Brownian flow systems, i.e. multidimensional Brownian motion with regulating barriers, can model queueing and inventory systems in which the behavior of different queues is correlated because of shared input processes. The behavior of such systems is typically difficult to describe exactly. We show how Brownian models of such systems, conditioned on one queue length exceeding a large value, decompose asymptotically into smaller subsystems. This conditioning induces a change in drift of the system's net input process and its components. The results here are analogous to results for jump-Markov queues recently obtained by Shwartz and Weiss. The Brownian setting leads to a simple description of the component processes' asymptotic behaviour, as well as to explicit distributional results.

We consider a controlled i.i.d. process, where several i.i.d. sources are sampled sequentially. At each time instant, a controller determines from which source to obtain the next sample. Any causal sampling policy, possibly history-dependent, may be employed. The purpose is to characterize the extremal large deviations of the sample mean, namely to obtain asymptotic rate bounds (similar to and extending Cramér's theorem) which hold uniformly over all sampling policies. Lower and upper bounds are obtained, and it is shown that in many (but not all) cases stationary sampling policies are sufficient to obtain the extremal large deviations rates. These results are applied to a hypothesis testing problem, where data samples may be sequentially chosen from several i.i.d. sources (representing different types of experiments). The analysis provides asymptotic estimates for the error probabilities, corresponding both to optimal and to worst-case sampling policies.

This paper considers the joint limiting behavior of sums and maxima of stationary discrete-valued processes. The asymptotic behavior is a cross between a central limit theorem and asymptotic bounds for the distribution of the maxima. Some applications and simulations are also included.

Suppose n possibly censored survival times are observed under an independent censoring model, in which the observed times are generated as the minimum of independent positive failure and censor random variables. A practical difficulty arises when the largest observation is censored since then the usual non-parametric estimator of the distribution of the survival time is improper. We calculate the probability that this occurs and give necessary and sufficient conditions for this probability to converge to 0 as n →∞. As an application, we show that if this probability is 0, asymptotically, then a consistent estimator for the mean failure time can be found. An almost sure version of the problem is also considered.

In this paper we introduce a multilevel birth-death particle system and consider its diffusion approximation which can be characterized as a M([R+)-valued process. The tightness of rescaled processes is proved and we show that the limiting M(R+)-valued process is the unique solution of the M([R+)-valued martingale problem for the limiting generator. We also study the moment structures of the limiting diffusion process.

A simple model for the intensity of infection during an epidemic in a closed population is studied. It is shown that the size of an epidemic (i.e. the number of persons infected) and the cumulative force of an epidemic (i.e. the amount of infectiousness that has to be avoided by a person that will stay uninfected during the entire epidemic) satisfy an equation of balance. Under general conditions, small deviances from this balance are, in large populations, asymptotically mixed normally distributed. For some special epidemic models the size of an asymptotically large epidemic is asymptotically normally distributed.

Let X1, X2, · ··, Xn be a sequence of n random variables taking values in the ξ -letter alphabet . We consider the number N = N(n, k) of non-overlapping occurrences of a fixed k-letter word under (a) i.i.d. and (b) stationary Markovian hypotheses on the sequence , and use the Stein–Chen method to obtain Poisson approximations for the same. In each case, results and couplings from Barbour et al. (1992) are used to show that the total variation distance between the distribution of N and that of an appropriate Poisson random variable is of order (roughly) O(kS(k)), where S(k) denotes the stationary probability of the word in question. These results vastly improve on the approximations obtained in Godbole (1991). In the Markov case, we also make use of recently obtained eigenvalue bounds on convergence to stationarity due to Diaconis and Stroock (1991) and Fill (1991).

In this note, the strong limiting behaviour of busy periods in GI/G/1 queues is studied under the condition that the traffic intensity equals unity.

We consider the likelihood ratio tests to detect an epidemic alternative in the following two cases of normal observations: (1) the alternative specifies a square wave drift in the mean value of an i.i.d. sequence; (2) the alternative permits a square wave drift in the intercept of a simple linear regression model. To develop the approximations for the significance levels leads us to consider boundary-crossing problems of some two-dimensional discrete-time Gaussian fields. By the method which was proposed originally by Woodroofe (1976) and adapted to study maxima of some random fields by Siegmund (1988), some large deviations for the conditional non-linear boundary-crossing probabilities are developed. Some results of Monte Carlo experiments confirm the accuracy of these approximations.

Limits in distribution of maxima of independent stochastic processes are characterized in terms of spectral functions acting on a Poisson point process.

It is known (Weizsäcker and Winkler (1990)) that for bounded predictable functions H and a Poisson process with jump times exists almost surely, and that in this case both limits are equal. Here we relax the boundedness condition on H. Our tool is a law of large numbers for local L2-martingales. We show by examples that our condition is close to optimal. Furthermore we indicate a generalization to point processes on more general spaces. The above property is called PASTA (‘Poisson arrivals see time averages') and is heavily used in queueing theory.

