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.

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.

The usual concept of asymptotic independence, as discussed in the context of extreme value theory, requires the distribution of the coordinatewise sample maxima under suitable centering and scaling to converge to a product measure. However, this definition is too broad to conclude anything interesting about the tail behavior of the product of two random variables that are asymptotically independent. Here we introduce a new concept of asymptotic independence which allows us to study the tail behavior of products. We carefully discuss equivalent formulations of asymptotic independence. We then use the concept in the study of a network traffic model. The usual infinite source Poisson network model assumes that sources begin data transmissions at Poisson time points and continue for random lengths of time. It is assumed that the data transmissions proceed at a constant, nonrandom rate over the entire length of the transmission. However, analysis of network data suggests that the transmission rate is also random with a regularly varying tail. So, we modify the usual model to allow transmission sources to transmit at a random rate over the length of the transmission. We assume that the rate and the time have finite mean, infinite variance and possess asymptotic independence, as defined in the paper. We finally prove a limit theorem for the input process showing that the centered cumulative process under a suitable scaling converges to a totally skewed stable Lévy motion in the sense of finite-dimensional distributions.

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).

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

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.

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.

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.

We introduce a class of increasing processes with independent increments that reduce to gamma processes under the further condition of stationarity. Each such process can be reduced to a simple gamma process by a stochastic integral transformation. Applications to deformation laws of materials such as concrete are mentioned.

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.

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.

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.

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

The self-exciting point process with exponential exciting function is investigated using the immigration birth representation of the process. The counting distribution is derived explicitly and some simpler interval properties are given. In particular, it is shown that the serial covariances of the interval sequence decrease monotonically to zero.

Martin and Walker ((1997) J. Appl. Prob.34, 657–670) proposed the power-law ρ(v) = c|v|-β, |v| ≥ 1, as a correlation model for stationary time series with long-memory dependence. A straightforward proof of their conjecture on the permissible range of c is given, and various other models for long-range dependence are discussed. In particular, the Cauchy family ρ(v) = (1 + |v/c|α)-β/α allows for the simultaneous fitting of both the long-term and short-term correlation structure within a simple analytical model. The note closes with hints at the fast and exact simulation of fractional Gaussian noise and related processes.

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.

This paper focuses on the stability of open queueing systems under stationary ergodic assumptions. It defines a set of conditions, the monotone separable framework, ensuring that the stability region is given by the following saturation rule: ‘saturate' the queues which are fed by the external arrival stream; look at the ‘intensity' μ of the departure stream in this saturated system; then stability holds whenever the intensity of the arrival process, say λ satisfies the condition λ < μ, whereas the network is unstable if λ > μ. Whenever the stability condition is satisfied, it is also shown that certain state variables associated with the network admit a finite stationary regime which is constructed pathwise using a Loynes-type backward argument. This framework involves two main pathwise properties, external monotonicity and separability, which are satisfied by several classical queueing networks. The main tool for the proof of this rule is subadditive ergodic theory. It is shown that, for various problems, the proposed method provides an alternative to the methods based on Harris recurrence and regeneration; this is particularly true in the Markov case, where we show that the distributional assumptions commonly made on service or arrival times so as to ensure Harris recurrence can in fact be relaxed.

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.

In this paper we propose an autoregressive representation for a particular type of stationary Gamma(θ–1, v) process whose n-dimensional joint distributions have Laplace transform |In + θSnVn|–v, where Sn = diag(s1, · ··, sn), Vn is an n × n positive definite matrix with elements υ ij = p|i–j|i2, i, j = 1, ···, n and p is the lag-1 autocorrelation of the gamma process. We also generalize the two-parameter NEAR(1) model of Lawrance and Lewis (1981) to an exponential first-order autoregressive model with three parameters. The correlation structure and higher-order properties of the two proposed models are also given.

Ordinary survival models implicitly assume that all individuals in a group have the same risk of death. It may, however, be relevant to consider the group as heterogeneous, i.e. a mixture of individuals with different risks. For example, after an operation each individual may have constant hazard of death. If risk factors are not included, the group shows decreasing hazard. This offers two fundamentally different interpretations of the same data. For instance, Weibull distributions with shape parameter less than 1 can be generated as mixtures of constant individual hazards. In a proportional hazards model, neglect of a subset of the important covariates leads to biased estimates of the other regression coefficients. Different choices of distributions for the unobserved covariates are discussed, including binary, gamma, inverse Gaussian and positive stable distributions, which show both qualitative and quantitative differences. For instance, the heterogeneity distribution can be either identifiable or unidentifiable. Both mathematical and interpretational consequences of the choice of distribution are considered. Heterogeneity can be evaluated by the variance of the logarithm of the mixture distribution. Examples include occupational mortality, myocardial infarction and diabetes.