We introduce and study a new model: zero-automatic queues. Roughly, zero-automatic queues are characterized by a special buffering mechanism evolving like a random walk on some infinite group or monoid. The salient result is that all stable zero-automatic queues have a product form stationary distribution and a Poisson output process. When considering the two simplest and extremal cases of zero-automatic queues, we recover the simple M/M/1 queue and Gelenbe's G-queue with positive and negative customers.

We characterize the Lyapunov exponent and ergodicity of nonlinear stochastic recursion models, including nonlinear AR-GARCH models, in terms of an easily defined, uniformly ergodic process. Properties of this latter process, known as the collapsed process, also determine the existence of moments for the stochastic recursion when it is stationary. As a result, both the stability of a given model and the existence of its moments may be evaluated with relative ease. The method of proof involves piggybacking a Foster-Lyapunov drift condition on certain characteristic behavior of the collapsed process.

In this paper telegraph processes on geodesic lines of the Poincaré half-space and Poincaré disk are introduced and the behavior of their hyperbolic distances examined. Explicit distributions of the processes are obtained and the related governing equations derived. By means of the processes on geodesic lines, planar random motions (with independent components) in the Poincaré half-space and disk are defined and their hyperbolic random distances studied. The limiting case of one-dimensional and planar motions together with their hyperbolic distances is discussed with the aim of establishing connections with the well-known stochastic representations of hyperbolic Brownian motion. Extensions of motions with finite velocity to the three-dimensional space are also hinted at, in the final section.

Under the unifying umbrella of a general result of Penrose and Yukich (Annals of Applied Probability13 (2003), 277-303) we give laws of large numbers (in the Lp sense) for the total power-weighted length of several nearest-neighbour-type graphs on random point sets in ℝd, d ∈ ℕ. Some of these results are known; some are new. We give limiting constants explicitly, where previously they have been evaluated in less generality or not at all. The graphs we consider include the k-nearest-neighbours graph, the Gabriel graph, the minimal directed spanning forest, and the on-line nearest-neighbour graph.

Based on the concept of multipower variation we establish a class of easily computable and robust estimators for the integrated volatility, especially including the squared integrated volatility, in Lévy-type stochastic volatility models. We derive consistency and feasible distributional results for the estimators. Furthermore, we discuss the applications to time-changed CGMY, normal inverse Gaussian, and hyperbolic models with and without leverage, where the time-changes are based on integrated Cox-Ingersoll-Ross or Ornstein-Uhlenbeck-type processes. We deduce which type of market microstructure does not affect the estimates.

Assume that the surplus of an insurer follows a compound Poisson surplus process. When the surplus is below zero or the insurer is on deficit, the insurer could borrow money at a debit interest rate to pay claims. Meanwhile, the insurer will repay the debts from her premium income. The negative surplus may return to a positive level. However, when the negative surplus is below a certain critical level, the surplus is no longer able to be positive. Absolute ruin occurs at this moment. In this paper, we study absolute ruin questions by defining an expected discounted penalty function at absolute ruin. The function includes the absolute ruin probability, the Laplace transform of the time to absolute ruin, the deficit at absolute ruin, the surplus just before absolute ruin, and many other quantities related to absolute ruin. First, we derive a system of integro-differential equations satisfied by the function and obtain a defective renewal equation that links the integro-differential equations in the system. Second, we show that when the initial surplus goes to infinity, the absolute ruin probability and the classical ruin probability are asymptotically equal for heavy-tailed claims while the ratio of the absolute ruin probability to the classical ruin probability goes to a positive constant that is less than one for light-tailed claims. Finally, we give explicit expressions for the function for exponential claims.

The Palm version of a stationary random subset of a d-dimensional grid is contructed using the two-step change-of-origin and change-of-measure method. An elementary proof is given of the fact that the Palm version is characterized by point-stationarity (distributional invariance under bijective shifts of the origin from a point of the set to another point of the set).

In the classical risk model with initial capital u, let τ(u) be the time of ruin, X+(u) be the risk reserve just before ruin, and Y+(u) be the deficit at ruin. Gerber and Shiu (1998) defined the function mδ(u) =E[e−δ τ(u)w(X+(u), Y+(u)) 1 (τ(u) < ∞)], where δ ≥ 0 can be interpreted as a force of interest and w(r,s) as a penalty function, meaning that mδ(u) is the expected discounted penalty payable at ruin. This function is known to satisfy a defective renewal equation, but easy explicit formulae for mδ(u) are only available for certain special cases for the claim size distribution. Approximations thus arise by approximating the desired mδ(u) by that associated with one of these special cases. In this paper a functional approach is taken, giving rise to first-order correction terms for the above approximations.

We consider a constant rate traffic which shares a buffer with a random cross traffic. A first come first served or priority service discipline is applied at the buffer. After service at the first buffer the constant rate traffic moves to a play-out buffer. Both buffers provide service at constant rate and infinite waiting room. We investigate logarithmic large and moderate deviation asymptotics for the tail probabilities of the steady-state queue length distribution at the play-out buffer for long-range dependent cross traffic in critical loading. We characterize the asymptotic behavior of the cross traffic which leads to a large queue length at the play-out buffer and compare it to the one for renewal cross traffic.

