In a coverage problem introduced by Dvoretzky (1956) the circle is covered with lengths The associated indexed martingale can be considered as a set of random densities with respect to Lebesgue measure, of which the limit is a random measure. The total masses of the set constitute a new martingale. From a theorem of Shepp (1972), this martingale does not converge in a space if . If 0 < α < 1 it converges in all spaces , and in this paper we demonstrate that it converges in all spaces Lp for p > 2. We also obtain quite precise estimates for the moments of the total mass of the limit measure, showing that the characteristic function of this total mass is an entire function of order 1/(1 − α). Thus in some sense the mass is distributed in a bounded domain.

Let be a real-valued, homogeneous, and isotropic random field indexed in . When restricted to those indices with , the Euclidean length of , equal to r (a positive constant), then the random field resides on the surface of a sphere of radius r. Using a modified stratified spherical sampling plan (Brown (1993)) on the sphere, define to be a realization of the random process and to be the cardinality of . Without specifying the dependence structure of nor the marginal distribution of the , conditions for asymptotic normality of the standardized sample mean, , are given. The conditions on and are motivated by the ideas and results for dependent stationary sequences.

Under the assumptions of the neutral infinite alleles model, K (the total number of alleles present in a sample) is sufficient for estimating θ (the mutation rate). This is a direct result of the Ewens sampling formula, which gives a consistent, asymptotically normal estimator for θ based on K. It is shown that the same estimator used to estimate θ under neutrality is consistent and asymptotically normal, even when the assumption of selective neutrality is violated.

Shot noise processes form an important class of stochastic processes modeling phenomena which occur as shocks to a system and with effects that diminish over time. In this paper we present extreme value results for two cases — a homogeneous Poisson process of shocks and a non-homogeneous Poisson process with periodic intensity function. Shocks occur with a random amplitude having either a gamma or Weibull density and dissipate via a compactly supported impulse response function. This work continues work of Hsing and Teugels (1989) and Doney and O'Brien (1991) to the case of random amplitudes.

Many applications of smoothed perturbation analysis lead to estimators with hazard rate functions of underlying distributions. A key assumption used in proving unbiasedness of the resulting estimator is that the hazard rate function be bounded, a restrictive assumption which excludes all distributions with finite support. Here, we prove through a simple example that this assumption can in fact be removed.

We consider the distribution of the free coordinates of a time-homogeneous Markov process at the time of its first passage into a prescribed stopping set. This calculation (for an uncontrolled process) is of interest because under some circumstances it enables one to calculate the optimal control for a related controlled process. Scaling assumptions are made which allow the application of large deviation techniques. However, the first-order evaluation obtained by these techniques is often too crude to be useful, and the second-order correction term must be calculated. An expression for this correction term as an integral over time is obtained in Equation (20). The integration can be performed in some cases to yield the conclusions of Theorems 1 and 2, expressed in Equations (7) and (9). Theorem 1 gives the probability density of the state vector (to the required degree of approximation) at a prescribed time for a class of processes we may reasonably term linear. Theorem 2 evaluates (without any assumption of linearity) the ratio of this density to the probability density of the coordinates under general stopping rules.

We study a fundamental feature of the generalized semi-Markov processes (GSMPs), called event coupling. The event coupling reflects the logical behavior of a GSMP that specifies which events can be affected by any given event. Based on the event-coupling property, GSMPs can be classified into three classes: the strongly coupled, the hierarchically coupled, and the decomposable GSMPs. The event-coupling property on a sample path of a GSMP can be represented by the event-coupling trees. With the event-coupling tree, we can quantify the effect of a single perturbation on a performance measure by using realization factors. A set of equations that specifies the realization factors is derived. We show that the sensitivity of steady-state performance with respect to a parameter of an event lifetime distribution can be obtained by a simple formula based on realization factors and that the sample-path performance sensitivity converges to the sensitivity of the steady-state performance with probability one as the length of the sample path goes to infinity. This generalizes the existing results of perturbation analysis of queueing networks to GSMPs.

The use of the risk-neutral probability measure has proved to be very powerful for computing the prices of contingent claims in the context of complete markets, or the prices of redundant securities when the assumption of complete markets is relaxed. We show here that many other probability measures can be defined in the same way to solve different asset-pricing problems, in particular option pricing. Moreover, these probability measure changes are in fact associated with numéraire changes, this feature, besides providing a financial interpretation, permits efficient selection of the numéraire appropriate for the pricing of a given contingent claim and also permits exhibition of the hedging portfolio, which is in many respects more important than the valuation itself.

The key theorem of general numéraire change is illustrated by many examples, among which the extension to a stochastic interest rates framework of the Margrabe formula, Geske formula, etc.

This paper is concerned with the distibutions and the moments of the area and the local time of a random walk, called the Bernoulli meander. The limit behavior of the distributions and the moments is determined in the case where the number of steps in the random walk tends to infinity. The results of this paper yield explicit formulas for the distributions and the moments of the area and the local time for the Brownian meander.

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.

