We extend the work of Antunović et al. (2016) on competing types in preferential attachment models to include cases where the types have different fitnesses, which may be either multiplicative or additive. We show that, depending on the values of the parameters of the models, there are different possible limiting behaviours depending on the zeros of a certain function. In particular, we show the existence of choices of the parameters where one type is favoured both by having higher fitness and by the type of attachment mechanism, but the other type has a positive probability of dominating the network in the limit.

This paper presents a degree-two probability lower bound for the union of an arbitrary set of events in an arbitrary probability space. The bound is designed in terms of the first-degree Bonferroni summation and pairwise joint probabilities of events, which are represented as weights of edges in a Hamilton-type circuit in a connected graph. The proposed lower bound strengthens the Dawson–Sankoff lower bound in the same way that Hunter and Worsley's degree-two upper bound improves the degree-two Bonferroni-type optimal upper bound. It can be applied to statistical inference in time series and outlier diagnoses as well as the study of dose response curves.

A discrete-time SIS model is presented that allows individuals in the population to vary in terms of their susceptibility to infection and their rate of recovery. This model is a generalisation of the metapopulation model presented in McVinish and Pollett (2010). The main result of the paper is a central limit theorem showing that fluctuations in the proportion of infected individuals around the limiting proportion converges to a Gaussian random variable when appropriately rescaled. In contrast to the case where there is no variation amongst individuals, the limiting Gaussian distribution has a nonzero mean.

Birth–death processes form a natural class where ideas and results on large deviations can be tested. We derive a large-deviation principle under an assumption that the rate of jump down (death) grows asymptotically linearly with the population size, while the rate of jump up (birth) grows sublinearly. We establish a large-deviation principle under various forms of scaling of the underlying process and the corresponding normalization of the logarithm of the large-deviation probabilities. The results show interesting features of dependence of the rate functional upon the parameters of the process and the forms of scaling and normalization.

We present a minimum counterexample to the conjecture that a negatively associated random variable has an ultra-log-concave rank sequence. The rank sequence does not in fact even need to be unimodal.

We consider the problem of stochastic comparison of general GARCH-like processes for different parameters and different distributions of the innovations. We identify several stochastic orders that are propagated from the innovations to the GARCH process itself, and we discuss their interpretations. We focus on the convex order and show that in the case of symmetric innovations it is also propagated to the cumulated sums of the GARCH process. More generally, we discuss multivariate comparison results related to the multivariate convex and supermodular orders. Finally, we discuss ordering with respect to the parameters in the GARCH(1, 1) case.

In this paper we address the question of finding the point which maximizes the pth moment of the exit time of planar Brownian motion from a given domain. We present a geometrical method for excluding parts of the domain from consideration which makes use of a coupling argument and the conformal invariance of Brownian motion. In many cases the maximizing point can be localized to a relatively small region. Several illustrative examples are presented.

The asymptotic behaviour of many locally branching epidemic models can, at least to first order, be deduced from the limit theory of two branching processes. The first is Whittle's (1955) branching approximation to the early stages of the epidemic, the phase in which approximately exponential growth takes place. The second is the susceptibility approximation; the backward branching process that approximates the history of the contacts that would lead to an individual becoming infected. The simplest coupling arguments for demonstrating the closeness of these branching process approximations do not keep the processes identical for quite long enough. Thus, arguments showing that the differences are unimportant are also needed. In this paper we show that, for some models, couplings can be constructed that are sufficiently accurate for this extra step to be dispensed with.

This note corrects an error in the definition of the rate function in Jacquier, Pakkanen, and Stone (2018) and slightly simplifies some proofs.

We investigate the probability that an insurance portfolio gets ruined within a finite time period under the assumption that the r largest claims are (partly) reinsured. We show that for regularly varying claim sizes the probability of ruin after reinsurance is also regularly varying in terms of the initial capital, and derive an explicit asymptotic expression for the latter. We establish this result by leveraging recent developments on sample-path large deviations for heavy tails. Our results allow, on the asymptotic level, for an explicit comparison between two well-known large-claim reinsurance contracts, namely LCR and ECOMOR. Finally, we assess the accuracy of the resulting approximations using state-of-the-art rare event simulation techniques.

We study an optimal reinsurance problem for a diffusion model, in which the drift of the claim follows an Ornstein–Uhlenbeck process. The aim of the insurer is to maximize the expected exponential utility of its terminal wealth. We consider two cases: full information and partial information. Full information occurs when the insurer directly observes the drift; partial information occurs when the insurer observes only its claims. By applying stochastic control and by solving the corresponding Hamilton–Jacobi–Bellman equations, we find the value function and the optimal reinsurance strategy under both full and partial information. We determine a relationship between the value function and reinsurance strategy under full information with the value function and reinsurance strategy under partial information.

