The distribution of the final size, K, in a general SIR epidemic model is considered in a situation when the critical parameter λ is close to 1. It is shown that with a ‘critical scaling’ λ ≈ 1 + a / n1/3, m ≈ bn1/3, where n is the initial number of susceptibles and m is the initial number of infected, then K / n2/3 has a limit distribution when n → ∞. It can be described as that of T, the first passage time of a Wiener process to a parabolic barrier b + at − t2/2. The proof is based on a diffusion approximation. Moreover, it is shown that the distribution of T can be expressed analytically in terms of Airy functions using the spectral representation connected with Airy's differential equation.

In this short paper, we show how fluctuation identities for Lévy processes with no positive jumps yield the distribution of the present value of dividends paid until ruin in a Lévy insurance risk model with a dividend barrier.

In this paper we consider limit theorems for a random walk in a random environment, (Xn). Known results (recurrence-transience criteria, law of large numbers) in the case of independent environments are naturally extended to the case where the environments are only supposed to be stationary and ergodic. Furthermore, if ‘the fluctuations of the random transition probabilities around are small’, we show that there exists an invariant probability measure for ‘the environments seen from the position of (Xn)’. In the case of uniquely ergodic (therefore non-independent) environments, this measure exists as soon as (Xn) is transient so that the ‘slow diffusion phenomenon’ does not appear as it does in the independent case. Thus, under regularity conditions, we prove that, in this case, the random walk satisfies a central limit theorem for any fixed environment.

Majority systems are encountered in both decision theory and reliability theory. In decision theory for example a jury or committee employing a majority rule will make the ‘correct' decision if a majority of the individuals do so. In reliability theory some coherent systems function if and only if a majority of the components work properly. In this paper results concerning the reliability of majority systems are developed which are applicable in both areas. Two models incorporating dependence between individuals or components in majority systems are introduced, and various monotonicity results for their reliability functions are established. Comparisons are also made between direct (or simple) and indirect majority systems.

Fast stable methods for inverting multidimensional Laplace transforms have been developed in recent years by Abate, Whitt and others. We use these methods here to compute numerically the first-passage-time distribution for a spectrally one-sided Lévy process; the basic algorithm is not easy to apply, and we have to develop a variant of it. The numerical performance is as good as the original algorithm.

For finite, homogeneous, continuous-time Markov chains having a unique stationary distribution, we derive perturbation bounds which demonstrate the connection between the sensitivity to perturbations and the rate of exponential convergence to stationarity. Our perturbation bounds substantially improve upon the known results. We also discuss convergence bounds for chains with diagonalizable generators and investigate the relationship between the rate of convergence and the sensitivity of the eigenvalues of the generator; special attention is given to reversible chains.

We consider the problem of twenty questions with noisy answers, in which we seek to find a target by repeatedly choosing a set, asking an oracle whether the target lies in this set, and obtaining an answer corrupted by noise. Starting with a prior distribution on the target's location, we seek to minimize the expected entropy of the posterior distribution. We formulate this problem as a dynamic program and show that any policy optimizing the one-step expected reduction in entropy is also optimal over the full horizon. Two such Bayes optimal policies are presented: one generalizes the probabilistic bisection policy due to Horstein and the other asks a deterministic set of questions. We study the structural properties of the latter, and illustrate its use in a computer vision application.

Long-range dependence has usually been defined in terms of covariance properties relevant only to second-order stationary processes. Here we provide new definitions, almost equivalent to the original ones in that domain of applicability, which are useful for processes which may not be second-order stationary, or indeed have infinite variances. The ready applicability of this formulation for categorizing the behaviour for various infinite variance models is shown.

Lamperti's transformation, an isometry between self-similar and stationary processes, is used to solve some problems of linear estimation of continuous-time, self-similar processes. These problems include causal whitening and innovations representations on the positive real line, as well as prediction from certain finite and semi-infinite intervals. The method is applied to the specific case of fractional Brownian motion (FBM), yielding alternate derivations of known prediction results, along with some novel whitening and interpolation formulae. Some associated insights into the problem of discrete prediction are also explored. Closed-form expressions for the spectra and spectral factorization of the stationary processes associated with the FBM are obtained as part of this development.

We study a stochastic differential game between two insurance companies who employ reinsurance to reduce the risk of exposure. Under the assumption that the companies have large insurance portfolios compared to any individual claim size, their surplus processes can be approximated by stochastic differential equations. We formulate competition between the two companies as a game with a single payoff function which depends on the surplus processes. One company chooses a dynamic reinsurance strategy in order to maximize this expected payoff, while the other company simultaneously chooses a dynamic reinsurance strategy so as to minimize the same quantity. We describe the Nash equilibrium of this stochastic differential game and solve it explicitly for the case of maximizing/minimizing the exit probability.

