In this paper we examine the extremal tail probabilities of moving sums in a marked Poisson random field. These sums are computed by adding up the weighted occurrences of events lying within a scanning set of fixed shape and size. We also provide an alternative representation of the constants of the asymptotic formulae in terms of the occupation measure of the conditional local random field at zero, and extend these representations to the constants of asymptotic tail probabilities of Gaussian random fields.

By using properties of up- and downcrossings of the sample functions of the workload process and of the attained waiting-time process for a G/G/1 queueing model, a direct proof of a theorem proved by Sakasegawa and Wolff is given.

Consider a circle of circumference 1. Throw n points at random onto this circle and append to each of these points a clockwise arc of length s. The resulting random set is a union of a random number of connected components, each with specific size. Using tools designed by Steutel, we compute the joint distribution of the lengths of the connected components. Asymptotic results are presented when n goes to ∞ and s to 0 jointly according to different regimes.

A central limit theorem for cumulative processes was first derived by Smith (1955). No remainder term was given. We use a different approach to obtain such a term here. The rate of convergence is the same as that in the central limit theorems for sequences of independent random variables.

Given two independent Poisson point processes Φ(1), Φ(2) in , the AB Poisson Boolean model is the graph with the points of Φ(1) as vertices and with edges between any pair of points for which the intersection of balls of radius 2r centered at these points contains at least one point of Φ(2). This is a generalization of the AB percolation model on discrete lattices. We show the existence of percolation for all d ≥ 2 and derive bounds for a critical intensity. We also provide a characterization for this critical intensity when d = 2. To study the connectivity problem, we consider independent Poisson point processes of intensities n and τn in the unit cube. The AB random geometric graph is defined as above but with balls of radius r. We derive a weak law result for the largest nearest-neighbor distance and almost-sure asymptotic bounds for the connectivity threshold.

In this note we give a counterexample which shows that the µc-rule is not optimal in the second node of the tandem queue. This counterexample contradicts the interchange argument in Nain [1] and Nain et al. [2].

Queueing networks have been rather restricted in order to have product form distributions for network states. Recently, several new models have appeared and enlarged this class of product form networks. In this paper, we consider another new type of queueing network with concurrent batch movements in terms of such product form results. A joint distribution of the requested batch sizes for departures and the batch sizes of the corresponding arrivals may be arbitrary. Under a certain modification of the network and mild regularity conditions, we give necessary and sufficient conditions for the network state to have the product form distribution, which is shown to provide an upper bound for the one in the original network. It is shown that two special settings satisfy these conditions. Algorithms to calculate their stationary distributions are considered, with numerical examples.

The set covariance of a dead leaves model, constructed from hard spheres of constant diameter, is calculated analytically. The calculation is based on the covariance of a single sphere and on the pair correlation function of the centres of the spheres. There exist applications in the field of random sequential adsorption and in the interpretation of small-angle scattering experiments.

Coalescents with multiple collisions (also called Λ-coalescents or simple exchangeable coalescents) are used as models of genealogies. We study a new class of Markovian coalescent processes connected to a population model with immigration. Consider an infinite population with immigration labelled at each generation by N := {1, 2, …}. Some ancestral lineages cannot be followed backwards after some time because their ancestor is outside the population. The individuals with an immigrant ancestor constitute a distinguished family and we define exchangeable distinguished coalescent processes as a model for genealogy with immigration, focusing on simple distinguished coalescents, i.e. such that when a coagulation occurs all the blocks involved merge as a single block. These processes are characterized by two finite measures on [0, 1] denoted by M = (Λ0, Λ1). We call them M-coalescents. We show by martingale arguments that the condition of coming down from infinity for the M-coalescent coincides with that obtained by Schweinsberg for the Λ-coalescent. In the same vein as Bertoin and Le Gall, M-coalescents are associated with some stochastic flows. The superprocess embedded can be viewed as a generalized Fleming-Viot process with immigration. The measures Λ0 and Λ1 respectively specify the reproduction and the immigration. The coming down from infinity of the M-coalescent will be interpreted as the initial types extinction: after a certain time all individuals are immigrant children.

In this paper, asymptotic estimates for the blocking probability of a call pertaining to a given route in a large multi-rate circuit-switched network are given. Concentrating on low load and critical load conditions, these estimates are essentially derived by using probability change techniques applied to the distribution of the number of occupied links. Such estimates for blocking probabilities are also given a uniform expression applicable to both load regimes. This uniform expression is numerically validated via simple examples.

The distribution of a fixed sum of independent and identically distributed random variables with the modified geometric distribution is the same as the distribution obtained by the compounding by a binomial distribution of either a negative binomial distribution or a Pascal distribution. This result can be used to obtain three summations for the game score probabilities of a two-person game, and leads to the consideration of various ways of dividing up the trials of the game. The game score probabilities are then used to consider the ‘fairness’ of four games and to analyse various methods of ‘setting’ (or ‘tie-breaking’) the games.

Let X be a Markov process on the line. Under certain conditions it is possible to find a diffusion process which is an approximation to X in the following sense:

(1) X can be embedded in that is there are stopping times (Tt) such that {Xt, t ≥ 0} and have the same distribution;

(2) for each t, E{Tt} = t.

We call the well-timed diffusion approximation to X, and suggest that it is useful for approximating quantities like the first-exit probabilities and expected first-exit times of X.