Subcritical Markov branching processes {Zt} die out sooner or later, say at time T < ∞. We give results for the path to extinction {ZuT, 0 ≤ u ≤ 1} that include its finite dimensional distributions and the asymptotic behaviour of xu−1ZuT, as Z0=x → ∞. The limit reflects an interplay of branching and extreme value theory. Then we consider the population on the verge of extinction, as modelled by ZT-u, u > 0, and show that as Z0= x → ∞ this process converges to a Markov process {Yu}, which we describe completely. Emphasis is on continuous time processes, those in discrete time displaying a more complex behaviour, related to Martin boundary theory.

We consider a weighted random walk model defined as follows. An n-step random walk on the integers with distribution Pn is weighted by giving the path S=(S0,…,Sn) a probability proportional to where the function f is the so-called fitness function. In the case of power-type fitness, we prove the convergence of the renormalized path to a deterministic function with exponential speed. This function is a solution to a variational problem. In the case of the simple symmetric random walk, explicit computations are done. Our result relies on large deviations techniques and Varadhan's integral lemma. We then study an application of this model to mutation-selection dynamics on the integers where a random walk operates the mutation. This dynamics is the infinite-population limit of that of mutation-selection genetic algorithms. We prove that the population grows to ∞ and make explicit its growth speed. This is a toy model for modelling the effect of stronger selection at ∞ for genetic algorithms taking place in a noncompact space.

We consider the Voronoi tessellation based on a stationary Poisson process N in ℝd. We provide a complete and explicit description of the Palm distribution describing N as seen from a randomly chosen (typical) point on a k-face of the tessellation. In particular, we compute the joint distribution of the d−k+1 neighbours of the k-face containing the typical point. Using this result as well as a fundamental general relationship between Palm probabilities, we then derive some properties of the typical k-face and its neighbours. Generalizing recent results of Muche (2005), we finally provide the joint distribution of the typical edge (typical 1-face) and its neighbours.

The random surface measure of a stationary Boolean model with grains from the convex ring is considered. A sufficient condition and a necessary condition for the existence of the density of the second-order moment measure of are given and a representation of this density is derived. As applications, the surface pair correlation functions of a Boolean model with spheres and a Boolean model with randomly oriented right circular cylinders in ℝ3 are determined.

In this paper we consider a discrete-time process which grows according to a random walk with nonnegative increments between crash times at which it collapses to 0. We assume that the probability of crashing depends on the level of the process. We study the stochastic stability of this growth-collapse process. Special emphasis is given to the case in which the probability of crashing tends to 0 as the level of the process increases. In particular, we show that the process may exhibit long-range dependence and that the crash sizes may have a power law distribution.

This paper investigates new properties concerning the multifractal structure of a class of random self-similar measures. These measures include the well-known Mandelbrot multiplicative cascades, sometimes called independent random cascades. We evaluate the scale at which the multifractal structure of these measures becomes discernible. The value of this scale is obtained through what we call the growth speed in Hölder singularity sets of a Borel measure. This growth speed yields new information on the multifractal behavior of the rescaled copies involved in the structure of statistically self-similar measures. Our results are useful in understanding the multifractal nature of various heterogeneous jump processes.

In this paper we propose a new genetic algorithm specifically based on mutation and selection in order to maximize a fitness function. This mutation-selection algorithm behaves as a gradient algorithm which converges to local maxima. In order to obtain convergence to global maxima we propose a new algorithm which is built by randomly perturbing the selection operator of the gradient-like algorithm. The perturbation is controlled by only one parameter: that which allows the selection pressure to be governed. We use the Markov model of the perturbed algorithm to prove its convergence to global maxima. The arguments used in the proofs are based on Freidlin and Wentzell's (1984) theory and large deviation techniques also applied in simulated annealing. Our main results are that (i) when the population size is greater than a critical value, the control of the selection pressure ensures the convergence to the global maxima of the fitness function, and (ii) the convergence also occurs when the population is the smallest possible, i.e. 1.

In this paper we introduce certain Hankel matrices that can be used to study ME (matrix exponential) distributions, in particular to compute their ME orders. The Hankel matrices for a given ME probability distribution can be constructed if one of the following five types of information about the distribution is available: (i) an ME representation, (ii) its moments, (iii) the derivatives of its distribution function, (iv) its Laplace-Stieltjes transform, or (v) its distribution function. Using the Hankel matrices, a necessary and sufficient condition for a probability distribution to be an ME distribution is found and a method of computing the ME order of the ME distribution developed. Implications for the PH (phase-type) order of PH distributions are examined. The relationship between the ME order, the PH order, and a lower bound on the PH order given by Aldous and Shepp (1987) is discussed in numerical examples.

We derive a new compound Poisson distribution with explicit parameters to approximate the number of overlapping occurrences of any set of words in a Markovian sequence. Using the Chen-Stein method, we provide a bound for the approximation error. This error converges to 0 under the rare event condition, even for overlapping families, which improves previous results. As a consequence, we also propose Poisson approximations for the declumped count and the number of competing renewals.