We consider the problem of finding an optimal betting strategy for a house-banked casino card game that is played for several coups before reshuffling. The sampling without replacement makes it possible to take advantage of the changes in the expected value as the deck is depleted, making large bets when the game is advantageous. Using such a strategy, which is easy to implement, is known as card counting. We consider the case of a large number of decks, making an approximation to continuous time possible. A limit law of the return process is found and the optimal card counting strategy is derived. This continuous-time strategy is shown to be a natural analog of the discrete-time strategy where the so-called effects of removal are replaced by the infinitesimal generator of the card process.

Recently, the relevation transformation has received further attention from researchers, and some interesting results have been developed. It is well known that the active redundancy at component level results in a more reliable coherent system than that at system level. However, the lack of study of this problem with relevation redundancy prevents us from fully understanding such a generalization of the active redundancy. In this note we deal with relevation redundancy to coherent systems of homogeneous components. Typically, for a series system of independent components, we have proved that the lifetime of a system with relevation redundancy at component level is larger than that with relevation redundancy at system level in the sense of the usual stochastic order and the likelihood ratio order, respectively. For a coherent system with dependent components, we have developed a sufficient condition in terms of the domination function to the usual stochastic order between the system lifetime with redundancy at component level and that at system level.

We study the bijection between binary Galton-Watson trees in continuous time and their exploration process, both in the subcritical and in the supercritical cases. We then take the limit over renormalized quantities, as the size of the population tends to ∞. We thus deduce Delmas' generalization of the second Ray-Knight theorem.

Examining possibilities for the coexistence of two competing populations is a classic problem which dates back to the earliest ‘predator-prey’ models. In this paper we study this problem in the context of a model introduced in Björnberg et al. (2012) for the spread of a virus infection in a population of healthy cells. The infected cells may be seen as a population of ‘predators’ and the healthy cells as a population of ‘prey’. We show that, depending on the parameters defining the model, there may or may not be coexistence of the two populations, and we give precise criteria for this.

In this paper we analyze players' long-run behavior in evolutionary coordination games with imperfect monitoring in a large population. Players can observe signals corresponding to other players' unseen actions and use the proposed simple or maximum likelihood estimation algorithm to extract information from the signals. In the simple learning process we find conditions for the risk-dominant and the non-risk-dominant equilibria to emerge alone in the long run. Furthermore, we find that the two equilibria can coexist in the long run. In contrast, the coexistence of the two equilibria is the only limit distribution under the maximum likelihood estimation learning algorithm. We also analyze the long-run equilibria of other 2x2 symmetric games under imperfect monitoring.

We consider an r component system having an arbitrary binary monotone structure function. We suppose that shocks occur according to a point process and that, independent of what has already occurred, each new shock is one of r different types, with respective probabilities p1, …, pr. We further suppose that there are given integers n1, …, nr such that component i fails (and remains failed) when there have been a total of ni type-i shocks. Letting L be the time at which the system fails, we are interested in using simulation to estimate E[L], E[L2], and P(L > t). We show how to efficiently accomplish this when the point process is (i) a Poisson, (ii) a renewal, and (iii) a Hawkes process.

With the rapid adoption of transgenic crops, gene flow from transgenic crops to wild relatives through pollen dispersal is of significant concern and warrants both empirical and theoretical studies to assess the risk of introduction of transgenes into wild populations. We propose to use the (biased) voter model in a heterogeneous environment to investigate the effects of recurrent gene flow from transgenic crop to wild relatives. The model is defined on the d-dimensional integer lattice that is divided into two parts, Δ and Zd \ Δ. Individuals carrying the transgene and individuals carrying the wild type gene compete according to the evolution rules of a (biased) voter model on Zd \ Δ, while the process is conditioned to have only individuals carrying the transgene on Δ. Our main findings suggest that unless transgenes confer increased fitness in wild relatives, introgression of transgenes into populations of wild plants is slow and may even be reversible without intervention. Our study also addresses the effects of different spatial planting patterns of transgenic crops on the rate of introgression.

Beurling's inequality is proved to have an interpretation in terms of ingredients that do not depend on the effective numerical contribution of transition probabilities of circuit processes (those processes whose transition law is expressed by a denumerable class of directed weighted circuits). Connections with reversibility for Sk-state, k≧2, circuit processes, where S is a denumerable set, are revealed.