We consider a classic competing-species model with the rates changed to include Gaussian white noise. We show that if the noise is not too large, then the stochastic version is ergodic. An explicit relation between the noise and the original competing-species parameters gives a sufficient condition for ergodicity.

Onwards from the mid-twentieth century, the stochastic filtering problem has caught the attention of thousands of mathematicians, engineers, statisticians, and computer scientists. Its applications span the whole spectrum of human endeavour, including satellite tracking, credit risk estimation, human genome analysis, and speech recognition. Stochastic filtering has engendered a surprising number of mathematical techniques for its treatment and has played an important role in the development of new research areas, including stochastic partial differential equations, stochastic geometry, rough paths theory, and Malliavin calculus. It also spearheaded research in areas of classical mathematics, such as Lie algebras, control theory, and information theory. The aim of this paper is to give a brief historical account of the subject concentrating on the continuous-time framework.

In this paper we study two distributions, namely the distribution of the waiting times until given numbers of occurrences of compound patterns and the distribution of the numbers of occurrences of compound patterns in a fixed number of trials. We elucidate the interrelation between these two distributions in terms of the generating functions. We provide perspectives on the problems related to compound patterns in statistics and probability. As an application, the waiting time problem of counting runs of specified lengths is considered in order to illustrate how the distributions of waiting times can be derived from our theoretical results.

This paper is concerned with the Birnbaum importance measure of a component in a binary coherent system. A representation for the Birnbaum importance of a component is obtained when the system consists of exchangeable dependent components. The results are closely related to the concept of the signature of a coherent system. Some examples are presented to illustrate the results.

The reduced Markov branching process is a stochastic model for the genealogy of an unstructured biological population. Its limit behavior in the critical case is well studied for the Zolotarev-Slack regularity parameter α ∈ (0, 1]. We turn to the case of very heavy-tailed reproduction distribution α = 0 assuming that Zubkov's regularity condition holds with parameter β ∈ (0, ∞). Our main result gives a new asymptotic pattern for the reduced branching process conditioned on nonextinction during a long time interval.

The Ehrenfest urn is a model for the diffusion of gases between two chambers. Classic research deals with this system as a Markovian model with a fixed number of balls, and derives the steady-state behavior as a binomial distribution (which can be approximated by a normal distribution). We study the gradual change for an urn containing n (a very large number) balls from the initial condition to the steady state. We look at the status of the urn after kn draws. We identify three phases of kn: the growing sublinear, the linear, and the superlinear. In the growing sublinear phase the amount of gas in each chamber is normally distributed, with parameters that are influenced by the initial conditions. In the linear phase a different normal distribution applies, in which the influence of the initial conditions is attenuated. The steady state is not a good approximation until a certain superlinear amount of time has elapsed. At the superlinear stage the mix is nearly perfect, with a nearly perfect symmetrical normal distribution in which the effect of the initial conditions is completely washed away. We give interpretations for how the results in different phases conjoin at the ‘seam lines’. In fact, these Gaussian phases are all manifestations of one master theorem. The results are obtained via martingale theory.

In this contribution we consider an M/M/1 queueing model with general server vacations. Transient and steady state analysis are carried out in discrete time by combinatorial methods. Using weak convergence of discrete-parameter Markov chains we also obtain formulas for the corresponding continuous-time queueing model. As a special case we discuss briefly a queueing system with a T-policy operating.

The survival distribution of a device subject to a sequence of shocks occurring randomly over time is studied by Esary, Marshall and Proschan (1973) and by A-Hameed and Proschan (1973), (1975). The present note treats the case in which shocks occur according to a homogeneous Poisson cluster process. It is shown that if [the device survives k shocks] = zk, 0 < z < 1, then the device exhibits a decreasing failure rate. A DFR preservation theorem is proved for completely monotonic . A counterexample to the IFR preservation theorem is given in which is strictly IFR while the failure rate is initially decreasing and then increasing.

The centred, periodic, stationary Gaussian process X(z), ≧ z ≧ 1 with covariances , appears when one studies the solutions of the vibrating string equation forced by noise, corresponding to the case of a finite string with the extremes tied together. The close relationship between this process and a Brownian bridge permits us to compute the distribution of the maximum excursion of the string at particular times.

