We show in this paper that if a stationary traffic source is regulated by a leaky bucket with leak rate ρ and bucket size σ, then the amount of information generated in successive time intervals is dominated, in the increasing convex ordering sense, by that of a Poisson arrival process with rate ρ/σ, with each arrival bringing an amount of information equal to σ. By exploiting this property, we then show that the mean value in the stationary regime of the content of a buffer drained at constant rate and fed with the superposition of regulated flows is less than the mean value of the same buffer fed with an adequate Poisson process, whose characteristics depend upon the regulated input flows.

Consider random vectors formed by a finite number of independent groups of independent and identically distributed random variables, where those of the last group are stochastically smaller than those of the other groups. Conditions are given such that certain functions, defined as suitable means of supermodular functions of the random variables of the vectors, are supermodular or increasing directionally convex. Comparisons based on the increasing convex order of supermodular functions of such random vectors are also investigated. Applications of the above results are then provided in risk theory, queueing theory, and reliability theory, with reference to (i) net stop-loss reinsurance premiums of portfolios from different groups of insureds, (ii) closed cyclic multiclass Gordon-Newell queueing networks, and (iii) reliability of series systems formed by units selected from different batches.

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.

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

It is shown that the sample paths of Poisson shot noise with heavy-tailed semiexponential distributions satisfy a large deviation principle with a rate function that is insensitive to the shot shape. This demonstrates that, on the scale of large deviations, paths to rare events do not depend on the shot shape.

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 a model in which we have k items to be sold. Potential buyers make offers in a sequential fashion. Once made, the offer is either rejected or marked for acceptance. Once k items have been marked, the items are then sold to the buyers whose offers were marked, but at a price equal to the minimum of the k marked offers. Assuming that the successive offers are independent and identically distributed according to a specified distribution and that there is a fixed cost incurred whenever an offer is rejected, we determine structural results about the optimal policy, present computational approaches for finding the optimal policy, and give some heuristic policies.

This paper investigates the probabilistic behaviour of the eigenvalue of the empirical transition matrix of a Markov chain which is of largest modulus other than 1, loosely called the second-largest eigenvalue. A central limit theorem is obtained for nonmultiple eigenvalues of the empirical transition matrix. When the Markov chain is actually a sequence of independent observations the distribution of the second-largest eigenvalue is determined and a test for independence is developed. The independence case is considered in more detail when the Markov chain has only two states, and some applications are given.

It is shown how to estimate centroids and other measures of location of an unknown embedded subset of a bounded specimen by random sectioning.

Consider an autoregressive-moving-average process of given order where it is known that a number of moving-average roots are of unit modulus. Such a situation might arise, for example, when a time series has been differenced to induce stationarity by removing a non-stationary polynomial or seasonal trend. A band-limited spectral estimation procedure is proposed for estimating the coefficients of such a process and the asymptotic properties of the estimators investigated. The asymptotic theory is illustrated with reference to simulated and real data. A preliminary investigation of the use of Akaike's AIC criterion and this procedure to determine the number of roots of unit modulus (in the case where this is unknown) is also carried out by means of simulation.

The proposed band-limited spectral estimation procedure can also be used to take account of other possible effects met in practice. These include, for example, the band-limited response of a recording device or trend-contaminated low-frequency components.

Consider a number of events in a probability space. Let X be a random variable that is the number of events that occur. Given some of the moments of the distribution of X, it is possible to obtain bounds on the probability that at least one event occurs. The best possible bounds are given here. If there are many equiprobable events that are d- wise independent, and d is even, then the probability that at least one event happens is at least 1 — O(µ–d/2), where μ = E(X).

The application of the generalised ballot theorem to queueing theory leads to elegant results for the simple M/G/1 queue. It is thought that such results are not possible for more general M/G/1-type queues. We, however, derive a batch ballot theorem which can be applied to derive the first passage distribution matrix, G, for the general M/G/1-type queue.

This paper is concerned with the multidimensional Markov chain {X(n)} = {X(x, n)}, considered by Borovkov [1], [2], [3], with the form X(n + 1) = ?(n) + ?(?(n), n), where the distribution of depends only on x. Sufficient conditions on moments of |?(x)| are established for the Markov chain {X(n)} to have rate of convergence results (i.e. geometric ergodicity or sub-geometric rates for a variety of rate functions).

The points of a non-stationary Poisson process with periodic intensity are independently shifted forward in time in such a way that the transformed process is stationary Poisson. The mean shift is shown to be minimal. The approach used is to consider an Mt/Gt/∞ queueing system where the arrival process is a non-stationary Poisson with periodic intensity function. A minimal service time distribution is constructed that yields a stationary Poisson departure process.

The paper studies a single-server two-queue priority system with changeover times and switching threshold. The server serves queue 1 exhaustively and does not remain at an empty queue if the other one is non-empty. It immediately switches from queue 2 to queue 1 when the length of the latter reaches some level M. Whenever service is changed from one queue to the other a changeover time is required. Arrivals are Poisson, service times and changeover times are independent and exponentially distributed. Using an analytic method we obtain the steady-state joint probability generating function of the lengths of the two queues. By means of this probability generating function some performance measures of the system such as mean length of queue and mean delay can be calculated.

Let X be a continuous-time strongly mixing or weakly dependent process and let T be a renewal process independent of X. We show general conditions under which the sampled process $(X_{T_i},T_i-T_{i-1})^{\top}$ is strongly mixing or weakly dependent. Moreover, we explicitly compute the strong mixing or weak dependence coefficients of the renewal sampled process and show that exponential or power decay of the coefficients of X is preserved (at least asymptotically). Our results imply that essentially all central limit theorems available in the literature for strongly mixing or weakly dependent processes can be applied when renewal sampled observations of the process X are at our disposal.

We present, discuss and prove an apparently unfamiliar result in the convergence of probability measures.

The information flow in discrete Markov systems provides a method for determining that such a system is ergodic. Estimates are obtained for the information flow in some classes of Markov systems and using these estimates criteria for the ergodicity of the systems are established.

In many endemic areas the level of infection of malaria exhibits a regular cycle. A stochastic model is formulated to describe endemic malaria and is used to justify a set of deterministic equations. The model incorporates the properties peculiar to malaria, i.e., in endemic areas malaria is not usually fatal and an individual can be and usually is reinfected. The theory of small oscillations is applied to the deterministic equations, and under the assumption of stable populations of hosts and vectors an expression is obtained for the period of oscillation of the infection.

A summation formula is given for computing the spectrum of intervals of a geometric branching Poisson process from the cumulant generating function of the associated counting process.