Birth and death processes can be constructed as projections of higher-dimensional Poisson processes. The existence and uniqueness in the strong sense of the solutions of the time change problem are obtained. It is shown that the solution of the time change problem is equivalent to the solution of the corresponding martingale problem. Moreover, the processes obtained by the projection method are ergodic under translations.

We give a unified presentation of stability results for stochastic vector difference equations based on various choices of binary operations and , assuming that are stationary and ergodic. In the scalar case, under standard addition and multiplication, the key condition for stability is E[log |A0|] < 0. In the generalizations, the condition takes the form γ< 0, where γis the limit of a subadditive process associated with . Under this and mild additional conditions, the process has a unique finite stationary distribution to which it converges from all initial conditions.

The variants of standard matrix algebra we consider replace the operations + and × with (max, +), (max,×), (min, +), or (min,×). In each case, the appropriate stability condition parallels that for the standard recursions, involving certain subadditive limits. Since these limits are difficult to evaluate, we provide bounds, thus giving alternative, computable conditions for stability.

The cumulative distribution of the finite sum of the binary sequence of order k is studied and some of its applications discussed. Certain properties of this sequence are investigated and uniformly superior bounds for the cumulative distribution under minimal information on the ‘success' probabilities are derived.

We derive an ESTA property for marked point processes similar to Wolff's PASTA property for ordinary (non-marked) point processes, via a stochastic integral approach. This new ESTA property allows us to extend a known result on the conditional PASTA property and to derive an ASTA property for batch arrival processes. We also present an application of our results.

The paper deals with processes in (or ), one of whose components is skip-free. We obtain identities for distributions of hitting times for the components of the process generalizing the well-known one for the one-dimensional case. These relations reflect the fact that in this case spatial and time coordinates play, in some sense, symmetric roles. They turn out to be useful for solving several problems. For example, they allow us to find the distribution of the number of jumps of the process, which fall in a fixed set before the skip-free component of the process hits a fixed level. Examples are given showing how our results can be applied to models in branching processes, queueing, and risk theory.

In this paper, we develop mathematical machinery for verifying that a broad class of general state space Markov chains reacts smoothly to certain types of perturbations in the underlying transition structure. Our main result provides conditions under which the stationary probability measure of an ergodic Harris-recurrent Markov chain is differentiable in a certain strong sense. The approach is based on likelihood ratio ‘change-of-measure' arguments, and leads directly to a ‘likelihood ratio gradient estimator' that can be computed numerically.

Neveu's exchange formula relates the Palm probabilities with respect to two jointly stationary simple point processes. We give a new proof of the exchange formula by using a simple result from discrete time stationary stochastic processes.

The first-passage problem for the one-dimensional Wiener process with drift in the presence of elastic boundaries is considered. We use the Kolmogorov backward equation with corresponding boundary conditions to derive explicit closed-form expressions for the expected value and the variance of the first-passage time. Special cases with pure absorbing and/or reflecting barriers arise for a certain choice of a parameter constellation.

Certain images arising in astrophysics and medicine are modelled as smooth random fields inside a fixed region, and experimenters are interested in the number of ‘peaks', or more generally, the topological structure of ‘hot-spots' present in such an image. This paper studies the Euler characteristic of the excursion set of a random field; the excursion set is the set of points where the image exceeds a fixed threshold, and the Euler characteristic counts the number of connected components in the excursion set minus the number of ‘holes'. For high thresholds the Euler characteristic is a measure of the number of peaks. The geometry of excursion sets has been studied by Adler (1981) who gives the expectation of two excursion set characteristics, called the DT (differential topology) and IG (integral geometry) characteristics, which equal the Euler characteristic provided the excursion set does not touch the boundary of the region. Worsley (1995) finds a boundary correction which gives the expectation of the Euler characteristic itself in two and three dimensions. The proof uses a representation of the Euler characteristic given by Hadwiger (1959). The purpose of this paper is to give a general result for any number of dimensions. The proof takes a different approach and uses a representation from Morse theory. Results are applied to the recently discovered anomalies in the cosmic microwave background radiation, thought to be the remnants of the creation of the universe.

Johnson–Mehl tessellations can be considered as the results of spatial birth–growth processes. It is interesting to know when such a birth–growth process is completed within a bounded region. This paper deals with the limiting distributions of the time of completion for various models of Johnson–Mehl tessellations in ℝd and k-dimensional sectional tessellations, where 1 ≦ k < d, by considering asymptotic coverage probabilities of the corresponding Boolean models. Random fractals as the results of birth–growth processes are also discussed in order to show that a birth–growth process does not necessarily lead to a Johnson–Mehl tessellation.

Records from are analyzed, where {Yj} is an independent sequence of random variables. Each Yj has a continuous distribution function Fj = Fλj for some distribution F and some λ j > 0. We study records, record times and related quantities for this sequence. Depending on the sequence of powers , a wide spectrum of behaviour is exhibited.

The partially observed control problem is considered for stochastic processes with control entering into the diffusion and the observation. The maximum principle is proved for the partially observable optimal control. A pure probabilistic approach is used, and the adjoint processes are characterized as solutions of related backward stochastic differential equations in finite-dimensional spaces. Most of the derivation is identified with that of the completely observable case.

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 (1993a)) on the sphere, define to be a realization of the random process and to be the cardinality of . A bootstrap algorithm is presented and conditions for strong uniform consistency of the bootstrap cumulative distribution function of the standardized sample mean, , are given. We illustrate the bootstrap algorithm with global land-area data.

We investigate the stationarity of minification processes when the marginal is a discrete distribution. There is a close relationship between the problem considered by Arnold and Isaacson (1976) and the stationarity in minification processes. We give a necessary and sufficient condition for a discrete distribution to be the marginal of a stationary minification process. Members of the Poisson and negative binomial families can be the marginals of stationary minification processes. The geometric minification process is studied in detail, and two characterizations of it based on the structure of the innovation process are given.

This work considers items (e.g. books, files) arranged in an array (e.g. shelf, tape) with N positions and assumes that items are requested according to a Markov chain (possibly, of higher order). After use, the requested item is returned to the leftmost position of the array. Successive applications of the procedure above give rise to a Markov chain on permutations. For equally likely items, the number of requests that makes this Markov chain close to its stationary state is estimated. To achieve that, a coupling argument and the total variation distance are used. Finally, for non-equally likely items and so-called p-correlated requests, the coupling time is presented as a function of the coupling time when requests are independent.

Formulas for the asymptotic failure rate, long-term average availability, and the limiting distribution of the number of long ‘outages' are obtained for a general class of two-state reliability models for maintained systems. The results extend known formulas for alternating renewal processes to a wider class of point processes that includes sequences of dependent or non-identically distributed operating and repair times.

We extend large exceedence results for i.i.d. -valued random variables to a class of uniformly recurrent Markov-additive processes and stationary strong-mixing processes. As in the i.i.d. case, the results are proved via large deviations estimates.

We introduce a stochastic process with discrete time and countable state space that is governed by a sequence of Markov matrices . Each Pm is used for a random number of steps Tm and is then replaced by Pm+1. Tm is a randomized stopping time that may depend on the most recent part of the state history. Thus the global character of the process is non-Markovian.

This process can be used to model the well-known simulated annealing optimization algorithm with randomized, partly state depending cooling schedules. Generalizing the concept of strong stationary times (Aldous and Diaconis [1]) we are able to show the existence of optimal schedules and to prove some desirable properties. This result is mainly of theoretical interest as the proofs do not yield an explicit algorithm to construct the optimal schedules.

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.