By a general shot noise process we mean a shot noise process in which the counting process of shots is arbitrary locally finite. Assuming that the counting process of shots satisfies a functional limit theorem in the Skorokhod space with a locally Hölder continuous Gaussian limit process, and that the response function is regularly varying at infinity, we prove that the corresponding general shot noise process satisfies a similar functional limit theorem with a different limit process and different normalization and centering functions. For instance, if the limit process for the counting process of shots is a Brownian motion, then the limit process for the general shot noise process is a Riemann–Liouville process. We specialize our result for five particular counting processes. Also, we investigate Hölder continuity of the limit processes for general shot noise processes.

A supercritical position-dependent Markov branching process has been used as an approximation to a model describing the initial geographical spread of a measles epidemic (Bartlett (1956)). Let α be its Malthusian parameter, ß its velocity of propagation, Z(A, t) the number of individuals in the set A at time t, and A√(ßt) = [√(ßt) r: r ∈ A]. The mean square convergence of the random variable W(A, t)= e–αtZ(A√(ßt), t) to a limit variable W(A) is established.

This paper deals with a class of discrete-time Markov chains for which the invariant measures can be expressed in terms of generalized continued fractions. The representation covers a wide class of stochastic models and is well suited for numerical applications. The results obtained can easily be extended to continuous-time Markov chains.

Characteristics of some collapsing supercritical stochastic population processes where an embedded Galton–Watson process can be isolated are shown to be identical to those of some associated subcritical process.

This paper treats stopping problems on Markov chains in which the OLA (one-step look ahead) policy is optimal. Its associated optimal value can be explicitly expressed by a potential for a charge function of the difference between the immediate reward and the one-step-after reward. As an application to the best choice problem, we shall obtain the value of three problems: the classical secretary problem, a problem with a refusal probability and a problem with a random number of objects.

We present a solution to a special system of Kolmogorov forward equations. We use this result to present a useful expression for the transition probabilities of the extended simple stochastic epidemic model and an epidemic model involving cross-infection between two otherwise isolated groups.

A stochastic model of a finite dam in which the epochs of input form a renewal process is considered. It is assumed that the quantities of input at different epochs and the inter-input times are two independent families of random variables whose characteristic functions are rational functions. The release rate is equal to unity. An imbedding equation is set up for the probability frequency governing the water level in the first wet period and the resulting equation is solved by Laplace transform technique. Explicit expressions relating to the moments of the random variables representing the number of occasions in which the dam becomes empty as well as the total duration of the dry period in any arbitrary interval of time are indicated for negative exponentially distributed inter-input times.

Start with a necklace consisting of one white bead and one black bead, and add new beads one at a time by inserting each new bead between a randomly chosen adjacent pair of old beads, with the proviso that the new bead will be white if and only if both beads of the adjacent pair are black. Let Wn denote the number of white beads when the total number of beads is n. We show that EWn = n/3 and, with c2 = 2/45, that (Wn − n/3) / c√n is asymptotically standard normal. We find that, for all r ≥ 1 and n > 2r, the rth cumulant of the distribution of Wn is of the form nhr. We find the expected numbers of gaps of given length between white beads, and examine the asymptotics of the longest gaps.

Homogeneous additive processes on a finite or semi-infinite interval have been studied in many forms. Wald's identity for the first passage process on the finite interval (see for example Miller, 1961), the waiting time process of Lindley (1952), and a variety of problems in the theory of queues, dams, and inventories come to mind. These processes have been treated by and large by methods in the complex plane. Lindley's discrete parameter process on the continuum, for example, described by where the ξn are independent identically distributed random variables, has been discussed by Wiener-Hopf methods in recent years by Lindley (1952), Smith (1953), Kemperman (1961), Keilson (1961), and many others. A review of earlier studies is given by Kemperman.

In this note a renewal density theorem in the multi-dimensional case is formulated and proved. Let f(x) be the density function of a p-dimensional random variable with positive mean vector μ and positive-definite covariance matrix Σ, let hn(x) be the n-fold convolution of f(x) with itself, and set Then for arbitrary choice of integers k1, …, kp–1 distinct or not in the set (1, 2, …, p), it is shown that under certain conditions as all elements in the vector x = (x1, …, xp) become large. In the above expression μ‵ is interpreted as a row vector and μ a column vector. An application to the theory of a class of age-dependent branching processes is also presented.