We consider a diffusing particle in one dimension that is subject to a time-dependent drift or potential field. A reflecting barrier constrains the particle's position to the half-line X ≥ 0. Such models arise naturally in the study of queues with time-dependent arrival rates, as well as in advection-diffusion problems of mathematical physics. We solve for the probability distribution of the particle as a function of space and time. Then we do a detailed study of the asymptotic properties of the solution, for various ranges of space and time. We also relate our asymptotic results to those obtained by probabilistic approaches, such as central limit theorems and large deviations. We consider drifts that are either piecewise constant or linear functions of time.

In the literature on mixed Poisson processes, a number of characterisation properties have been studied. As a new characterisation property for mixed Poisson processes, we show that normalised event occurrence times are the order statistics of independent uniform random variables on (0,1). Berman's theorem on l p -isotropic sequences is applied to prove the results.

Let X t be a Feller (branching) diffusion with drift αx. We consider new processes, the probability measures of which are obtained from that of X via changes of measure involving suitably normalized exponential functions of with λ > 0. The new processes can be thought of as ‘self-reinforcing’ versions of the old.

Depending on the values of α, T and λ, the process under the new measure is shown to exhibit explosion in finite time. We also obtain a number of other results related to the new processes.

Since the Feller diffusion is also the total mass process of a superprocess, we relate the finite-time explosion property to the behaviour of superprocesses with local self-interaction, and raise some interesting questions for these.

This article describes new estimates for the second largest eigenvalue in absolute value of reversible and ergodic Markov chains on finite state spaces. These estimates apply when the stationary distribution assigns a probability higher than 0.702 to some given state of the chain. Geometric tools are used. The bounds mainly involve the isoperimetric constant of the chain, and hence generalize famous results obtained for the second eigenvalue. Comparison estimates are also established, using the isoperimetric constant of a reference chain. These results apply to the Metropolis-Hastings algorithm in order to solve minimization problems, when the probability of obtaining the solution from the algorithm can be chosen beforehand. For these dynamics, robust bounds are obtained at moderate levels of concentration.

We consider the uncovered set (i.e. the complement of the union of growing random intervals) in the one-dimensional Johnson-Mehl model. Let S(z,L) be the number of components of this set at time z > 0 which intersect (0, L]. An explicit bound is known for the total variation distance between the distribution of S(z,L) and a Poisson distribution, but due to clumping of the components the bound can be rather large. We here give a bound for the total variation distance between the distribution of S(z,L) and a simple compound Poisson distribution (a Pólya-Aeppli distribution). The bound is derived by interpreting S(z,L) as the number of visits to a ‘rare’ set by a Markov chain, and applying results on compound Poisson approximation for Markov chains by Erhardsson. It is shown that under a mild condition, if z→∞ and L→∞ in a proper fashion, then both the Pólya-Aeppli and the Poisson approximation error bounds converge to 0, but the convergence of the former is much faster.

Let τr denote the hitting time of B(0:r) for a multidimensional diffusion process. We give verifiable criteria for finiteness/infiniteness of As an application we exhibit classes of diffusion processes which are recurrent but is infinite for all p > 0, |x| > r > 0; this includes the two-dimensional Brownian motion and the reflecting Brownian motion in a wedge with a certain parameter α = 0.

We prove some limit theorems for contiunous time and state branching processes. The non-degenerate limit laws are obtained in critical and non-critical cases by conditioning or introducing immigration processes. The limit laws in non-critical cases are characterized in terms of the cononical measure of the cumulant semigroup. The proofs are based on estimates of the cumulant semigroup derived from the forward and backward equations, which are easier than the proffs in the classical setting.

This paper considers positive recurrent Markov chains where the probability of remaining in the current state is arbitrarily close to 1. Specifically, conditions are given which ensure the non-existence of central limit theorems for ergodic averages of functionals of the chain. The results are motivated by applications for Metropolis–Hastings algorithms which are constructed in terms of a rejection probability (where a rejection involves remaining at the current state). Two examples for commonly used algorithms are given, for the independence sampler and the Metropolis-adjusted Langevin algorithm. The examples are rather specialized, although, in both cases, the problems which arise are typical of problems commonly occurring for the particular algorithm being used.

A random spatial coverage process whose generating point process is homogeneous Poisson, and whose attached random sets are independent and identically distributed, is called a Boolean model. Motivated by Błaszczyszyn et al. [1], distributional and higher moment properties of the size of clumps (connected clusters of overlapping sets) in this model are derived. This provides some complements to the result on the finiteness of the first moment presented in Hall [3]. The key idea is to construct a certain coupling process for a multitype branching process that dominates the clump size.

Using the Girsanov transformation we derive estimates for the accuracy of piecewise approximations for one-sided and two-sided boundary crossing probabilities. We demonstrate that piecewise linear approximations can be calculated using repeated numerical integration. As an illustrative example we consider the case of one-sided and two-sided square-root boundaries for which we also present analytical representations in a form of infinite power series.

