A carrier-borne epidemic is considered in which the carriers, subject to a death process, infect susceptibles by random allocation rather than the classical homogeneous mixing process. An explicit solution for the probability generating function (p.g.f.) of the process is obtained, and a probabilistic analysis of carrier models provided. The sizes and durations of the random allocation and classical carrier epidemics are compared. The strongest comparisons concern sample path results based on the probabilistic analysis; this also gives a sounder basis for computational work.

Shock models based on Poisson processes have been used to derive univariate and multivariate exponential distributions. But in many applications, Poisson processes are not realistic models of physical shock processes because they have independent increments; expanded models that allow for possibly dependent increments are of interest. In this paper, univariate and bivariate Pólya urn schemes are used to derive models of shock sources. The life distributions obtained from these models form a large parametric family that includes the exponential distribution. Even in the univariate case these life distributions have not been widely used, though they form a large and flexible family. In the bivariate case, the family includes the bivariate exponential distributions of Marshall and Olkin as a special case.

Faddy (1990) has conjectured that the variability of a pure birth process is increased, relative to the linear case, if the birth rates are convex and decreased if they are concave. We prove the conjecture by relating variability to the correlation structure of certain more informative versions of the process. A correlation inequality due to Harris (1977) is used to derive the necessary positive and negative correlation results.

This paper considers several models for biological processes in which animate individuals live and die as members of groups which can split to form smaller groups. Resulting distributions of individuals over groups are compared and contrasted. In particular, two qualitatively different types of distributions are identified. It is clear that distinguishing between models giving rise to the same distribution types is difficult. Implications for more complex models are discussed and avenues for further research are outlined.

Let be the Brownian motion process starting at the origin, its primitive and Ut = (Xt+x + ty, Bt + y), , the associated bidimensional process starting from a point . In this paper we present an elementary procedure for re-deriving the formula of Lefebvre (1989) giving the Laplace–Fourier transform of the distribution of the couple (σ α, Uσa), as well as Lachal's (1991) formulae giving the explicit Laplace–Fourier transform of the law of the couple (σ ab, Uσab), where σ α and σ ab denote respectively the first hitting time of from the right and the first hitting time of the double-sided barrier by the process . This method, which unifies and considerably simplifies the proofs of these results, is in fact a ‘vectorial' extension of the classical technique of Darling and Siegert (1953). It rests on an essential observation (Lachal (1992)) of the Markovian character of the bidimensional process .

Using the same procedure, we subsequently determine the Laplace–Fourier transform of the conjoint law of the quadruplet (σ α, Uσa, σb, Uσb).

We introduce and define for the first time the concept of a non-homogeneous Markov system in a stochastic environment (S-NHMS). The problem of finding the expected population structure in an S-NHMS is studied, and important properties among the basic parameters of the S-NHMS are established. Moreover, we study the problem of maintaining the relative sizes of the states in a stochastic environment applying control in the input process. Among other things, we provide the probability of maintaining any vector of relative state sizes. Also strategies for attaining in an optimal way a desired relative structure are designed, with the use of a given algorithm. Finally, an illustration is provided of the present results in a manpower system.

A continuous-time Markov chain on the non-negative integers is called skip-free to the left (right) if the governing infinitesimal generator A = (aij) has the property that aij = 0 for j ≦ i ‒ 2 (i ≦ j – 2). If a Markov chain is skip-free both to the left and to the right, it is called a birth-death process. Quasi-limiting distributions of birth–death processes have been studied in detail in their own right and from the standpoint of finite approximations. In this paper, we generalize, to some extent, results for birth-death processes to Markov chains that are skip-free to the left in continuous time. In particular the decay parameter of skip-free Markov chains is shown to have a similar representation to the birth-death case and a result on convergence of finite quasi-limiting distributions is obtained.

This paper describes a set of stochastic processes that is useful for modeling and analyzing a new genetic mapping method. Some of the processes are Markov chains, and some are best described as functions of Markov chains. The central issue is boundary-crossing probabilities, which correspond to p-values for the existence of genes for particular traits. The methods elaborated by Aldous (1989) provide very accurate approximate p-values, as spot-checked against simulations.

We prove Lévy's theorem concerning positiveness of transition probabilities of Markov processes when the state space is countable and an invariant probability distribution exists. Our approach relies on the representation of transition probabilities in terms of the directed circuits that occur along the sample paths.