This paper discusses the asymptotic behavior of the Mt/G/c queue having a Poisson arrival process with a general deterministic intensity. Since traditional equilibrium does not always exist, other notions of asymptotic stability are introduced and investigated. For the periodic case, limit theorems are proved complementing Harrison and Lemoine (1977) and Lemoine (1981).

We study an optimal investment problem in a continuous-time framework where the interest rates follow Cox-Ingersoll-Ross dynamics. Closed form formulae for the optimal investment strategy are obtained by assuming the completeness of financial markets and the CRRA utility function. In particular, we study the behaviour of the solution when time approaches the terminal date.

The distance from Gaussianity of the shot noise process is considered, where ti are the random times of a Poisson process with average density λ(t). With F(x) the distribution function of x(t) and G(x) that of a normal process with the same mean and variance as x(t) it is shown that where If the process x(t) is stationary with λ(t) =λ and h(t, τ) = h(t – τ) and the function h(t) is bandlimited by ωc, then the above yields

The present paper generalises some results for spectrally negative Lévy processes to the setting of Markov additive processes (MAPs). A prominent role is assumed by the first passage times, which will be determined in terms of their Laplace transforms. These have the form of a phase-type distribution, with a rate matrix that can be regarded as an inverse function of the cumulant matrix. A numerically stable iteration to compute this matrix is given. The theory is first developed for MAPs without positive jumps and then extended to include positive jumps having phase-type distributions. Numerical and analytical examples show agreement with existing results in special cases.

Burn-in is a procedure used for eliminating weak components in a mixed population. In this paper we focus on general mixed populations. Three types of results are established. First, it is shown that any mixed population displays a type of monotonicity property which is appropriate for burn-in. Second, it is shown that if, asymptotically, components have constant failure rates, then the mixed population will also asymptotically have a constant failure rate and this will correspond to the rate of the strongest subpopulation of the mixture. Finally, it is shown for a reasonable cost function, that if one mixture distribution dominates another in a strong sense, the resulting mixture of the dominant distribution will have larger optimal burn-in time.

The M/M/∞ queue in a random environment is an infinite-server queue where arrival and service rates are stochastic processes. Here we study the steady-state behavior of such a system. Explicit results are obtained for the factorial moments, the impossibility of a ‘matrix-Poisson' steady-state distribution is demonstrated and two numerical examples are presented.

A simple method is proposed for the generation of successive ‘nearest neighbours' to a given origin in an n-dimensional Poisson process. It is shown that the method provides efficient simulation of random Voronoi polytopes. Results are given of simulation studies in two and three dimensions.

Given an i.i.d. sequence X1,X2, … with common distribution function (d.f.) F, the usual non-parametric estimator of F is the e.d.f. Fn; where Uo is the d.f. of the unit mass at zero. An admissible perturbation of the e.d.f., say , is obtained if Uo is replaced by a d.f. , where is a sequence of d.f.'s converging weakly to Uo. Such perturbed e.d.f.′s arise quite naturally as integrals of non-parametric density estimators, e.g. as . It is shown that if F satisfies some smoothness conditions and the perturbation is not too drastic then ‘has the Chung–Smirnov property'; i.e., with probability one, 1. But if the perturbation is too vigorous then this property is lost: e.g., if F is the uniform distribution and Hn is the d.f. of the unit mass at n–α then the above lim sup is ≦ 1 or = ∞, depending on whether or

A Poisson stream of customers arrives at a service center which consists of two single-server queues in parallel. The service times of the customers are exponentially distributed, and both servers serve at the same rate. Arriving customers join the shortest of the two queues, with ties broken in any plausible manner. No jockeying between the queues is allowed. Employing linear programming techniques, we calculate bounds for the probability distribution of the number of customers in the system, and its expected value in equilibrium. The bounds are asymptotically tight in heavy traffic.

Signature-based representations of the reliability functions of coherent systems with independent and identically distributed component lifetimes have proven very useful in studying the ageing characteristics of such systems and in comparing the performance of different systems under varied criteria. In this paper we consider extensions of these results to systems with heterogeneous components. New representation theorems are established for both the case of components with independent lifetimes and the case of component lifetimes under specific forms of dependence. These representations may be used to compare the performance of systems with homogeneous and heterogeneous components.