A biphase image, representing the normal and degenerated fibres in a vertical cross-section of a nerve, is considered. A random set model based on a Gibbs point process is proposed for the union of the two phases. A kind of independence between the degeneration process and the original fibres is defined and tested.

Let u ε(t, x) be the position at time t of a point x on a string, where the time variable t varies in an interval I: = [0, T], T is a fixed positive time, and the space variable x varies in an interval J. The string is performing forced vibrations and also under the influence of small stochastic perturbations of intensity ε. We consider two kinds of random perturbations, one in the form of initial white noise, and the other is a nonlinear random forcing which involves the formal derivative of a Brownian sheet. When J has finite endpoints, a Dirichlet boundary condition is imposed for the solutions of the resulting non-linear wave equation. Assuming that the initial conditions are of sufficient regularity, we analyze the deviations u ε(t, x) from u 0(t, x), the unperturbed position function, as the intensity of perturbation ε ↓ 0 in the uniform topology. We also discuss some continuity properties of the realization of the solutions u ε(t, x).

Simulation procedures for typical Johnson-Mehl crystals generated under various models for random nucleation are proposed. These procedures include algorithms for simulating spatio-time-inhomogeneous Poisson processes. Empirical results for a particular class of Johnson-Mehl tessellations in two and three dimensions show remarkably different crystals.

The present paper proposes a general approach for finding differential equations to evaluate probabilities of ruin in finite and infinite time. Attention is given to real-valued non-diffusion processes where a Markov structure is obtainable. Ruin is allowed to occur upon a jump or between the jumps. The key point is to define a process of conditional ruin probabilities and identify this process stopped at the time of ruin as a martingale. Using the theory of marked point processes together with the change-of-variable formula or the martingale representation theorem for point processes, we obtain differential equations for evaluating the probability of ruin.

Numerical illustrations are given by solving a partial differential equation numerically to obtain the probability of ruin over a finite time horizon.

Consider a sequence of possibly dependent random variables having the same marginal distribution F, whose tail 1−F is regularly varying at infinity with an unknown index − α < 0 which is to be estimated. For i.i.d. data or for dependent sequences with the same marginal satisfying mixing conditions, it is well known that Hill's estimator is consistent for α−1 and asymptotically normally distributed. The purpose of this paper is to emphasize the central role played by the tail empirical process for the problem of consistency. This approach allows us to easily prove Hill's estimator is consistent for infinite order moving averages of independent random variables. Our method also suffices to prove that, for the case of an AR model, the unknown index can be estimated using the residuals generated by the estimation of the autoregressive parameters.

A stationary (but not necessarily isotropic) Boolean model Y in the plane is considered as a model for overlapping particle systems. The primary grain (i.e. the typical particle) is assumed to be simply connected, but no convexity assumptions are made. A new method is presented to estimate the intensity y of the underlying Poisson process (i.e. the mean number of particles per unit area) from measurements on the union set Y. The method is based mainly on the concept of convexification of a non-convex set, it also produces an unbiased estimator for a (suitably defined) mean body of Y, which in turn makes it possible to estimate the mean grain of the particle process.

Models for epidemic spread of infections are formulated by defining intensities for relevant counting processes. It is assumed that an infected individual passes through k stages of infectivity. The times spent in the different stages are random. Many well-known models for the spread of infections can be described in this way. The models can also be applied to describe other processes of epidemic character (such as models for rumour spreading). Asymptotic results are derived both for the size and for the duration of the epidemic.

Non-parametric estimators of the distribution of the grain of the Boolean model are considered. The technique is based on the study of point processes of tangent points in different directions related to the Boolean model. Their second- and higher-order characteristics are used to estimate the mean body and the distribution of the typical grain. Central limit theorems for the improved estimator of the intensity and surface measures of the Boolean model are also proved.

Consider a homogeneous Poisson process in with density ρ, and add the origin as an extra point. Now connect any two points x and y of the process with probability g(x − y), independently of the point process and all other pairs, where g is a function which depends only on the Euclidean distance between x and y, and which is nonincreasing in the distance. We distinguish two critical densities in this model. The first is the infimum of all densities for which the cluster of the origin is infinite with positive probability, and the second is the infimum of all densities for which the expected size of the cluster of the origin is infinite. It is known that if , then the two critical densities are non-trivial, i.e. bounded away from 0 and ∞. It is also known that if g is of the form , for some r > 0, then the two critical densities coincide. In this paper we generalize this result and show that under the integrability condition mentioned above the two critical densities are always equal.

One result that is of both theoretical and practical importance regarding point processes is the method of thinning. The basic idea of this method is that under some conditions, there exists an embedded Poisson process in any point process such that all its arrival points form a sub-sequence of the Poisson process. We extend this result by showing that on the embedded Poisson process of a uni- or multi-variable marked point process in which interarrival time distributions may depend on the marks, one can define a Markov chain with a discrete state that characterizes the stage of the interarrival times. This implies that one can construct embedded Markov chains with countable state spaces for the state processes of many practical systems that can be modeled by such point processes.