We show that the transition p.d.f. of the Ornstein–Uhlenbeck process with a reflection condition at an assigned state S is related by integral-type equations to the free transition p.d.f., to the transition p.d.f. in the presence of an absorption condition at S, to the first-passage-time p.d.f. to S and to the probability current. Such equations, which are also useful for computational purposes, yield as an immediate consequence all known closed-form results for Wiener and Ornstein–Uhlenbeck processes.

This paper studies inspected systems with non-self-announcing failures where the rate of deterioration is governed by a Markov chain. We compute the lifetime distribution and availability when the system is inspected according to a periodic inspection policy. In doing so, we expose the role of certain transient distributions of the environment.

Discrete renewal theory is generalized to study the occurrence of a collection of patterns in random sequences, where a renewal is defined to be the occurrence of one of the patterns in the collection which does not overlap an earlier renewal. The action of restriction enzymes on DNA sequences provided motivation for this work. Related results of Guibas and Odlyzko are discussed.

We study the genealogical structure of samples from a population for which any given generation is made up of direct descendants from several previous generations. These occur in nature when there are seed banks or egg banks allowing an individual to leave offspring several generations in the future. We show how this temporal structure in the reproduction mechanism causes a decrease in the coalescence rate. We also investigate the effects of age-dependent neutral mutations. Our main result gives weak convergence of the scaled ancestral process, with the usual diffusion scaling, to a coalescent process which is equivalent to a time-changed version of Kingman's coalescent.

A general multivariate non-linear regression model is considered, including as special cases linear regression when the regression matrix is of less than full rank, simultaneous equations systems and regression on an unobservable predetermined variable. Given a time-series of observations at unit intervals we consider the estimation of the parameters, subject to non-linear constraints, by minimizing a criterion based on the Fourier-transformed model. We allow the residuals to be generated by a stationary, linear, process and establish asymptotic properties of our estimates.

1. In 1923 Eggenberger and Pólya introduced the following ‘urn scheme’ as a model for the development of a contagious phenomenon. A box contains b black and r red balls, and a ball is drawn from it at random with ‘double replacement’ (i.e. whatever ball is drawn, it is returned to the box together with a fresh ball of the same colour); the procedure is then continued indefinitely. A slightly more complicated version with m-fold replacement is sometimes discussed, but it will be sufficient for our purposes to keep m = 2 and it will be convenient further to simplify the scheme by taking b = r = 1 as the initial condition. We shall however generalise the scheme in another direction by allowing an arbitrary number k(≧2) of colours. Thus initially the box will contain k differently coloured balls and successive random drawings will be followed by double replacement as before. We write sn (a k-vector with jth component ) for the numerical composition of the box immediately after the nth replacement, so that and we observe that is a Markov process for which the state-space consists of all ordered k-ads of positive integers, the (constant) transition-probability matrix having elements determined by where Sn is the sum of the components of sn and (e(i))j = δij. We shall calculate the Martin boundary for this Markov process, and point out some applications to stochastic models for population growth.

The similarity between order statistics and record values motivated us, in this paper, to investigate some relationships between the order statistics and the record values. These relationships are employed to characterize a continuous distribution by some moment properties of the spacings of the record values, and hence obtain characterizations of the exponential distribution. Some of the well-known results follow trivially.

Let {X(t), V(t), t ≥ 0} be a telegraph process, with V(0+) = 1. The distribution of X(t) is derived for the general case of an alternating renewal process, describing the length of time a particle is moving to the right or to the left. The distributions of the first-crossing times of given levels a and −a are studied for M/G and for G/M processes.

In this paper we present a robust numerical method to compute the scale function W(q)(x) of a general spectrally negative Lévy process (X, P). The method is based on the Esscher transform of measure Pν under which X is taken and the scale function is determined. This change of measure makes it possible for the scale function to be bounded and, hence, makes numerical computation easy, fast, and stable. Working under the new measure Pν and using the method of Abate and Whitt (1992) and Choudhury, Lucantoni, and Whitt (1994), we give a fast stable numerical algorithm for the computation of W(q)(x).

Block replacement policy, under which items are replaced at constant intervals of time and at failure, is wasteful because sometimes almost new items are also replaced. In order to eliminate this kind of waste, the following replacement policy is suggested.

(1) Items are exchanged for new items at time kT (k ··· 1, 2, …).

(2) If items fail in [(k – 1) T, kT – v), they are replaced by new items, and if in [kT – v, kT), they are replaced by used items.

The results of this ‘extended block replacement policy with used items' are studied for the Erlang lifetime distribution.

We prove that, under rather general conditions, the law of a continuous Gaussian process represented by a stochastic integral of a deterministic kernel, with respect to a standard Wiener process, can be weakly approximated by the law of some processes constructed from a standard Poisson process. An example of a Gaussian process to which this result applies is the fractional Brownian motion with any Hurst parameter.

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.

