Motivated by recent studies of big samples, this work aims to construct a parametric model which is characterized by the following features: (i) a ‘local’ reinforcement, i.e. a reinforcement mechanism mainly based on the last observations, (ii) a random persistent fluctuation of the predictive mean, and (iii) a long-term almost sure convergence of the empirical mean to a deterministic limit, together with a chi-squared goodness-of-fit result for the limit probabilities. This triple purpose is achieved by the introduction of a new variant of the Eggenberger–Pólya urn, which we call the rescaled Pólya urn. We provide a complete asymptotic characterization of this model, pointing out that, for a certain choice of the parameters, it has properties different from the ones typically exhibited by the other urn models in the literature. Therefore, beyond the possible statistical application, this work could be interesting for those who are concerned with stochastic processes with reinforcement.

We study a stochastic compartmental susceptible–infected (SI) epidemic process on a configuration model random graph with a given degree distribution over a finite time interval. We split the population of graph vertices into two compartments, namely, S and I, denoting susceptible and infected vertices, respectively. In addition to the sizes of these two compartments, we keep track of the counts of SI-edges (those connecting a susceptible and an infected vertex) and SS-edges (those connecting two susceptible vertices). We describe the dynamical process in terms of these counts and present a functional central limit theorem (FCLT) for them as the number of vertices in the random graph grows to infinity. The FCLT asserts that the counts, when appropriately scaled, converge weakly to a continuous Gaussian vector semimartingale process in the space of vector-valued càdlàg functions endowed with the Skorokhod topology. We discuss applications of the FCLT in percolation theory and in modelling the spread of computer viruses. We also provide simulation results illustrating the FCLT for some common degree distributions.

We analyse an additive-increase and multiplicative-decrease (also known as growth–collapse) process that grows linearly in time and that, at Poisson epochs, experiences downward jumps that are (deterministically) proportional to its present position. For this process, and also for its reflected versions, we consider one- and two-sided exit problems that concern the identification of the laws of exit times from fixed intervals and half-lines. All proofs are based on a unified first-step analysis approach at the first jump epoch, which allows us to give explicit, yet involved, formulas for their Laplace transforms. All eight Laplace transforms can be described in terms of two so-called scale functions associated with the upward one-sided exit time and with the upward two-sided exit time. All other Laplace transforms can be obtained from the above scale functions by taking limits, derivatives, integrals, and combinations of these.

We consider the simultaneous propagation of two contagions over a social network. We assume a threshold model for the propagation of the two contagions and use the formal framework of discrete dynamical systems. In particular, we study an optimization problem where the goal is to minimize the total number of new infections subject to a budget constraint on the total number of available vaccinations for the contagions. While this problem has been considered in the literature for a single contagion, our work considers the simultaneous propagation of two contagions. This optimization problem is NP-hard. We present two main solution approaches for the problem, namely an integer linear programming (ILP) formulation to obtain optimal solutions and a heuristic based on a generalization of the set cover problem. We carry out a comprehensive experimental evaluation of our solution approaches using many real-world networks. The experimental results show that our heuristic algorithm produces solutions that are close to the optimal solution and runs several orders of magnitude faster than the ILP-based approach for obtaining optimal solutions. We also carry out sensitivity studies of our heuristic algorithm.

We derive formulae for the moments of the time of ruin in both ordinary and modified Sparre Andersen risk models without specifying either the inter-claim time distribution or the individual claim amount distribution. We illustrate the application of our results in the special case of exponentially distributed claims, as well as for the following ordinary models: the classical risk model, phase-type(2) risk models, and the Erlang($\mathscr{n}$) risk model. We also show how the key quantities for modified models can be found.