This paper establishes new Foster-type criteria for a Markov chain on a general state space to be Harris recurrent, positive Harris recurrent, or geometrically ergodic. The criteria are based on drift conditions involving stopping times rather than deterministic steps. Meyn and Tweedie (1994) developed similar criteria involving random-sized steps, independent of the Markov chain under study. They also posed an open problem of finding criteria involving stopping times. Our results essentially solve that problem. We also show that the assumption of ψ-irreducibility is not needed when stating our drift conditions for positive Harris recurrence or geometric ergodicity.

We derive asymptotic properties for a stochastic dynamic network model in a stochastic dynamic population. In the model, nodes give birth to new nodes until they die, each node being equipped with a social index given at birth. During the life of a node it creates edges to other nodes, nodes with high social index at higher rate, and edges disappear randomly in time. For this model, we derive a criterion for when a giant connected component exists after the process has evolved for a long period of time, assuming that the node population grows to infinity. We also obtain an explicit expression for the degree correlation ρ (of neighbouring nodes) which shows that ρ is always positive irrespective of parameter values in one of the two treated submodels, and may be either positive or negative in the other model, depending on the parameters.

The aim of this paper is to evaluate the performance of the optimal policy (the Gittins index policy) for open tax problems of the type considered by Klimov in the undiscounted limit. In this limit, the state-dependent part of the cost is linear in the state occupation numbers for the multi-armed bandit, but is quadratic for the tax problem. The discussion of the passage to the limit for the tax problem is believed to be largely new; the principal novelty is our evaluation of the matrix of the quadratic form. These results are confirmed by a dynamic programming analysis, which also suggests how the optimal policy should be modified when resources can be freely deployed only within workstations, rather than system-wide.

We introduce a class of discrete-time two-sex branching processes where the offspring probability distribution and the mating function are governed by an environmental process. It is assumed that the environmental process is formed by independent but not necessarily identically distributed random vectors. For such a class, we determine some relationships among the probability generating functions involved in the mathematical model and derive expressions for the main moments. Also, by considering different probabilistic approaches we establish several results concerning the extinction probability. A simulated example is presented as an illustration.

In this paper we study a single-period optimal portfolio problem in which the aim of the investor is to maximize the expected utility. We assume that the return of every security in the market is a mixture of some common underlying source of risks. A sufficient condition to order the optimal allocations is obtained, and it is shown that several models studied in the literature before are special cases of the proposed model. In the course of the analysis concepts in stochastic orders are employed, and a new characterization of the likelihood ratio order is obtained.

We examine the geometry of regions of maintainable structures arising in a Markov manpower model. The regions are described in terms of convex hulls, and it is shown that for systems divided into two or three grades these regions form an increasing sequence. It is also shown that the monotonie property fails quite drastically for a four-graded system.

An open problem is discussed for a related model.

We observe a realization of a stationary weighted Voronoi tessellation of the d-dimensional Euclidean space within a bounded observation window. Given a geometric characteristic of the typical cell, we use the minus-sampling technique to construct an unbiased estimator of the average value of this geometric characteristic. Under mild conditions on the weights of the cells, we establish variance asymptotics and the asymptotic normality of the unbiased estimator as the observation window tends to the whole space. Moreover, weak consistency is shown for this estimator.

In this paper we prove asymptotic upper bounds on the variance of the number of vertices and the missed area of inscribed random disc-polygons in smooth convex discs whose boundary is C+2. We also consider a circumscribed variant of this probability model in which the convex disc is approximated by the intersection of random circles.

We analyse in this paper the fluid weighted fair queueing system with two classes of customers, who arrive according to Poisson processes and require arbitrarily distributed service times. In a first step, we express the Laplace transform of the joint distribution of the workloads in the two virtual queues of the system by means of unknown Laplace transforms. Such an unknown Laplace transform is related to the distribution of the workload in one queue provided that the other queue is empty. We explicitly compute the unknown Laplace transforms by means of a Wiener—Hopf technique. The determination of the unknown Laplace transforms can be used to compute some performance measures characterizing the system (e.g. the mean waiting time for each class) which we compute in the exponential service case.

