circle is considered from the shadowing process point of view. A random number of shadow arcs are distributed on a circle. The length of each arc is a random variable which depends on the random diameter of a shadowing disk and its random location. Formulae are derived for the numerical determination of the moments of the measure of vacancy of arcs on the circle, for a special example. An approximation to the distribution of the measure of vacancy is also provided.

Place n arcs of equal length an uniformly at random on the circumference of a circle. We discuss the limiting distributions of the number of gaps, the length of the maximum gap and the uncovered proportion of the circle, depending on the asymptotic behaviour of an → 0.

The Percus–Yevick approximation of classical liquid theory is employed to obtain variances associated with random configurations of equal spheres in ℝ3. The idea is illustrated by considering the number of spheres, and also the amount of hard material, contained within a fixed cylinder — an application of some practical value.

We consider a bivariate stochastic process where one component is an ordinary time series and the other is a point process. In the stationary case, a useful measure of the association of the time series and the point process is provided by a conditional intensity function, ṙ11(x;u), which gives the intensity with which events occur near time t given that the time series takes on a value x at time t + u. In this paper we consider the estimation of the function ṙ11(x;u) and certain related functions that are also useful in partially characterizing the degree of interdependence of the time series and the point process. Histogram and smoothed histogram-type estimates are proposed and asymptotic distributions of these estimates are derived. We also discuss an application of the estimation theory to the analysis of some data from a neurophysiological study.

Let {X(t), N(t)}, – ∞< t <∞, be a stationary bivariate stochastic process where X(t) is an ordinary time series and N(t) is an orderly point process counting the number of points in (0, t]. Suppose values of {X(t), N(t)} are available for 0< t ≦T and let σ1, σ2, ···, σ N(T) denote the jump points of N(t) in (0, T]. For |v| < T, define mT12(υ)=∑ X (σj+υ)/T and μT12(υ)=∑ X (σj+υ)/∑1 where all summations are over indices j such that 0<σj, σj+υ≦T for some σj. The functions MT12(υ) and μT12(υ) are often useful in analyzing the covariation of the time series and point process. In this paper, we shall develop some statistical properties of the functions MT12(υ) and μT12(υ) and discuss some specific situations where it is useful to consider these functions.

When the number of candidates is unknown, the problem of selecting the best during a sequence of interviews has many reasonable solutions. A simple condition for admissibility is established and it is shown that the class of Bayes solutions obtained by treating the number of candidates as a random variable is by no means complete. On the other hand, there is a single improper prior distribution for which the extended Bayes solutions constitute the whole family of admissible procedures.

For a sequence of random variables {Xn, n ≧ 0}, optimal stopping is considered over stopping times T constrained so that ET ≦ α, for some fixed α > 0. It is shown that under certain circumstances a Lagrangian approach may be used to reduce the problem to an unconstrained optimal stopping problem of a conventional type. The optimal value of the natural dual problem is shown to be equal to the optimal value of the original (primal) problem when certain randomised stopping times are permitted. Two examples are considered in detail.

In a previous paper, Natvig [4], we suggested a new measure of the importance of a component in a coherent system and derived some of its properties. The measure is for the case of components not undergoing repair proportional to the expected reduction in remaining system lifetime due to the failure of the component. In the present paper, we arrive at the whole distribution of this reduction in remaining system lifetime. Furthermore, for the case where components have proportional hazards, and are not repaired, a speculation of another measure is given. This measure is proportional to the derivative of the expected total lifetime of a new system with respect to the inverse of the component's proportional hazard rate.

A class of Lyapunov functions is used to demonstrate that strategy stability occurs in complex randomly mating diploid populations. Strategies close to the evolutionarily stable strategy tend to fare better than more remote strategies. If convergence in mean strategy to an evolutionarily stable strategy is not possible, evolution will continue until all strategies in use lie on a unique face of the convex hull of available strategies.

The results obtained are also relevant to the haploid parthenogenetic case.

Let X1 ··· Xn be i.i.d.r.v.'s uniformly distributed on [0, l]. Let X(1) ··· X(n) be the ordered r.v.'s 0≦ X(1) ≦ X(2)···≦ X(n) ≦ l. We obtain the n + 1 intervals (X(i), X(i+1))ni=0, with X(0) = 0, X(n+1)= l of length Li = X(i+1) – X(i). Let N(c) be the number of intervals of length Li < c. We have where The study of Nc is of some interest in molecular biology and was studied in [3] where the author found a complicated expression for the probability for a randomly chosen segment to have a length less than or equal to c; he then obtains an approximation. From the experimental point of view, it seems more reasonable to work with the mean since one can obtain an unbiased estimator that way.

A complete characterization of elementary pure birth processes is given by means of record counting processes from independent (non-identically) distributed random variables.

As a consequence of the convergence of the age distribution, several other limit results follow for supercritical Bellman-Harris processes. These concern the distribution of split times, the growth ‘rate' of the process, and the distribution of ancestral lifelengths.

A fundamental task of philologists is to construct the family tree (stemma) of preserved copies of ancient manuscripts. A simple probabilistic model based on random rooted trees is proposed to assist in the identification of the number of terminal copies. The model provides the distribution of the number of terminal vertices in a random tree. An application to stemma construction is given.