A simple argument demonstrates that the mean strategy of a diploid sexual population at evolutionary equilibrium can be expected to be an evolutionarily stable strategy (ESS) in the formal sense. This result follows under a wide set of models of genetic inheritance of strategy (including sexual selection) provided that the ESS is both attainable and maintainable.

A coherent system of components, the failure pattern of each of which being assumed to be modelled by an alternating renewal process, is considered. Formulae are derived for the probability that the system is available at a series of points and for the expected numbers of failures and repairs of the system in a fixed interval. It is shown how the model can be generalised to permit each component to exhibit partial availability in the failed state.

The sequential broken stick model has appeared in numerous contexts, including biology, physics, engineering and geology. Kolmogorov showed that under appropriate conditions, sequential breakage processes often yield a lognormal distribution of particle sizes. Of particular interest to ecologists is the observed variance of the logarithms of the sizes, which characterizes the evenness of an assemblage of species. We derive the first two moments for the logarithms of the sizes in terms of the underlying distribution used to determine the successive breakages. In particular, for a process yielding n pieces, the expected sample variance behaves asymptotically as log(n). These results also yield a new identity for moments of path lengths in random binary trees.

Several stochastic models have been proposed to describe the kinetic theory of reversible chemical reactions. However, in macroscopic systems the effects of stochastic variability are often outweighed by mean effects. In the present paper we show that some observed phenomena can be explained quite adequately by a stochastic model in which the stochastic variability is not negligible in comparison with mean effects. Our argument involves approximations to a stochastic model for competing chemical reactions.

We consider the problem of maximizing the probability of choosing the largest from a sequence of N observations when N is a bounded random variable. The present paper gives a necessary and sufficient condition, based on the distribution of N, for the optimal stopping rule to have a particularly simple form: what Rasmussen and Robbins (1975) call an s(r) rule. A second result indicates that optimal stopping rules for this problem can, with one restriction, take virtually any form.

The concept of a learning model (or random system with complete connections) with continuous time parameter is introduced on the basis of the notion of a multivariate point process possessing an intensity. The stepwise transition probabilities in terms of the intensity are derived and a Monte Carlo method for simulating a sample is presented. By modelling the intensity process various types of learning models can be built. We propose a linear learning model which comprises the continuous-time Markov process as well as Hawkes's mutually exciting point process. We study the asymptotic behaviour of this linear model in terms of explosion or extinction and of convergence of some estimates. We close with some numerical results from computer simulations.

In this paper we examine a continuous model of job search. An individual is seeking employment continuously over time. There is a search cost of c monetary units per unit time. Job offers are received randomly over time according to a renewal process. The wage offers are assumed to be positive, independent and identically distributed random variables. The only decision the searcher must make is when to stop searching and accept an offer. The objective is to determine a stopping rule that is optimal under the following optimality criteria:

1. (a) maximum expected net return,

2. (b) maximum expected discounted net return.

We investigate the case in which the stream of job offers is determined by a Poisson process. We show that under the two optimality criteria defined above an optimal stopping rule is determined by a single critical value ξ. The optimal strategy is to accept the first job offer that exceeds ξ. In the second part of the article, the search model in which offers are received according to a general renewal process is examined.

An expression is derived for the proportion of quadrilaterals among the polygons formed by random lines in the plane. Its value is 0.381 466….

This note obtains the distribution of N(a), the number of independent random variables each distributed uniformly on (0, 1) which must be added for the sum first to exceed some positive constant a.

In This paper we answer the following question. Is there any a priori reason for supposing that there is no more than one set of ARMA model parameters minimising the one-step-ahead prediction error when the true system is not in the model set?

