We consider the Λ-coalescent processes with a positive frequency of singleton clusters. The class in focus covers, for instance, the beta(a, b)-coalescents with a > 1. We show that some large-sample properties of these processes can be derived by coupling the coalescent with an increasing Lévy process (subordinator), and by exploiting parallels with the theory of regenerative composition structures. In particular, we discuss the limit distributions of the absorption time and the number of collisions.

Consider an independent site percolation model on Z d , with parameter p ∈ (0, 1), where all long-range connections in the axis directions are allowed. In this work we show that, given any parameter p, there exists an integer K(p) such that all binary sequences (words) ξ ∈ {0, 1} N can be seen simultaneously, almost surely, even if all connections with length larger than K(p) are suppressed. We also show some results concerning how K(p) should scale with p as p goes to 0. Related results are also obtained for the question of whether or not almost all words are seen.

The signature is an important structural characteristic of a coherent system. Its computation, however, is often rather involved and complex. We analyze several cases where this complexity can be considerably reduced. These are the cases when a ‘large’ coherent system is obtained as a series, parallel, or recurrent structure built from ‘small’ modules with known signature. Corresponding formulae can be obtained in terms of cumulative notions of signatures. An algebraic closure property of families of homogeneous polynomials plays a substantial role in our derivations.

We establish heavy-traffic limits for nearly deterministic queues, such as the G/D/n many-server queue. Since waiting times before starting service in the G/D/n queue are equivalent to waiting times in an associated Gn /D/1 model, where the Gn interarrival times are the sum of n consecutive interarrival times in the original model, we focus on the Gn /D/1 model and the generalization to Gn /Gn /1, where ‘cyclic thinning’ is applied to both the arrival and service processes. We establish different limits in two cases: (i) when (1 − ρn )√n → β as n → ∞ and (ii) when (1 − ρn )n → β as n → ∞, where ρn is the traffic intensity in model n. The nearly deterministic feature leads to interesting nonstandard scaling.

In this paper we deal with an M/G/1 vacation system with the sojourn time (wait plus service) limit and two typical vacation rules, i.e. multiple and single vacation rules. Using the level crossing approach, explicit expressions for the steady-state distributions of the virtual waiting times are obtained in vacation systems with exponential and constant service times, a general vacation time, and two vacation rules.

Signature-based representations of the reliability functions of coherent systems with independent and identically distributed component lifetimes have proven very useful in studying the ageing characteristics of such systems and in comparing the performance of different systems under varied criteria. In this paper we consider extensions of these results to systems with heterogeneous components. New representation theorems are established for both the case of components with independent lifetimes and the case of component lifetimes under specific forms of dependence. These representations may be used to compare the performance of systems with homogeneous and heterogeneous components.

We consider statistical inference for a parametric cooperative sequential adsorption model for spatial time series data, based on maximum likelihood. We establish asymptotic normality of the maximum likelihood estimator in the thermodynamic limit. We also perform and discuss some numerical simulations of the model, which illustrate the procedure for creating confidence intervals for large samples.

Consider the region L = {(x, y): 0 ≤ y ≤ Clog(1 + x), x > 0} for a constant C > 0. We study the percolation and coverage properties of this region. For the coverage properties, we place a Poisson point process of intensity λ on the entire half space R+ x R and associated with each Poisson point we place a box of a random side length ρ. Depending on the tail behaviour of the random variable ρ we exhibit a phase transition in the intensity for the eventual coverage of the region L. For the percolation properties, we place a Poisson point process of intensity λ on the region R2. At each point of the process we centre a box of a random side length ρ. In the case ρ ≤ R for some fixed R > 0 we study the critical intensity λc of the percolation on L.

A new formula for continuum percolation on the Euclidean space R d (d ≥ 2), which is analogous to Russo's formula for bond or site percolation, is proved. Using this formula, we prove the equivalence between uniqueness of the infinite cluster and continuous differentiability of the mean number of clusters per Poisson point (or free energy). This yields a new proof for uniqueness of the infinite cluster since the continuous differentiability of free energy has been proved by Bezuidenhout, Grimmett and Löffler (1998); a consequence of this new proof gives the continuity of connectivity functions.

This paper is concerned with the behaviour of a Lévy process when it crosses over a positive level, u, starting from 0, both as u becomes large and as u becomes small. Our main focus is on the time, τu, it takes the process to transit above the level, and in particular, on the stability of this passage time; thus, essentially, whether or not τu behaves linearly as u ↓ 0 or u → ∞. We also consider the conditional stability of τu when the process drifts to -∞ almost surely. This provides information relevant to quantities associated with the ruin of an insurance risk process, which we analyse under a Cramér condition.

