A stationary partition based on a stationary point process N in ℝd is an ℝd -valued random field π={π(x): x∈ℝd } such that both π(y)∈N for each y∈ℝd and the random partition {{y∈ℝd : π(y)=x}: x∈N} is stationary jointly with N. Stationary partitions may be considered as general versions of the stationary random tessellations studied in stochastic geometry. As in the special case of the Voronoi tessellation, a stationary partition can be used to relate the underlying stationary probability measure to the associated Palm probability measure of N. In doing so, we will develop some basic theory for stationary partitions and extend properties of stationary tessellations to our more general case. One basic idea is that the stationary measure is (up to a shift) a weighted version of the Palm measure, where the weight is the volume of the typical cell. We will make systematic use of a known modified probability measure. Finally, we use our approach to extend some recent results on the shift coupling of the stationary distribution and the Palm distribution.

Let F be a probability distribution with support on the nonnegative integers. We describe two algorithms for generating a stationary random graph, with vertex set ℤ, in which the degrees of the vertices are independent, identically distributed random variables with distribution F. Focus is on an algorithm generating a graph in which, initially, a random number of ‘stubs’ with distribution F is attached to each vertex. Each stub is then randomly assigned a direction (left or right) and the edge configuration obtained by pairing stubs pointing to each other, first exhausting all possible connections between nearest neighbors, then linking second-nearest neighbors, and so on. Under the assumption that F has finite mean, it is shown that this algorithm leads to a well-defined configuration, but that the expected length of the shortest edge attached to a given vertex is infinite. It is also shown that any stationary algorithm for pairing stubs with random, independent directions causes the total length of the edges attached to a given vertex to have infinite mean. Connections to the problem of constructing finitary isomorphisms between Bernoulli shifts are discussed.

We discuss the limit behavior of the partial sums process of stationary solutions to the (autoregressive) AR(1) equation X t = a t X t−1 + εt with random (renewal-reward) coefficient, a t , taking independent, identically distributed values A j ∈ [0,1] on consecutive intervals of a stationary renewal process with heavy-tailed interrenewal distribution, and independent, identically distributed innovations, εt , belonging to the domain of attraction of an α-stable law (0 < α ≤ 2, α ≠ 1). Under suitable conditions on the tail parameter of the interrenewal distribution and the singularity parameter of the distribution of A j near the unit root a = 1, we show that the partial sums process of X t converges to a λ-stable Lévy process with index λ < α. The paper extends the result of Leipus and Surgailis (2003) from the case of finite-variance X t to that of infinite-variance X t .

Consider a sequence {X k , k ≥ 1} of random variables on (−∞, ∞). Results on the asymptotic tail probabilities of the quantities , and S (n) = max0 ≤ k ≤ n S k , with X 0 = 0 and n ≥ 1, are well known in the case where the random variables are independent with a heavy-tailed (subexponential) distribution. In this paper we investigate the validity of these results under more general assumptions. We consider extensions under the assumptions of having long-tailed distributions (the class L) and having the class D ∩ L, where D is the class of distribution functions with dominatedly varying tails. Some results are also given in the case where X k , k ≥ 1, are not necessarily identically distributed and/or independent.

Fractional Laplace motion is obtained by subordinating fractional Brownian motion to a gamma process. Used recently to model hydraulic conductivity fields in geophysics, it might also prove useful in modeling financial time series. Its one-dimensional distributions are scale mixtures of normal laws, where the stochastic variance has the generalized gamma distribution. These one-dimensional distributions are more peaked at the mode than is a Gaussian distribution, and their tails are heavier. In this paper we derive the basic properties of the process, including a new property called stochastic self-similarity. We also study the corresponding fractional Laplace noise, which may exhibit long-range dependence. Finally, we discuss practical methods for simulation.

We consider the stability of the longest-queue-first scheduling policy (LQF), a natural and low-complexity scheduling policy, for a generalized switch model. Unlike that of common scheduling policies, the stability of LQF depends on the variance of the arrival processes in addition to their average intensities. We identify new sufficient conditions for LQF to be throughput optimal for independent, identically distributed arrival processes. Deterministic fluid analogs, proved to be powerful in the analysis of stability in queueing networks, do not adequately characterize the stability of LQF. We combine properties of diffusion-scaled sample path functionals and local fluid limits into a sharper characterization of stability.

Gaussian wave models have been successfully used since the early 1950s to describe the development of random sea waves, particularly as input to dynamic simulation of the safety of ships and offshore structures. A drawback of the Gaussian model is that it produces stochastically symmetric waves, which is an unrealistic feature and can lead to unconservative safety estimates. The Gaussian model describes the height of the sea surface at each point as a function of time and space. The Lagrange wave model describes the horizontal and vertical movements of individual water particles as functions of time and original location. This model is physically based, and a stochastic version has recently been advocated as a realistic model for asymmetric water waves. Since the stochastic Lagrange model treats both the vertical and the horizontal movements as Gaussian processes, it can be analysed using methods from the Gaussian theory. In this paper we present an analysis of the stochastic properties of the first-order stochastic Lagrange waves model, both as functions of time and as functions of space. A Slepian model for the description of the random shape of individual waves is also presented and analysed.