Poisson-like behavior for event count data is ubiquitous in nature. At the same time, differencing of such counts arises in the course of data processing in a variety of areas of application. As a result, the Skellam distribution – defined as the distribution of the difference of two independent Poisson random variables – is a natural candidate for approximating the difference of Poisson-like event counts. However, in many contexts strict independence, whether between counts or among events within counts, is not a tenable assumption. Here we characterize the accuracy in approximating the difference of Poisson-like counts by a Skellam random variable. Our results fully generalize existing, more limited, results in this direction and, at the same time, our derivations are significantly more concise and elegant. We illustrate the potential impact of these results in the context of problems from network analysis and image processing, where various forms of weak dependence can be expected.

More than half a century ago, it was proved that the increasing failure rate (IFR) property is preserved under the formation of k-out-of-n systems (order statistics) when the lifetimes of the components are independent and have a common absolutely continuous distribution function. However, this property has not yet been proved in the discrete case. Here we give a proof based on the log-concavity property of the function $f({{\mathrm{e}}}^x)$. Furthermore, we extend this property to general distribution functions and general coherent systems under some conditions.

It is shown that the two descriptions of the ages of alleles corresponding to the two formulations of the stationary infinitely-many-neutral-alleles diffusion model discussed by Ethier (1990a) are equivalent.

We consider a stochastic graph generated by a continuous-time birth-and-death process with exponentially distributed waiting times. The vertices are the living particles, directed edges go from mothers to daughters. The size and the structure of the connected components are investigated. Furthermore, the number of connected components is determined.

The equilibrium probability distribution over the set of absorbing states of a reducible Markov chain is specified a priori and it is required to obtain the constrained sub-space or feasible region for all possible initial probability distributions over the set of transient states. This is called the inverse problem. It is shown that a feasible region exists for the choice of equilibrium distribution. Two different cases are studied: Case I, where the number of transient states exceeds that of the absorbing states and Case II, the converse. The approach is via the use of generalised inverses and numerical examples are given.

A stationary doubly stochastic Poisson sequence is considered. Asymptotic properties of the best linear estimates of the intensities are investigated.

Suppose that n arcs with random lengths having distributions F1, F2, · ··, Fn are placed uniformly and independently on a circle. This paper presents inequalities which tell how certain distributions and probabilities change as the variability of the distributions Fl, F2, ··, Fn is increased. A distribution F is considered to be more variable than G if f h(x)dF(x) ≧ h(x)dG(x) for all convex functions h.

In this paper we present a stability criterion for processor-sharing queues, in which the throughput may depend on the number of customers in the system (such as in the case of interferences between users). Such a system is represented by a point measure-valued stochastic recursion keeping track of the remaining processing times of the customers.

A point process procedure can be used to study reservoir-induced seismicity (RIS), in which the intensity function representing earthquake hazard is a combination of three terms: a constant background term, an ETAS (epidemic-type aftershock sequence) term for aftershocks, and a time function derived from observation of water levels of a reservoir. This paper presents the results of such a study of the seismicity in the vicinity of the Tarbela reservoir in Pakistan. Making allowance for changes in detection capability and the background seismicity related to tectonic activity, earthquakes of magnitude ≥ 2.0, occurring between May 1978 and January 1982 and whose epicentres were within 100 km of the reservoir, were used in this analysis. Several different intensities were compared via their Akaike information criterion (AIC) values relative to those of a Poisson process. The results demonstrate that the seismicity within 20 km of the reservoir correlates with water levels of the reservoir, namely, active periods occur about 250 days after the appearance of low water levels. This suggests that unloading the reservoir activates the seismicity beneath it. Seasonal variations of the seismicity in an area up to 100 km from the reservoir were also found, but these could not be adequately interpreted by an appropriate RIS mechanism.

This paper is an extension of Davis (1965) by allowing immigration. Mean square convergence is proved for a random variable in a branching diffusion process allowing immigration.

We study a system of interacting renewal processes which is a model for neuronal activity. We show that the system possesses an exponentially large number (with respect to the number of neurons in the network) of limiting configurations of the ‘firing neurons’. These we call patterns. Furthermore, under certain conditions of symmetry we find an algorithm to control limiting patterns by means of the connection parameters.