Points are independently and uniformly distributed onto the unit interval. The first n—1 points subdivide the interval into n subintervals. For 1 we find a necessary and sufficient condition on {ln} for the events [Xn belongs to the ln th largest subinterval] to occur infinitely often or finitely often with probability 1. We also determine when the weak and strong laws of large numbers hold for the length of the ln th largest subinterval. The strong law of large numbers and the central limit theorem are shown to be valid for the number of times by time n the events [Xr belongs to lr th largest subinterval] occur when these events occur infinitely often.

Under appropriate long-range dependence conditions, it is well known that the joint distribution of the number of exceedances of several high levels is asymptotically compound Poisson. Here we investigate the structure of a cluster of exceedances for stationary sequences satisfying a suitable local dependence condition, under which it is only necessary to get certain limiting probabilities, easy to compute, in order to obtain limiting results for the highest order statistics, exceedance counts and upcrossing counts.

In this paper, we consider kth-order two-state Markov chains {Xi} with stationary transition probabilities. For k = 1, we construct in detail an upper bound for the total variation d(Sn, Y) = Σx |𝐏(Sn = x) − 𝐏(Y = x)|, where Sn = X1 + · ··+ Xn and Y is a compound Poisson random variable. We also show that, under certain conditions, d(Sn, Y) converges to 0 as n tends to ∞. For k = 2, the corresponding results are given without derivation. For general k ≧ 3, a conjecture is proposed.

It is well known that most commonly used discrete distributions fail to belong to the domain of maximal attraction for any extreme value distribution. Despite this negative finding, C. W. Anderson showed that for a class of discrete distributions including the negative binomial class, it is possible to asymptotically bound the distribution of the maximum. In this paper we extend Anderson's result to discrete-valued processes satisfying the usual mixing conditions for extreme value results for dependent stationary processes. We apply our result to obtain bounds for the distribution of the maximum based on negative binomial autoregressive processes introduced by E. McKenzie and Al-Osh and Alzaid. A simulation study illustrates the bounds for small sample sizes.

Let X1, X2, · ·· be independent and identically distributed random variables such that ΕΧ1 < 0 and P(X1 ≥ 0) ≥ 0. Fix M ≥ 0 and let T = inf {n: X1 + X2 + · ·· + Xn ≥ M} (T = +∞, if for every n = 1,2, ···). In this paper we consider the estimation of the level-crossing probabilities P(T <∞) and , by using Monte Carlo simulation and especially importance sampling techniques. When using importance sampling, precision and efficiency of the estimation depend crucially on the choice of the simulation distribution. For this choice we introduce a new criterion which is of the type of large deviations theory; consequently, the basic large deviations theory is the main mathematical tool of this paper. We allow a wide class of possible simulation distributions and, considering the case that M →∞, we prove asymptotic optimality results for the simulation of the probabilities P(T <∞) and . The paper ends with an example.

We generalize a two-type mutation process in which particles reproduce by binary fission, inheriting the parental type, but which can mutate with small probability during their lifetimes to the opposite type. The generalization allows an arbitrary offspring distribution. The branching process structure of this scheme is exploited to obtain a variety of limit theorems, some of which extend known results for the binary case. In particular, practically usable asymptotic normality results are obtained when the initial population size is large.

We consider an increasing supercritical branching process in a random environment and obtain bounds on the Laplace transform and distribution function of the limiting random variable. There are two possibilities that can be distinguished depending on the nature of the component distributions of the environment. If the minimum family size of each is 1, the growth will be as a power depending on a parameter α. If the minimum family sizes of some are greater than 1, it will be exponential, depending on a parameter γ. We obtain bounds on the distribution function analogous to those found for the simple Galton-Watson case. It is not possible to obtain exact estimates and we are only able to obtain bounds to within ε of the parameters.

Cottrell et al. (1983) have indicated how ideas from the large deviations theory lead to fast simulation schemes that estimate the mean time taken by the slotted ALOHA protocol to saturate starting empty. Such fast simulation schemes are particularly useful when the attempt probability is small. The remaining time to saturation when the protocol has been operating for a time is more accurately described by the quasi-stationary exit time from the stable regime. The purpose of this article is to prove that the ratio of the quasi-stationary exit time to the exit time starting empty approaches 1 as the attempt probability becomes small.

Let X(1) ≦ X(2) ≦ ·· ·≦ X(N(t)) be the order statistics of the first N(t) elements from a sequence of independent identically distributed random variables, where {N(t); t ≧ 0} is a renewal counting process independent of the sequence of X's. We give a complete description of the asymptotic distribution of sums made from the top kt extreme values, for any sequence kt such that kt → ∞, kt/t → 0 as t → ∞. We discuss applications to reinsurance policies based on large claims.