In the last few years there has been renewed interest in the classical control problem of de Finetti (1957) for the case where the underlying source of randomness is a spectrally negative Lévy process. In particular, a significant step forward was made by Loeffen (2008), who showed that a natural and very general condition on the underlying Lévy process which allows one to proceed with the analysis of the associated Hamilton-Jacobi-Bellman equation is that its Lévy measure is absolutely continuous, having completely monotone density. In this paper we consider de Finetti's control problem, but with the restriction that control strategies are absolutely continuous with respect to the Lebesgue measure. This problem has been considered by Asmussen and Taksar (1997), Jeanblanc-Picqué and Shiryaev (1995), and Boguslavskaya (2006) in the diffusive case, and Gerber and Shiu (2006) for the case of a Cramér-Lundberg process with exponentially distributed jumps. We show the robustness of the condition that the underlying Lévy measure has a completely monotone density and establish an explicit optimal strategy for this case that envelopes the aforementioned existing results. The explicit optimal strategy in question is the so-called refraction strategy.

It is possible to interpret the classical central limit theorem for sums of independent random variables as a convergence rate result for the law of large numbers. For example, if Xi, i = 1, 2, 3, ··· are independent and identically distributed random variables with EXi = μ, var Xi = σ2 < ∞ and then the central limit theorem can be written in the form This provides information on the rate of convergence in the strong law as . (“a.s.” denotes almost sure convergence.) It is our object in this paper to discuss analogues for the super-critical Galton-Watson process.

This paper applies the theory of semi-Markov processes to the construction of a stochastic model for interpreting data obtained from clinical trials. The model characterizes the patient as being in one of a finite number of states at any given time with an arbitrary probability distribution to describe the length of stay in a state. Transitions between states are assumed to be chosen according to a stationary finite Markov chain.

Other attempts have been made to develop stochastic models of clinical trials. However, these have all been essentially Markovian with constant transition probabilities which implies that the distribution of time spent during a visit to a state is exponential (or geometric for discrete Markov chains). Markov models need also to assume that the transitions in the state of a patient depend only on absolute time whereas the semi-Markov model assumes that transitions depend on time relative to a patient. Thus the models are applicable to degenerative diseases (cancer, acute leukemia), while Markov models with time dependent transition probabilities are applicable to colds and epidemic diseases. In this paper the Laplace transforms are obtained for (i) probability of being in a state at time t, (ii) probability distribution to reach absorption state and (iii) the probability distribution of the first passage times to go from initial states to transient or absorbing states, transient to transient, and transient to absorbing. The model is applied to a clinical study of acute leukemia in which patients have been treated with methotrexate and 6-mercaptopurine. The agreement between the data and the model is very good.

In this paper we study the properties of total time on test transforms of order n and examine their applications in reliability analysis. It is shown that the successive transforms produce either distributions with increasing or bathtub-shaped failure rates or distributions with decreasing or upside bathtub-shaped failure rates. The ageing properties of the baseline distribution is compared with those of transformed distributions, and a partial order based on nth-order transforms and their implications are discussed.

In this paper we propose a stochastic process, which is a Cox-Ingersoll-Ross process with Hawkes jumps. It can be seen as a generalization of the classical Cox-Ingersoll-Ross process and the classical Hawkes process with exponential exciting function. Our model is a special case of the affine point processes. We obtain Laplace transforms and limit theorems, including the law of large numbers, central limit theorems, and large deviations.

A continuous-time model with stationary increments for asset price {Pt} is an extension of the symmetric subordinator model of Heyde (1999), and allows for skewness of returns. In the setting of independent variance-gamma-distributed returns the model resembles closely that of Madan, Carr, and Chang (1998). A simple choice of parameters renders {e−rtPt} a familiar martingale. We then specify the activity time process, {Tt}, for which {Tt − t} is asymptotically self-similar and {τt}, with τt = Tt − Tt−1, is gamma distributed. This results in a skew variance-gamma distribution for each log price increment (return) Xt and a model for {Xt} which incorporates long-range dependence in squared returns. Our approach mirrors that for the (symmetric) Student process model of Heyde and Leonenko (2005), to which the present work is intended as a complement and a sequel. One intention is to compare, partly on the basis of fitting to data, versions of the general model wherein the returns have either (symmetric) t-distributions or variance-gamma distributions.

An urn contains A balls of each of N colours. At random n balls are drawn in succession without replacement, with replacement or with replacement together with S new balls of the same colour. Let Xk be the number of drawn balls having colour k, k = 1, …, N. For a given function f the characteristic function of the random variable ZM = f(X1)+ … + f(XM), M ≦ N, is derived. A limit theorem for ZM when M, N, n → ∞is proved by a general method. The theorem covers many special cases discussed separately in the literature. As applications of the theorem limit distributions are obtained for some occupancy problems and for dispersion statistics for the binomial, Poisson and negative-binomial distribution.

Consider the random uniform placement of a finite number of arcs on the circle, where the arc lengths are sampled from a distribution on (0, 1). We provide exact formulae for the probability that the circle is completely covered and for the distribution of the number of uncovered gaps, extending Stevens's (1939) formulae for the case of fixed equal arc lengths. A special class of arc length distributions is considered, and exact probabilities of coverage are tabulated for the uniform distribution on (0, 1). Some asymptotic results for the number of gaps are also given.