Necessary and sufficient conditions for the so-called Hurst effect are given in the case of a weakly dependent stationary sequence of random variables perturbed by a trend. As a consequence of this general result it is shown that the Hurst effect is present in the case of weakly dependent random variables with a small monotonic trend of the form f(n) = c(m + n)ß, where m is an arbitrary non-negative parameter and c is not 0. For – ½ < ß < 0 the Hurst exponent is shown to be precisely given by 1 + ß. For ß ≦ – ½ and for ß = 0 the Hurst exponent is 0.5, while for ß > 0 it is 1. This simple mathematical model, motivated by empirical evidence in various geophysical records, demonstrates the presence of the Hurst effect in a direction not explored before.

A secretary problem which allows the applicant to refuse an offer of employment with a fixed probability and admits backward solicitations of previous interviewees with known probability of successful solicitation is considered. The optimal strategy that maximizes the probability of employing the best applicant is derived. Two types of probability of successful solicitation, constant and geometric, are discussed in detail.

A Borel–Cantelli-type property in terms of one-step transition probabilities is given for events like {|Xn+1| > a + ε, |Xn|≦a}, a and ε being two positive numbers. Applications to normed sums of i.i.d. random variables with infinite mean and branching processes in varying environment with or without immigration are derived.

This note is concerned with N, the time at which a random walk first exits from [0,∞), and M, the maximum of the random walk up to time N. In the case that the random walk has zero mean and finite variance, simple proofs are given of asymptotic estimates for P{M > x}, P{N ≦ ux2|M > x} and P{M ≦ v √n|N > n}.

Majorisation is used to compare discrete distributions in terms of randomness or, equivalently, informativeness. Some properties of measures of randomness are discussed. The effect on degree of randomness of a change in parameter value is investigated for some distributions including the binomial and Poisson.

This paper considers a finite-capacity storage model defined on a Markov chain {Xn; n = 0, 1, ·· ·}, having state space J ⊆ {1, 2, ·· ·}. If Xn = j, then there is a random ‘input' Vn(j) (a negative input implying a demand) of ‘type' j, having a distribution function Fj(·). We assume that {Vn(j)} is an i.i.d. sequence of random variables, taken to be independent of {Xn} and of {Vn (k)}, for k ≠ j. Here, the random variables Vn(j) represent instantaneous ‘inputs' of type j for our storage model. Within this framework, we establish certain limit distributions for the joint processes (Zn, Xn) and (Zn, Qn, Ln), where Zn (defined in (1.2)) is the level of storage at time n, Qn (defined in (1.3)) is the cumulative overflow at time n, and Ln (defined in (1.4)) is the cumulative demand lost due to shortage of supply up to time n. In addition, an expression for the time-dependent distribution of (Zn, Xn) is obtained.

By assuming the proportionality of the intensity functions at each time point for a continuous-time non-homogeneous Markov process, strong ergodicity for the process is determined through strong ergodicity of a related discrete-time Markov process. For processes having proportional intensities, strong ergodicity implies having the limiting matrix L satisfy L · P(s, t) = L, where P(s, t) is the matrix of transition functions.

A discrete-time network with multiple transitions is defined. It is characterized by the independence of the service processes at different nodes. That network is shown to have the familiar quasi-reversibility properties.

This paper is on conditioned weak limit theorems for imbedded waiting-time processes of an M/G/1 queue. More specifically we study functional limit theorems for the actual waiting-time process conditioned by the event that the number of customers in a busy period exceeds n or equals n. Attention is also paid to the actual waiting-time process with random time index.

Combined with the existing literature on the subject this paper gives a complete account of the conditioned limit theorems for the actual waiting-time process of an M/G/1 queue for arbitrary traffic intensity and for a rather general class of service-time distributions.

The limit processes that occur are Brownian excursion and meander, while in the case of random time index also the following limit occurs: Brownian excursion divided by an independent and uniform (0, 1) distributed random variable.

In a symmetric sequence of Bernoulli trials we define each successive l outcomes as a pattern, and look at the first time that any pattern repeats. Asymptotic expressions, as l →∞, of the distribution and the expectation of this waiting time are derived.