Balls are drawn with replacement from an urn containing m distinguishable balls until a match is noted. The distribution of the number of drawings required is considered in the case where only the last k balls drawn are remembered. The asymptotic behavior of this distribution, as m and k become large, is investigated. Two further variants of the problem are suggested.

The departure process from the GI/G/1 queue is shown to be a semi-Markov process imbedded at departure points with a two-dimensional state space. Transition probabilities for this process are defined and derived from the distributions of the arrival and service processes. The one step transition probabilities and a stationary distribution are obtained for the imbedded two-dimensional Markov chain.

The mathematical model we consider here is the classical Bienaymé–Galton–Watson branching process modified with immigration in the state zero.

We study properties of the waiting time to explosion of the supercritical modified process, i.e. that time until all beginning cycles which die out have disappeared. We then derive the expected total progeny of a cycle and show how higher moments can be computed. With a view to applications the main goal is to show that any statistical inference from observed cycle lengths or estimates of total progeny on the fertility rate of the process must be treated with care. As an example we discuss population experiments with trout.

Let {W(t), 0≦t<∞} be the standard Wiener process. The computation schemes developed in the past are not computationally efficient for the absorption probabilities of the type P{sup0≦t≦TW(t) − f(t) ≧ 0} when either T is large or f(0) > 0 is small. This paper gives an efficient and accurate algorithm to compute such probabilities, and some applications to other Gaussian stochastic processes are discussed.

For a GI/G/1 queue a limit theorem is obtained for the total input of work during (0, t]. This result is then used to obtain similar theorems for the waiting and idle times and for busy periods initiated by large service loads. Some of the results contained in the paper have recently been proved by other authors in more general settings. The intent of this work is to show how they may be obtained in lesser generality using simpler techniques.

This paper provides a quasi-likelihood or minimum-contrast-type method for the parameter estimation of random fields in the frequency domain based on higher-order information. The estimation technique uses the spectral density of the general kth order and allows for possible long-range dependence in the random fields. To avoid bias due to edge effects, data tapering is incorporated into the method. The suggested minimum contrast functional is linear with respect to the periodogram of kth order, hence kernel estimation for the spectral densities is not needed. Furthermore, discretization is not required in the estimation of continuously observed random fields. The consistency and asymptotic normality of the resulting estimators are established. Illustrative applications of the method to some problems in mathematical finance and signal detection are given.

The paper shows that the use of both types of random noise, white noise and Poisson noise, can be justified when using an innovations approach. The historical background for this is sketched, and then several methods of whitening dependent time series are outlined, including a mixture of Gaussian white noise and a compound Poisson process: this appears as a natural extension of the Gaussian white noise model for the prediction errors of a non-Gaussian time series. A statistical method for the identification of non-linear time series models with noise made up of a mixture of Gaussian white noise and a compound Poisson noise is presented. The method is applied to financial time series data (dollar-yen exchange rate data), and illustrated via six models.

This note presents a method of evaluating the distribution of a path integral for Markov chains on a countable state space.

A random mapping (Tn;q) of a finite set V, V = {1,2,…,n}, into itself assigns independently to each i ∊ V its unique image j ∊ V with probability q if i = j and with probability P = (1-q)/(n−1) if i ≠ j. Three versions of epidemic processes on a random digraph GT representing (Tn;q) are studied. The exact probability distributions of the total number of infected elements as well as the threshold functions for these epidemic processes are determined.

Natvig's measure of importance of system components is defined and investigated, using martingale notions and methods. Most results are proved in the case of dependent component life lengths. The sum of the ‘absolute importances' of the components is studied as a quantity expressing the ‘share of failures' in the destruction of the system, contrasted to the ‘share of time’.

In the study of the busy period for a single server queueing system, three variables that have been investigated individually or at most in pairs are:

1. 1. The duration of the busy period.

2. 2. The number of customers served during the busy period.

3. 3. The maximum number of customers in the queue during the busy period.

Let Xi, i = 1, 2, 3, … be a sequence of independent and identically distributed random variables. Write and for x ≧ 0 define M(x) + 1 is then the first passage time out of the interval (– ∞, x] for the random walk process Sn.