In this paper we study players' long-run behaviors in evolutionary coordination games with imperfect monitoring. In each time period, signals corresponding to players' underlying actions, instead of the actions themselves, are observed. A boundedly rational quasi-Bayesian learning process is proposed to extract information from the realized signals. We find that players' long-run behaviors depend not only on the correlations between actions and signals, but on the initial probabilities of risk-dominant and non-risk-dominant equilibria being chosen. The conditions under which risk-dominant equilibrium, non-risk-dominant equilibrium, and the coexistence of both equilibria emerges in the long run are shown. In some situations, the number of limiting distributions grows unboundedly as the population size grows to infinity.

In this paper, we analyze a queueing system characterized by a space-time arrival process of customers served by a countable set of servers. Customers arrive at points in space and the server stations have space-dependent processing rates. The workload is seen as a Radon measure and the server stations can adapt their power allocation to the current workload. We derive the stability region of the queueing system in the usual stationary ergodic framework. The analysis of this stability region gives some counter-intuitive results. Some specific subclasses of policy are also studied. Wireless communications networks is a natural field of application for the model.

A continuous-time model with stationary increments for asset price {P t } is an extension of the symmetric subordinator model of Heyde (1999), and allows for skewness of returns. In the setting of independent variance-gamma-distributed returns the model resembles closely that of Madan, Carr, and Chang (1998). A simple choice of parameters renders {e−rt P t } a familiar martingale. We then specify the activity time process, {T t }, for which {T t − t} is asymptotically self-similar and {τt }, with τt = T t − T t−1, is gamma distributed. This results in a skew variance-gamma distribution for each log price increment (return) X t and a model for {X t } which incorporates long-range dependence in squared returns. Our approach mirrors that for the (symmetric) Student process model of Heyde and Leonenko (2005), to which the present work is intended as a complement and a sequel. One intention is to compare, partly on the basis of fitting to data, versions of the general model wherein the returns have either (symmetric) t-distributions or variance-gamma distributions.

In this paper, we extend the integer-valued model class to give a nonnegative integer-valued bilinear process, denoted by INBL(p,q,m,n), similar to the real-valued bilinear model. We demonstrate the existence of this strictly stationary process and give an existence condition for it. The estimation problem is discussed in the context of a particular simple case. The method of moments is applied and the asymptotic joint distribution of the estimators is given: it turns out to be a normal distribution. We present numerical examples and applications of the model to real time series data on meningitis and Escherichia coli infections.

We consider a buying-selling problem when two stops of a sequence of independent random variables are required. An optimal stopping rule and the value of a game are obtained.

We study identities for the distribution of the number of edges at time t back (i.e. measured backwards) in a coalescent tree whose subtrees have no mutations. This distribution is important in the infinitely-many-alleles model of mutation, where every mutation is unique. The model includes, as a special case, the number of edges in a coalescent tree at time t back when mutation is ignored. The identities take the form of expected values of functions of Z t =eiX t , where X t is distributed as standard Brownian motion. Associated identities are also found for the distributions of the time to the most recent common ancestor, the time until loss of ancestral lines by coalescence or mutation, and the age of a mutation. Hypergeometric functions play an important role in the identities. The identities are of mathematical interest, as well as potentially being formulae to use for numerical integration or simulation to compute distributions that are usually expressed as alternating-sign series expansions, which are difficult to compute.

In this short note we show how new fluctuation identities and their associated asymptotics, given in Vigon (2002), Klüppelberg et al. (2004) and Doney and Kyprianou (2006), provide the basis for establishing, in an elementary way, asymptotic overshoot and undershoot distribitions for a general class of Lévy insurance risk processes. The results bring the earlier conclusions of Asmussen and Klüppelberg (1996) for the Cramér-Lundberg process into greater generality.

Classical extreme value theory for stationary sequences of random variables can to a large extent be paraphrased as the study of exceedances over a high threshold. A special role within the description of the temporal dependence between such exceedances is played by the extremal index. Parts of this theory can be generalized not only to random variables on an arbitrary state space hitting certain failure sets, but even to a triangular array of rare events on an abstract probability space. In the case of M4 (maxima of multivariate moving maxima) processes, the arguments take a simple and direct form.

In this article, we study a particular example of general random tessellation, the dead leaves model. This model, first studied by the mathematical morphology school, is defined as a sequential superimposition of random closed sets, and provides the natural tool to study the occlusion phenomenon, an essential ingredient in the formation of visual images. We generalize certain results of G. Matheron and, in particular, compute the probability of n compact sets being included in visible parts. This result characterizes the distribution of the boundary of the dead leaves tessellation.

We consider a stationary moving average process with random coefficients, , generated by an array, {C t,k , t ∈ Z, k ≥ 0}, of random variables and a heavy-tailed sequence, {Z t , t ∈ Z}. We analyze the limit behavior using a point process analysis. As applications of our results we compare the limiting behavior of the moving average process with random coefficients with that of a standard MA(∞) process.

We discuss two Monte Carlo algorithms for finding the global maximum of a simple random walk with negative drift. This problem can be used to connect the analysis of random input Monte Carlo algorithms with ideas and principles from mathematical statistics.

Methods using gambling teams and martingales are developed and applied to find formulae for the expected value and the generating function of the waiting time to observation of an element of a finite collection of patterns in a sequence generated by a two-state Markov chain of first, or higher, order.

Let {X(s , t), s = (s 1, s 2) ∈ ℝ2, t ∈ ℝ} be a stationary random field defined over a discrete lattice. In this paper, we consider a set of domain of attraction criteria giving the notion of extremal index for random fields. Together with the extremal-types theorem given by Leadbetter and Rootzen (1997), this will give a characterization of the limiting distribution of the maximum of such random fields.