Individuals in communities in which different strains of pathogen are circulating can acquire resistance by accumulating immunity to each strain. After considering susceptibility, models of infection and immunity are defined for vector-borne diseases such as malaria and trypanosomiasis. For these models the prevalence of infection, the number of infections per individual, and the mean duration of infection, increase rapidly in young individuals, but decrease in older individuals as immunity is acquired to the various strains of pathogen; the mean interval between successive infections lengthens with age. The bivariate Poisson distribution is shown to be a close approximation to some stochastic processes. The models explain observed cross-sectional patterns of age prevalence, and longitudinal patterns in which individuals typically continue to become infected as they age, albeit with decreasing frequency. In these models the time spent infected depends on parasite diversity, as well as the inoculation and recovery rates. It is shown that control measures can cause an increase in the number of infections and the prevalence of infection in older individuals, and in the average prevalence in the community, even when strain-specific immunity is life-long.

In the epidemic with removal with range r, each site z, once infected, remains so for a period of time Tz, the variables Tz being i.i.d. with mean μ. While infected, a site infects its healthy r-neighbours independently at total rate α. After infection, sites become immune. We show that the critical rate of infection αc (r), above which an epidemic starting from a single site may continue forever, converges to μ–1 as r →∞.

A general multi-type branching process where new individuals immigrate according to some point process is considered. An intrinsic submartingale is defined and a convergence result for processes counted with random characteristics is obtained. Some examples are given.

We consider the mathematical properties of a time-discrete stochastic process describing explosive proliferation of DNA repeats in human genetic diseases. The process is constructed using a cascade of Galton–Watson branching processes. The main results concern the probability of absorption and the supergeometric growth of the process in the supercritical case. Examples of simulations are provided.

The asymptotic behaviour of a superadditive bisexual Galton–Watson branching process is studied. Sufficient conditions for the almost sure and L1 convergence of the suitably normed process are given. Finally, a first approach to the study of the L1 convergence for a superadditive bisexual Galton–Watson branching process under the Z log+Z condition is considered.

We prove a monotonicity property for a function of general square integrable pairs of martingales which is useful in fractal-based algorithms for compression of image data.

Let P be an infinite irreducible stochastic matrix, recurrent positive and stochastically monotone and Pn be any n × n stochastic matrix with Pn ≧ Tn, where Tn denotes the n × n northwest corner truncation of P. These assumptions imply the existence of limit distributions π and πn for P and Pn respectively. We show that if the Markov chain with transition probability matrix P meets the further condition of geometric recurrence then the exact convergence rate of πn to π can be expressed in terms of the radius of convergence of the generating function of π. As an application of the preceding result, we deal with the random walk on a half line and prove that the assumption of geometric recurrence can be relaxed. We also show that if the i.i.d. input sequence (A(m)) is such that we can find a real number r0 > 1 with , then the exact convergence rate of πn to π is characterized by r0. Moreover, when the generating function of A is not defined for |z| > 1, we derive an upper bound for the distance between πn and π based on the moments of A.

This work concerns controlled Markov chains with denumerable state space, (possibly) unbounded cost function, and an expected average cost criterion. Under a Lyapunov function condition, together with mild continuity-compactness assumptions, a simple necessary and sufficient criterion is given so that the relative value functions and differential costs produced by the value iteration scheme converge pointwise to the solution of the optimality equation; this criterion is applied to obtain convergence results when the cost function is bounded below or bounded above.

We provide an explicit matrix analytic solution for finite quasi birth and death (QBD) processes, directly expressed in terms of process parameters. We show that this solution has the same asymptotic complexity of previously proposed non-explicit solutions and is more general than some of them. Moreover, it can be easily extended to the case of generalized QBD processes.

Based on a semi-Markov process J(t), t ≧ 0, a reward process Z(t), t ≧ 0, is introduced where it is assumed that the reward function, p(k, x) is nonlinear; if the reward function is linear, i.e. ρ (k, x) = kx, the reward process Z(t), t ≧ 0, becomes the classical one, which has been considered by many authors. An explicit formula for E(Z(t)) is given in terms of the moments of the sojourn time distribution at t, when the reward function is a polynomial.