We continue the study of the discrete-time risk model introduced by Picard et al. (2003). The cumulative loss process (St)t∊ℕ has independent and stationary increments, the increments per unit of time having nonnegative integer values with distribution {ai, i ∊ ℕ and mean ā. The premium receipt process (ck)k∊ℕ is deterministic, nonnegative and nonuniform; in addition, we assume it to be regular in order for there to exist a constant c > ā such that the deviation is bounded as the time t varies. We are interested in whether or not ruin occurs within a finite time. If T is the time of ruin, we obtain P(T = ∞) as the limit of P(T > t) as t → ∞, firstly in the particular case where c = 1/d for some positive d ∊ ℕ, and then in the general case for positive c under the condition that a0 > ½.

We study a search model in which offers of random size are received randomly over time. The arrival times form a point process of a certain type, and the offer size distribution may depend on the corresponding arrival time. The search costs have a time-dependent cost rate. The objective is to stop the search process such that the expected discounted net reward (associated with the maximal offer received so far) is maximized. A stopping time σ is suggested, and conditions are specified under which σ turns out to be optimal.

Multidimensional Markov processes in continuous time with asymptotically linear mean change per unit of time are studied as randomly perturbed linear differential equations. Conditions for exponential and polynomial growth rates with stable type distribution are given. From these conditions results on branching models of populations with stabilizing reproduction for near-supercritical and near-critical cases follow.

Patch clamp recordings from ion channels often show periods of repetitive activity, known as bursts, which are noticeably separated from each other by periods of inactivity. Depending on the type of channel, such recordings may exhibit (conductance) levels between the closed (zero) level and the fully open level. Properties of bursts are less subject to problems that arise from recording than are properties for individual sojourns at different levels, and study of bursting behaviour provides important information about the finer structure of the underlying channel gating process. For a general finite state space continuous-time Markov chain model allowing one or more nonzero conductance levels, the present paper establishes results about the semi-Markov structure of a single burst and of a sequence of bursts, and uses this in a unified approach to properties of both theoretical and empirical bursts. The distribution and moments of particular burst properties, including the total charge transfer, the total sojourn time and the total number of visits to specified conductance levels during a burst, are derived. Various extensions are also described.

A model is presented for a bounded growth population subjected to random-sized emigrations that occur due to population pressure.

The deterministic growth component examined in detail is defined by a Prendiville process. Results are obtained for the times between emigration events and for the population increase between emigrations. Some information is obtained about the mean time to extinction and also for the mean population level when the emigration-size distribution is negative exponential.

We introduce a class of stock models that interpolates between exponential Lévy models based on Brownian subordination and certain stochastic volatility models with Lévy-driven volatility, such as the Barndorff-Nielsen–Shephard model. The driving process in our model is a Brownian motion subordinated to a business time which is obtained by convolution of a Lévy subordinator with a deterministic kernel. We motivate several choices of the kernel that lead to volatility clusters while maintaining the sudden extreme movements of the stock. Moreover, we discuss some statistical and path properties of the models, prove absence of arbitrage and incompleteness, and explain how to price vanilla options by simulation and fast Fourier transform methods.

In this paper we consider the probabilities of finite- and infinite-time absolute ruins in the renewal risk model with constant premium rate and constant force of interest. In the particular case of the compound Poisson model, explicit asymptotic expressions for the finite- and infinite-time absolute ruin probabilities are given. For the general renewal risk model, we present an asymptotic expression for the infinite-time absolute ruin probability. Conditional distributions of Poisson processes and probabilistic techniques regarding randomly weighted sums are employed in the course of this study.

A continuous-time Markov chain on the non-negative integers is called skip-free to the right (left) if only unit increments to the right (left) are permitted. If a Markov chain is skip-free both to the right and to the left, it is called a birth–death process. Karlin and McGregor (1959) showed that if a continuous-time Markov chain is monotone in the sense of likelihood ratio ordering then it must be an (extended) birth–death process. This paper proves that if an irreducible Markov chain in continuous time is monotone in the sense of hazard rate (reversed hazard rate) ordering then it must be skip-free to the right (left). A birth–death process is then characterized as a continuous-time Markov chain that is monotone in the sense of both hazard rate and reversed hazard rate orderings. As an application, the first-passage-time distributions of such Markov chains are also studied.