We investigate stochastic comparisons of parallel systems (corresponding to the largest-order statistics) with respect to the reversed hazard rate and likelihood ratio orders for the proportional reversed hazard rate (PRHR) model. As applications of the main results, we obtain the equivalent characterizations of stochastic comparisons with respect to the reversed hazard rate and likelihood rate orders for the exponentiated generalized gamma and exponentiated Pareto distributions. Our results recover and strengthen some recent results in the literature.

Given a sequence (Mk, Qk)k ≥ 1 of independent and identically distributed random vectors with nonnegative components, we consider the recursive Markov chain (Xn)n ≥ 0, defined by the random difference equation Xn = MnXn - 1 + Qn for n ≥ 1, where X0 is independent of (Mk, Qk)k ≥ 1. Criteria for the null recurrence/transience are provided in the situation where (Xn)n ≥ 0 is contractive in the sense that M1 ⋯ Mn → 0 almost surely, yet occasional large values of the Qn overcompensate the contractive behavior so that positive recurrence fails to hold. We also investigate the attractor set of (Xn)n ≥ 0 under the sole assumption that this chain is locally contractive and recurrent.

A continuous-state population-size-dependent branching process {Xt} is a modification of the Jiřina process. We prove that such a process arises as the limit of a sequence of suitably scaled population-size-dependent branching processes with discrete states. The extinction problem for the population Xt is discussed, and the limit distribution of Xt / t obtained when Xt tends to infinity.

We consider a continuous, infinitely divisible random field in Rd given as an integral of a kernel function with respect to a Lévy basis with convolution equivalent Lévy measure. For a large class of such random fields we compute the asymptotic probability that the supremum of the field exceeds the level x as x → ∞. Our main result is that the asymptotic probability is equivalent to the right tail of the underlying Lévy measure.

A general age-dependent branching process is studied in which the population is distributed in N colonies. Migration from one colony to another is possible and no particular geometrical configuration is assumed for the colonies. Asymptotic expressions for the first two moments are given in several different cases.

Gerber and Shiu (1997) have studied the joint density of the time of ruin, the surplus immediately before ruin, and the deficit at ruin in the classical model of collective risk theory. More recently, their results have been generalised for risk models where the interarrival density for claims is nonexponential, but belongs to the Erlang family. Here we obtain generalisations of the Gerber-Shiu (1997) results that are valid in a general Sparre Andersen model, i.e. for any interclaim density. In particular, we obtain a generalisation of the key formula in that paper. Our results are made more concrete for the case where the distribution between claim arrivals is phase-type or the integrated tail distribution associated with the claim size distribution belongs to the class of subexponential distributions. Furthermore, we obtain conditions for finiteness of the joint moments of the surplus before ruin and the deficit at ruin in the Sparre Andersen model.

Autoregressive processes are intensively studied in statistics and other fields of applied stochastics. For many applications, the overshoot and the threshold time are of special interest. When the upward innovations are in the class of phase-type distributions, we determine the joint distribution of these two quantities and apply this result to problems of optimal stopping. Using a principle of continuous fit, this leads to explicit solutions.

The probability generating functional (p.g.fl.) of a non-homogeneous Poisson cluster process is characterized in Ammann and Thall (1977) via a decomposition of the KLM measure of the process. This p.g.fl. representation is utilized in the present article to show that the family 𝒟 of Poisson cluster processes with a.s. finite clusters is invariant under a class of cluster transformations. Explicit expressions for the finite-dimensional count distributions, product moment measures, and the distribution of clusters are derived in terms of the KLM measure. It is also shown that an element of 𝒟 has no multiple events iff the points of each cluster are a.s. distinct.

Consider a slotted communication channel which carries packetized voice and which can transmit exactly one packet every timeslot. Assume that every conversation routed over this channel generates exactly one packet every N timeslots. We study, for the case of an infinite buffer and an N-periodic strongly interchangeable input process, buffer behavior and packet delays as functions of the number of calls routed over the channel, as long as that number is less than or equal to N. Among our results are a simple algorithm which computes the marginal distribution of the buffer content and an algorithm with complexity of order N4 which computes the distribution of the maximal buffer content (over a period of N timeslots).