The distribution of the sample quantiles of random processes is important for the pricing of some of the so-called financial ‘look-back' options. In this paper a representation of the distribution of the α-quantile of an additive renewal reward process is obtained as the sum of the supremum and the infimum of two rescaled independent copies of the process. This representation has already been proved for processes with stationary and independent increments. As an example, the distribution of the α-quantile of a randomly observed Brownian motion is obtained.

In the random walk whose state space is a subset of the non-negative integers explicit representations for the generating functions of the n-step transition and the first return probabilities are obtained. These representations involve the Stieltjes transform of the spectral measure of the process and the corresponding orthogonal polynomials. Several examples are given in order to illustrate the application of the results.

Prediction for autoregressive sequences with finite second moment and of general order is considered. It is shown that the best predictor with time reversed is linear if and only if the innovations are Gaussian. The connection to time reversibility is also discussed.

In this paper statistical properties of estimators of drift parameters for diffusion processes are studied by modern numerical methods for stochastic differential equations. This is a particularly useful method for discrete time samples, where estimators can be constructed by making discrete time approximations to the stochastic integrals appearing in the maximum likelihood estimators for continuously observed diffusions. A review is given of the necessary theory for parameter estimation for diffusion processes and for simulation of diffusion processes. Three examples are studied.

Processes that exhibit repeatability in their kth-order moments are frequently studied in signal analysis. Such repeatability can be conveniently expressed with the help of almost periodicity. In particular, almost periodically correlated (APC) processes play an important role in the analysis of repeatable signals. This paper presents a study of asymptotic distributions of the estimator of the spectral covariance function for APC processes. It is demonstrated that, for a large class of APC processes, the functional central limit theorem holds.

A bookmaker makes a book on a horse race: he offers odds against the various horses winning the race, and gamblers accept bets at those odds when they find the odds attractive. The book at a particular time consists of the bookmaker's winnings according to the different outcomes of the race if the race were run at that time. We consider strategies the bookmaker might adopt when deciding how to alter his quoted odds as bets accumulate. The bookmaker is assumed to behave conservatively in the sense that he tries to minimise his expected maximum loss over all possible outcomes of the race.

In this article we define the s-order equilibrium distribution of a life distribution and establish some interesting properties regarding moments for s-order equilibrium distributions. Results are expected to be useful in reliability and renewal processes.

Given a finite collection of strings of letters from a fixed alphabet, it is of interest, in the contexts of data compression and DNA sequencing, to find the length of the shortest string which contains each of the given strings as a consecutive substring. In order to analyze the average behavior of the optimal superstring length, substrings of specified lengths are considered with the letters selected independently at random. An asymptotic expression is obtained for the savings from compression, i.e. the difference between the uncompressed (concatenated) length and the optimal superstring length.

In this paper conditions for the convergence of a class of simulated annealing algorithms for continuous global optimization are given. The previous literature about the subject gives results for the convergence of algorithms in which the next candidate point is generated according to a probability distribution whose support is the whole feasible set. A class of possible cooling schedules has been introduced in order to remove this restriction.

A random vibration model is investigated in this paper. The model is formulated as a cosine function with a constant frequency and a random walk phase. We show that this model is second-order stationary and can be rewritten as a vector-valued AR(1) model as well as a scalar ARMA(2, 1) model. The linear innovation sequence of the AR(1) model is shown to be a martingale difference sequence while the linear innovation sequence of the ARMA(2, 1) model is only an uncorrelated sequence. A non-linear predictor is derived from the AR(1) model while a linear predictor is derived from the ARMA(2, 1) model. We deduce that the non-linear predictor of this model has less mean square error than that of the linear predictor. This has significance, for example, for predicting seasonal phenomena with this model. In addition, the limit distributions of the sample mean, the finite Fourier transforms and the autocovariance functions are derived using a martingale approach. The limit distribution of autocovariance functions differs from the classical result given by Bartlett's formula.