We consider a classical risk model and its diffusion approximation, where the individual claims are reinsured by a reinsurance treaty with deductible b ∈ [0, b̃]. Here b = b̃ means ‘no reinsurance’ and b= 0 means ‘full reinsurance’. In addition, the insurer is allowed to invest in a riskless asset with some constant interest rate m > 0. The cedent can choose an adapted reinsurance strategy {b t }t≥0, i.e. the parameter can be changed continuously. If the surplus process becomes negative, the cedent has to inject additional capital. Our aim is to minimise the expected discounted capital injections over all admissible reinsurance strategies. We find an explicit expression for the value function and the optimal strategy using the Hamilton-Jacobi-Bellman approach in the case of a diffusion approximation. In the case of the classical risk model, we show the existence of a ‘weak’ solution and calculate the value function numerically.

The almost-sure existence of a polymer probability in the infinite volume limit is readily obtained under general conditions of weak disorder from standard theory on multiplicative cascades or branching random walks. However, speculations in the case of strong disorder have been mixed. In this note existence of an infinite volume probability is established at critical strong disorder for which one has convergence in probability. Some calculations in support of a specific formula for the almost-sure asymptotic variance of the polymer path under strong disorder are also provided.

We study four discrete-time stochastic systems on N, modeling processes of rumor spreading. The involved individuals can either have an active or a passive role, speaking up or asking for the rumor. The appetite for spreading or hearing the rumor is represented by a set of random variables whose distributions may depend on the individuals. Our goal is to understand - based on the distribution of the random variables - whether the probability of having an infinite set of individuals knowing the rumor is positive or not.

In this paper we consider an extension of the two-dimensional risk model introduced in Avram, Palmowski and Pistorius (2008a). To this end, we assume that there are two insurers. The first insurer is subject to claims arising from two independent compound Poisson processes. The second insurer, which can be viewed as a different line of business of the same insurer or as a reinsurer, covers a proportion of the claims arising from one of these two compound Poisson processes. We derive the Laplace transform of the time until ruin of at least one insurer when the claim sizes follow a general distribution. The surplus level of the first insurer when the second insurer is ruined first is discussed at the end in connection with some open problems.

We consider a stochastic control model for a queueing system driven by a two-dimensional fractional Brownian motion with Hurst parameter 0 < H < 1. In particular, when H > ½, this model serves to approximate a controlled two-station tandem queueing model with heavy-tailed ON/OFF sources in heavy traffic. We establish the weak convergence results for the distribution of the state process and construct an explicit stationary state process associated with given controls. Based on suitable coupling arguments, we show that each state process couples with its stationary counterpart and we use it to represent the long-run average cost functional in terms of the stationary process. Finally, we establish the existence result of an optimal control, which turns out to be independent of the initial data.

We are concerned with the variation of the supercritical nearest-neighbours contact process such that first infection occurs at a lower rate; it is known that the process survives with positive probability. Regarding the rightmost infected of the process started from one site infected and conditioned to survive, we specify a sequence of space-time points at which its behaviour regenerates and, thus, obtain the corresponding strong law and central limit theorem. We also extend complete convergence in this case.

In this paper we study first passage times of (reflected) Ornstein-Uhlenbeck processes over compound Poisson-type boundaries. In fact, we extend the results of first rendezvous times of (reflected) Brownian motion and compound Poisson-type processes in Perry, Stadje and Zacks (2004) to the (reflected) Ornstein-Uhlenbeck case.

We consider the model of Deijfen, Häggström and Bagley (2004) for competing growth of two infection types in Rd, based on the Richardson model on Zd. Stochastic ball-shaped infection outbursts transmit the infection type of the center to all points of the ball that are not yet infected. Relevant parameters of the model are the initial infection configuration, the (type-dependent) growth rates, and the radius distribution of the infection outbursts. The main question is that of coexistence: Which values of the parameters allow the unbounded growth of both types with positive probability? Deijfen, Häggström and Bagley (2004) conjectured that the initial configuration is basically irrelevant for this question, and gave a proof for this under strong assumptions on the radius distribution, which, e.g. do not include the case of a deterministic radius. Here we give a proof that does not rely on these assumptions. One of the tools to be used is a slight generalization of the model with immune regions and delayed initial infection configurations.

In this note we revisit the discussion on minimal repair in heterogeneous populations in Finkelstein (2004). We consider the corresponding stochastic intensities (intensity processes) for items in heterogeneous populations given available information on their operational history, i.e. the failure (repair) times and the time since the last failure (repair). Based on the improved definitions, the setup of Finkelstein (2004) is modified and the main results are corrected in accordance with the updating procedure for the conditional frailty distribution.

In this paper we generalize existing results for the steady-state distribution of growth-collapse processes with independent exponential intercollapse times to the case where they have a general distribution on the positive real line having a finite mean. In order to compute the moments of the stationary distribution, no further assumptions are needed. However, in order to compute the stationary distribution, the price that we are required to pay is the restriction of the collapse ratio distribution from a general distribution concentrated on the unit interval to minus-log-phase-type distributions. A random variable has such a distribution if the negative of its natural logarithm has a phase-type distribution. Thus, this family of distributions is dense in the family of all distributions concentrated on the unit interval. The approach is to first study a certain Markov-modulated shot noise process from which the steady-state distribution for the related growth-collapse model can be inferred via level crossing arguments.