We consider lumpability for continuous-time Markov chains and provide a simple probabilistic proof of necessary and sufficient conditions for strong lumpability, valid in circumstances not covered by known theory. We also consider the following marginalisability problem. Let {X{t)} = {(X1(t), X2(t), · ··, Xm(t))} be a continuous-time Markov chain. Under what conditions are the marginal processes {X1(t)}, {X2(t)}, · ··, {Xm(t)} also continuous-time Markov chains? We show that this is related to lumpability and, if no two of the marginal processes can jump simultaneously, then they are continuous-time Markov chains if and only if they are mutually independent. Applications to ion channel modelling and birth–death processes are discussed briefly.

We consider a large class of random walks on the discrete circle Z/(n), defined in terms of a piecewise Lipschitz function, and motivated by the ‘generation gap' process of Diaconis. For such walks, we show that the time until convergence to stationarity is bounded independently of n. Our techniques involve Fourier analysis and a comparison of the random walks on Z/(n) with a random walk on the continuous circle S1.

The paper investigates stochastic processes directed by a randomized time process. A new family of directing processes called Hougaard processes is introduced. Monotonicity properties preserved under subordination, and dependence among processes directed by a common randomized time are studied. Results for processes subordinated to Poisson and stable processes are presented. Potential applications to shock models and threshold models are also discussed. Only Markov processes are considered.

Continuous-time Markov chain models have been widely considered for the gating behaviour of a single ion channel. In such models the state space is usually partitioned into two classes, designated ‘open' and ‘closed', and there is ‘aggregation' in that it is possible to observe only which class the process is in at any given time. Hawkes et al. (1990) have derived an expression for the density function of the exact distribution of an observed open time in such an aggregated Markov model, where brief sojourns in either the open or the closed class are unobservable. This paper extends their result to single ion channel models based on aggregated semi-Markov processes, giving a more direct derivation which is probabilistic and exhibits clearly the combinatorial content.

This paper is concerned with the problem of estimation for the diffusion coefficient of a diffusion process on R, in a non-parametric situation. The drift function can be unknown and considered as a nuisance parameter. We propose an estimator of σ based on discrete observation of the diffusion X throughout a given finite time interval. We describe the asymptotic behaviour of this estimator when the step of discretization tends to zero. We prove consistency and asymptotic normality, the rate of convergence to the normal law being a random variable linked to the local time of the diffusion or to its suitable discrete approximation. This can also be interpreted as a convergence to a mixture of normal law.

We obtain a single formula which, when its components are adequately chosen, transforms itself into the main formulas of the Palm theory of point processes: Little's L = λW formula [10], Brumelle's H = λG formula [5], Neveu's exchange formula [14], Palm inversion formula and Miyazawa's rate conservation law [12]. It also contains various extensions of the above formulas and some new ones.

We examine the main properties of the Markov chain Xt = T(Xt– 1) + σ(Xt– 1)ε t. Under general and tractable assumptions, we derive bounds for the tails of the stationary density of the process {Xt} in terms of the common density of the ε t's.

Using a simple characterization of the Linnik distribution, discrete-time processes having a stationary Linnik distribution are constructed. The processes are structurally related to exponential processes introduced by Arnold (1989), Lawrance and Lewis (1981) and Gaver and Lewis (1980). Multivariate versions of the processes are also described. These Linnik models appear to be viable alternatives to stable processes as models for temporal changes in stock prices.

We consider the likelihood ratio tests to detect an epidemic alternative in the following two cases of normal observations: (1) the alternative specifies a square wave drift in the mean value of an i.i.d. sequence; (2) the alternative permits a square wave drift in the intercept of a simple linear regression model. To develop the approximations for the significance levels leads us to consider boundary-crossing problems of some two-dimensional discrete-time Gaussian fields. By the method which was proposed originally by Woodroofe (1976) and adapted to study maxima of some random fields by Siegmund (1988), some large deviations for the conditional non-linear boundary-crossing probabilities are developed. Some results of Monte Carlo experiments confirm the accuracy of these approximations.

If X is a Brownian motion with drift and γ = inf{t > 0: Mt = t} we derive the joint density of the triple {U, γ, Δ}, where and Δ= γ —Xγ. In the case δ = 0 it follows easily from this that Δ has an Exp(2) distribution and this in turn implies the rather surprising result that if τ= inf{t > 0: Xt = Mt = t}, then Pr{τ = 0} = 0 and . We also derive various other distributional results involving the pair (X, M), including for example the distribution of ; in particular we show that, in case δ. = 1, when Pr{0 < τ < ∞} = 1, the ratio τ+/τ has the arc-sine distribution.

For an independent percolation model on , where is a homogeneous tree and is a one-dimensional lattice, it is shown, by verifying that the triangle condition is satisfied, that the percolation probability θ (p) is a continuous function of p at the critical point pc, and the critical exponents , γ, δ, and Δ exist and take their mean-field values. Some analogous results for Markov fields on are also obtained.