The estimation of the parameters of a discrete-time semi-Markov process is considered, when the data consist of records of a large number of individuals observed in a specified short period of time. Such a problem may arise in the modelling of intra-urban mobility. Methods of estimation of the parameters are suggested, and a numerical example discussed for the case when the observation period consists of three consecutive time points.

In this paper we derive conditions on the internal wear process under which the resulting time to failure model will be of the simple collapsible form when the usage accumulation history is available. We suppose that failure occurs when internal wear crosses a certain threshold or a traumatic event causes the item to fail. We model the infinitesimal increment in internal wear as a function of time, accumulated internal wear, and usage history, and we derive conditions on this function to get a collapsible model for the distribution of time to failure given the usage history. We reach the conclusion that collapsible models form the subset of accelerated failure time models with time-varying covariates for which the time transformation function satisfies certain simple properties.

We determine the exact large-buffer asymptotics for a mixture of light-tailed and heavy-tailed input flows. Earlier studies have found a ‘reduced-load equivalence’ in situations where the peak rate of the heavy-tailed flows plus the mean rate of the light-tailed flows is larger than the service rate. In that case, the workload is asymptotically equivalent to that in a reduced system, which consists of a certain ‘dominant’ subset of the heavy-tailed flows, with the service rate reduced by the mean rate of all other flows. In the present paper, we focus on the opposite case where the peak rate of the heavy-tailed flows plus the mean rate of the light-tailed flows is smaller than the service rate. Under mild assumptions, we prove that the workload distribution is asymptotically equivalent to that in a somewhat ‘dual’ reduced system, multiplied by a certain prefactor. The reduced system now consists of only the light-tailed flows, with the service rate reduced by the peak rate of the heavy-tailed flows. The prefactor represents the probability that the heavy-tailed flows have sent at their peak rate for more than a certain amount of time, which may be interpreted as the ‘time to overflow’ for the light-tailed flows in the reduced system. The results provide crucial insight into the typical overflow scenario.

A simple asymmetric random walk on the integers is stopped when its range is of a given length. When and where is it stopped? Analogous questions can be stated for a Brownian motion. Such problems are studied using results for the classical ruin problem, yielding results for the cover time and the range, both for asymmetric random walks and Brownian motion with drift.

A stochastic dynamical context is developed for Bookstein's shape theory. It is shown how Bookstein's shape space for planar triangles arises naturally when the landmarks are moved around by a special Brownian motion on the general linear group of invertible (2×2) real matrices. Asymptotics for the Brownian transition density are used to suggest an exponential family of distributions, which is analogous to the von Mises-Fisher spherical distribution and which has already been studied by J. K. Jensen. The computer algebra implementation Itovsn3 (W. S. Kendall) of stochastic calculus is used to perform the calculations (some of which actually date back to work by Dyson on eigenvalues of random matrices and by Dynkin on Brownian motion on ellipsoids). An interesting feature of these calculations is that they include the first application (to the author's knowledge) of the Gröbner basis algorithm in a stochastic calculus context.

Let X(t) be the virtual waiting-time process of a stable M/G/1 queue. Let R(t) be the covariance function of the stationary process X(t), B(t) the busy-period distribution of X(t); and let E(t) = P{X(t) = 0|X(0) = 0}.

For X(t) some heavy-traffic results are given, among which are limiting expressions for R(t) and its derivatives and for B(t) and E(t).

These results are used to find the covariance function of stationary Brownian motion on [0, ∞).

Conditional stochastic ordering is concerned with the stochastic ordering of a pair of probability measures conditional on certain subsets or sub-σ -algebras. Some basic results of conditional stochastic ordering were proved by Whitt. We extend some of Whitt's results and prove a basic relation between stochastic ordering conditional on subsets and stochastic ordering conditional on σ -algebras. In the second part of the paper we consider the ordering of conditional expectations. There are several different formulations of this problem motivated by different types of applications.

We present the first class of perfect sampling (also known as exact simulation) algorithms for the steady-state distribution of non-Markovian loss systems. We use a variation of dominated coupling from the past. We first simulate a stationary infinite server system backwards in time and analyze the running time in heavy traffic. In particular, we are able to simulate stationary renewal marked point processes in unbounded regions. We then use the infinite server system as an upper bound process to simulate the loss system. The running time analysis of our perfect sampling algorithm for loss systems is performed in the quality-driven (QD) and the quality-and-efficiency-driven regimes. In both cases, we show that our algorithm achieves subexponential complexity as both the number of servers and the arrival rate increase. Moreover, in the QD regime, our algorithm achieves a nearly optimal rate of complexity.

As an alternative to conventional discrete time models for stochastic processes that fluctuate within the sampling interval, we propose difference equations containing non-integral lags. We discuss the problems of stability, identification and estimation, for which an approximate model is needed. Least squares applied to an approximate Fourier-transformed model yields estimators of the coefficients that are consistent with respect to the true model under some conditions. The conditions are weak when the model contains predetermined variables that obey an “aliasing condition”; estimators of the lags as well as coefficients can then be found that are consistent, efficient and satisfy a central limit theorem. Optimal estimators for stochastic difference-differential equations are also available.

Let ξ, ξ1, ξ2,… be a sequence of independent and identically distributed random variables, and let Sn=ξ1+⋯+ξn and Mn=maxk≤nSk. Let τ=min{n≥1: Sn≤0}. We assume that ξ has a heavy-tailed distribution and negative, finite mean E(ξ)<0. We find the asymptotics of P{Mτ ∈ (x, x+T]} as x→∞, for a fixed, positive constant T.