We consider a sequence of observations which is generated by a so-called hidden Markov model. An exponential smoothing procedure applied to such an observation sequence generates an inhomogeneous Markov process as a sequence of smoothed values. If the state sequence of the underlying hidden Markov model is moreover ergodic, then for two classes of smoothing functions the strong ergodicity of the sequence of smoothed values is proved. As a consequence a central limit theorem and a law of large numbers hold true for the smoothed values. The proof uses general results for so-called convergent inhomogeneous Markov processes. The procedure proposed by the author can be applied to some time series discussed in the literature.

In this paper, we consider a certain class of Markov renewal processes where the matrix of the transition kernel governing the Markov renewal process possesses some block-structured property, including repeating rows. Duality conditions and properties are obtained on two probabilistic measures which often play a key role in the analysis and computations of such a block-structured process. The method used here unifies two different concepts of duality. Applications of duality are also provided, including a characteristic theorem concerning recurrence and transience of a transition matrix with repeating rows and a batch arrival queueing model.

We prove a central limit theorem for the super-Brownian motion with immigration governed by another super-Brownian. The limit theorem leads to Gaussian random fields in dimensions d ≥ 3. For d = 3 the field is spatially uniform; for d ≥ 5 its covariance is given by the potential operator of the underlying Brownian motion; and for d = 4 it involves a mixture of the two kinds of fluctuations.

In earlier work, we investigated the dynamics of shape when rectangles are split into two. Further exploration, into the more general issues of Markovian sequences of rectangular shapes, has identified four particularly appealing problems. These problems, which lead to interesting invariant distributions on [0,1], have motivating links with the classical works of Blaschke, Crofton, D. G. Kendall, Rényi and Sulanke.

This paper deals with the recurrence of branching random walks on polynomially growing graphs. Amongst other things, we demonstrate the strong recurrence of tree indexed random walks determined by the resistance properties of spherically symmetric graphs. Several branching walk models are considered to show how the branching mechanism influences the recurrence behaviour.

In this paper, we give necessary and sufficient conditions to ensure the validity of confidence intervals, based on the central limit theorem, in simulations of highly reliable Markovian systems. We resort to simulations because of the frequently huge state space in practical systems. So far the literature has focused on the property of bounded relative error. In this paper we focus on ‘bounded normal approximation’ which asserts that the approximation of the normal law, suggested by the central limit theorem, does not deteriorate as the reliability of the system increases. Here we see that the set of systems with bounded normal approximation is (strictly) included in the set of systems with bounded relative error.

Previous study of the time to a common ancestor of all present-day individuals has focused on models in which each individual has just one parent in the previous generation. For example, ‘mitochondrial Eve’ is the most recent common ancestor (MRCA) when ancestry is defined only through maternal lines. In the standard Wright-Fisher model with population size n, the expected number of generations to the MRCA is about 2n, and the standard deviation of this time is also of order n. Here we study a two-parent analog of the Wright-Fisher model that defines ancestry using both parents. In this model, if the population size n is large, the number of generations, 𝒯n , back to a MRCA has a distribution that is concentrated around lgn (where lg denotes base-2 logarithm), in the sense that the ratio 𝒯n (lgn) converges in probability to 1 as n→∞. Also, continuing to trace back further into the past, at about 1.77 lgn generations before the present, all partial ancestry of the current population ends, in the following sense: with high probability for large n, in each generation at least 1.77lgn generations before the present, all individuals who have any descendants among the present-day individuals are actually ancestors of all present-day individuals.

We derive explicit closed expressions for the moment generating functions of whole collections of quantities associated with the waiting time till the occurrence of composite events in either discrete or continuous-time models. The discrete-time models are independent, or Markov-dependent, binary trials and the events of interest are collections of successes with the property that each two consecutive successes are separated by no more than a fixed number of failures. The continuous-time models are renewal processes and the relevant events are clusters of points. We provide a unifying technology for treating both the discrete and continuous-time cases. This is based on first embedding the problems into similar ones for suitably selected Markov chains or Markov renewal processes, and second, applying tools from the exponential family technology.

On the basis of the Karlin and McGregor result, which states that the transition probability functions of a birth and death process can be expressed via the introduction of an orthogonal polynomial system and a spectral measure, we investigate in this paper how the Laplace transforms and the distributions of different transient characteristics related to excursions of a birth and death process can be expressed by means of the basic orthogonal polynomial system and the spectral measure. This allows us in particular to give a probabilistic interpretation of the series introduced by Stieltjes to study the convergence of the fundamental continued fraction associated with the system. Throughout the paper, we pay special attention to the case when the birth and death process is ergodic. Under the assumption that the spectrum of the spectral measure is discrete, we show how the distributions of different random variables associated with excursions depend on the fundamental continued fraction, the orthogonal polynomial system and the spectral measure.

We consider the distribution of the number of generations to extinction in subcritical branching processes, with particular emphasis on applications to the spread of infectious diseases. We derive the generation distributions for processes with Bernoulli, geometric and Poisson offspring, and discuss some of their distributional and inferential properties. We present applications to the spread of infection in highly vaccinated populations, outbreaks of enteric fever, and person-to-person transmission of human monkeypox.