We determine the well-timed approximation in several special cases and give an asymptotic approximation for use in cases in which cannot be exactly determined.

We present two variations of a two-person, noncooperative stochastic game, inspired by the famous red-and-black gambling problem presented by Dubins and Savage. Two players each hold an integer amount of money and they each aim to win the other player's fortune. At every stage of the game they simultaneously bid an integer portion of their current fortune, and their probabilities of winning depend on these bids. We describe two different laws of motion specifying this dependency. In one version of the game, the players' probabilities of winning are proportional to their bets. In the other version, the probabilities of winning depend on the size of their bets and a weight parameter w. For each version we give a Nash equilibrium, in which the player for which the game is subfair (w ≤ ½) plays boldly and the player for which the game is superfair (w ≥ ½) plays timidly.

Consider a continuum percolation model in which, at each point of a d-dimensional Poisson process of rate λ, a ball of radius 1 is centred. We show that, for any d ≧ 3, there exists a phase where both the regions, occupied and vacant, contain unbounded components. The proof uses the concept of enhancement for the Boolean model, and along the way we prove that the critical intensity of a Boolean model defined on a slab is strictly larger than the critical intensity of a Boolean model defined on the whole space.

We consider the Voronoi tessellation of Euclidean space that is generated by an inhomogeneous Poisson point process whose intensity takes different constant values on sets of some finite partition of the space. Considering the Voronoi cells as marks associated with points of the point process, we prove that the intensity measure (mean measure) of the marked Poisson point process admits an approximate decomposition formula. The true value is approximated by a mixture of respective intensity measures for homogeneous models, while the explicit upper bound for the remainder term can be computed numerically for a large class of practical examples. By the Campbell formula, analogous approximate decompositions are deduced for the Palm distributions of individual cells. This approach makes possible the analysis of a wide class of inhomogeneous-Poisson Voronoi tessellations, by means of formulae and estimates already established for homogeneous cases. Our analysis applies also to the Poisson process modulated by an independent stationary random partition, in which case the error of the approximation of the double-stochastic-Poisson Voronoi tessellation depends on some integrated linear contact distribution functions of the boundaries of the partition elements.

For k-armed Bernoulli bandits with discounting, sharp comparisons are given between average optimal rewards for a gambler and for a ‘perfectly informed' gambler, over natural collections of prior distributions. Some of these comparisons are proved under general discounting, and others under non-increasing discount sequences. Connections are made between these comparisons and the concept of ‘regret' in the minimax approach to bandit processes. Identification of extremal cases in the sharp comparisons is emphasized.

Organisms adapt to fluctuating environments by regulating their dynamics, and by adjusting their phenotypes to environmental changes. We model population growth using multitype branching processes in random environments, where the offspring distribution of some organism having trait t ∈ in environment e ∈ ε is given by some (fixed) distribution ϒt,e on ℕ. Then, the phenotypes are attributed using a distribution (strategy) πt,e on the trait space . We look for the optimal strategy πt,e, t ∈ , e ∈ ε, maximizing the net growth rate or Lyapounov exponent, and characterize the set of optimal strategies. This is considered for various models of interest in biology: hereditary versus nonhereditary strategies and strategies involving or not involving a sensing mechanism. Our main results are obtained in the setting of nonhereditary strategies: thanks to a reduction to simple branching processes in a random environment, we derive an exact expression for the net growth rate and a characterization of optimal strategies. We also focus on typical genealogies, that is, we consider the problem of finding the typical lineage of a randomly chosen organism.

Let (X0, Y0), (X1, Y1), · ·· be a sequence of independent two-dimensional random vectors such that (X1, Y1), (X2, Y2), · ·· are i.i.d. Let {(Sn, Un)}n≧0 be the associated sum process, and define for t ≧ 0 Under suitable conditions on (X0, Y0) and (X1, Y1) we derive expansions up to vanishing terms, as t→∞, for EUT(t), Var UT(t) and Cov (UT(t), T(t)). Corresponding results will be obtained for EUN(t), Var UN(t) and Cov (UN(t), N(t)) when X0, Χ1 are both almost surely non-negative and

A three-parameter model of a random directed graph (digraph) is specified by the probability of ‘up arrows' from vertex i to vertex j where i < j, the probability of ‘down arrows' from i to j where i ≥ j, and the probability of bidirectional arrows between i and j. In this model, a phase transition—the abrupt appearance of a giant strongly connected component—takes place as the parameters cross a critical surface. The critical surface is determined explicitly. Before the giant component appears, almost surely all non-trivial components are small cycles. The asymptotic probability that the digraph contains no cycles of length 3 or more is computed explicitly. This model and its analysis are motivated by the theory of food webs in ecology.

We discuss the joint temporal and contemporaneous aggregation of N independent copies of random-coefficient AR(1) processes driven by independent and identically distributed innovations in the domain of normal attraction of an $\alpha$ -stable distribution, $0< \alpha \le 2$ , as both N and the time scale n tend to infinity, possibly at different rates. Assuming that the tail distribution function of the random autoregressive coefficient regularly varies at the unit root with exponent $\beta > 0$ , we show that, for $\beta < \max (\alpha, 1)$ , the joint aggregate displays a variety of stable and non-stable limit behaviors with stability index depending on $\alpha$ , $\beta$ and the mutual increase rate of N and n. The paper extends the results of Pilipauskaitė and Surgailis (2014) from $\alpha =2$ to $0 < \alpha < 2$ .