A model of a stochastic froth is introduced in which the rate of random coalescence of a pair of bubbles depends on an inverse power law of their sizes. The main question of interest is whether froths with a large number of bubbles can grow in a stable fashion; that is, whether under some time-varying change of scale the distributions of rescaled bubble sizes become approximately stationary. It is shown by way of a law of large numbers for the froths that the question can be re-interpreted in terms of a measure flow solving a nonlinear Boltzmann equation that represents an idealized deterministic froth. Froths turn out to be stable in the sense that there are scalings in which the rescaled measure flow is tight and, for a particular case, stable in the stronger sense that the rescaled flow converges to an equilibrium measure. Precise estimates are also given for the degree of tightness of the rescaled measure flows.

In this article we consider a Monte-Carlo-based method to filter partially observed diffusions observed at regular and discrete times. Given access only to Euler discretizations of the diffusion process, we present a new procedure which can return online estimates of the filtering distribution with no time-discretization bias and finite variance. Our approach is based upon a novel double application of the randomization methods of Rhee and Glynn (Operat. Res.63, 2015) along with the multilevel particle filter (MLPF) approach of Jasra et al. (SIAM J. Numer. Anal.55, 2017). A numerical comparison of our new approach with the MLPF, on a single processor, shows that similar errors are possible for a mild increase in computational cost. However, the new method scales strongly to arbitrarily many processors.

Our aim is to estimate the volume-weighted mean of the volumes of three-dimensional ‘particles’ (compact, not-necessarily-convex subsets) from plane sections of the particle population. The standard stereological technique is to place test lines in the plane section, and measure cubed intercept lengths with the two-dimensional particle profiles. This paper discusses more efficient estimators obtained by integrating over all possible placements of the test line. We prove that these estimators have smaller variance than the line transect estimators, and indeed are related to them by the Rao-Blackwell process. In the improved estimators, the cubed intercept length is replaced by a moment of the distance between two points in the section profile. This can be computed as a moment of the set covariance function, which in turn is computable using the fast Fourier transform. We also derive an isoperimetric-type inequality between the improved estimator and the area-weighted 3/2th moment of the profile areas. Finally, we present two practical applications to particles of silicon carbide and to synaptic boutons in brain tissue. We estimate the variance of the technique and the gain in efficiency over line transect techniques; the efficiency improvement appears to be as much as one order of magnitude.

The explicit form for the characteristic function of a stable distribution on the line is derived analytically by solving the associated functional equation and applying the theory of regular variation, without appeal to the general Lévy‒Khintchine integral representation of infinitely divisible distributions.

The relationship between the Wiener-Hopf factorisation of matrices and the solution of systems of certain operator equations is discussed in an algebraic setting. It is shown that the study of the waiting time process of the nth arriving group of the general single server bulk queue leads to equations of that type. This system of equations may be considered as an extension of Lindley's waiting-time equation.

We use the fact that the Palm measure of a stationary random measure is invariant to phase space change to generalize the light traffic formula initially obtained for stationary processes on a line to general spaces. This formula gives a first-order expansion for the expectation of a functional of the random measure when its intensity vanishes. This generalization leads to new algorithms for estimating gradients of functionals of geometrical random processes.

In this paper, we are concerned with preservation properties of first and second order by an operator L representable in terms of a stochastic process Z with non-decreasing right-continuous paths. We introduce the derived operator D of L and the derived process V of Z in order to characterize the preservation of absolute continuity and convexity. To obtain different characterizations of the preservation of convexity, we introduce two kinds of duality, the first referring to the process Z and the second to the derived process V. We illustrate the preceding results by considering some examples of interest both in probability and in approximation theory - namely, mixtures, centred subordinators, Bernstein polynomials and beta operators. In most of them, we find bidensities to describe the duality between the derived processes. A unified approach based on stochastic orders is given.

Intrinsic volumes are key functionals in convex geometry and have also appeared in several stochastic settings. Here we relate them to questions of regularity in Gaussian processes with regard to Itô–Nisio oscillation and metrization of GB/GC indexing sets. Various bounds and estimates are presented, and questions for further investigation are suggested. From alternate points of view, much of the discussion can be interpreted in terms of (i) random sets and (ii) properties of (deterministic) infinite-dimensional convex bodies.

A classical result about Markov jump processes states that a certain class of dynamical systems given by ordinary differential equations are obtained as the limit of a sequence of scaled Markov jump processes. This approach fails if the scaling cannot be carried out equally across all entities. In the present paper we present a convergence theorem for such an unequal scaling. In contrast to an equal scaling the limit process is not purely deterministic but still possesses randomness. We show that these processes constitute a rich subclass of piecewise-deterministic processes. Such processes apply in molecular biology where entities often occur in different scales of numbers.

In this paper we develop the theory of the W and Z scale functions for right-continuous (upwards skip-free) discrete-time, discrete-space random walks, along the lines of the analogous theory for spectrally negative Lévy processes. Notably, we introduce for the first time in this context the one- and two-parameter scale functions Z, which appear for example in the joint deficit at ruin and time of ruin problems of actuarial science. Comparisons are made between the various theories of scale functions as one makes time and/or space continuous.