We consider risk processes with reinsurance. A general family of reinsurance contracts is allowed, including proportional and excess-of-loss policies. Claim occurrence is regulated by a classical compound Poisson process or by a Markov-modulated compound Poisson process. We provide some large deviation results concerning these two risk processes in the small-claim case. Finally, we derive the so-called Lundberg estimate for the ruin probabilities and present a numerical example.

Wold's point process is briefly introduced and its forward equation is derived in terms of an integro-differential equation which is used to obtain the ‘renewal' function. An example of this family of processes is given in which each interval is exponentially distributed. The theory of diagonal expansion for a bivariate distribution is used to obtain the numerical estimate of the spectrum of counts.

We consider a real-valued random walk which drifts to -∞ and is such that the step distribution is heavy tailed, say, subexponential. We investigate the asymptotic tail behaviour of the distribution of the upwards first passage times. As an application, we obtain the exact rate of convergence for the ruin probability in finite time. Our result supplements similar theorems in risk theory.

We prove a central limit theorem for conditionally centred random fields, under a moment condition and strict positivity of the empirical variance per observation. We use a random normalization, which fits non-stationary situations. The theorem applies directly to Markov random fields, including the cases of phase transition and lack of stationarity. One consequence is the asymptotic normality of the maximum pseudo-likelihood estimator for Markov fields in complete generality.

We consider a fluid queue controlled by a semi-Markov process and we apply the Markov-renewal approach developed earlier in the context of quasi-birth-and-death processes and of Markovian fluid queues. We analyze two subfamilies of semi-Markov processes. In the first family, we assume that the intervals during which the input rate is negative have an exponential distribution. In the second family, we take the complementary case and assume that the intervals during which the input rate is positive have an exponential distribution. We thoroughly characterize the structure of the stationary distribution in both cases.

The properties of a measure of randomness on the class of discrete probability distributions are discussed. A result concerning finiteness is obtained for a family of measures which includes the entropy function.

We investigate the stationarity of minification processes when the marginal is a discrete distribution. There is a close relationship between the problem considered by Arnold and Isaacson (1976) and the stationarity in minification processes. We give a necessary and sufficient condition for a discrete distribution to be the marginal of a stationary minification process. Members of the Poisson and negative binomial families can be the marginals of stationary minification processes. The geometric minification process is studied in detail, and two characterizations of it based on the structure of the innovation process are given.

In this note we adopt the notation and terminology of Kingman (1966) without further comment. For the general single server queue one has For the queueing system GI/Ek/1 it is possible to make use of the particular nature of the service time distribution to evaluate the right-hand side of Equation (1) in terms of the k roots of a certain equation. This evaluation is carried out in detail in Prabhu (1965) to which reference should be made for the technicalities involved. A similar evaluation applies to the limiting distribution when it exists. However, the resulting expression again involves the k roots of a certain equation. In this note we draw attention to an alternative procedure which does not involve the calculation of roots. We remark that a similar, but slightly different, procedure can be used in the study of the queueing system Ek/GI/1. Details of this will be presented in a separate note.

In the present paper the solutions of two integral equations are derived. One of the integral equations dominates the mathematical description of the stochastic process {vn, n = 1,2, …}, recursively defined by K is a positive constant, τ1, τ2, …; Σ1, Σ2, …; are independent, non-negative variables, with τ1, τ2,…, identically distributed, similarly, the variables Σ1, Σ2, …, are identically distributed.

An M/M/s queueing system with a simple cost structure is considered, under the assumption that the system will close in a finite time after which any remaining customers will require extra overtime service costs. We determine the optimal policy for admitting customers to the queue, as a function of the time, t, to closing and the current queue length, i. It is shown to have the form: admit if and only if f1(t) ≦ i ≦ f2(t). The bounds f1(t) and f2(t) are specified, and it is shown under what conditions f1(t) = 0 (a control limit rule) or f2(t) = ∞ (an inverse control limit rule).

The present paper considers methods for computing moments connected with subcritical multitype branching processes. By stressing the interplay between moment and cumulant structure, it is possible to prove simultaneously the existence of the moments. The kinds of moments considered are those arising in (1) times to extinction, (2) ultimate numbers of particles starting from a single particle, and (3) equilibrium numbers of particles for subcritical processes experiencing immigration.