Competing patterns are compound patterns that compete to be the first to occur pattern-specific numbers of times. They represent a generalisation of the sooner waiting time problem and of start-up demonstration tests with both acceptance and rejection criteria. Through the use of finite Markov chain imbedding, the waiting time distribution of competing patterns in multistate trials that are Markovian of a general order is derived. Also obtained are probabilities that each particular competing pattern will be the first to occur its respective prescribed number of times, both in finite time and in the limit.

Expansions for the first-passage-time p.d.f. through a constant boundary and for its Laplace transform are derived in terms of probability currents for a temporally homogeneous diffusion process. Ultimate absorption and recurrence problems are also considered. The moments of the first-passage time are finally explicitly obtained.

We consider a consecutive-k-out-of-n system situated in a random environment where shocks occur and cause components to fail with certain probabilities. Under certain reasonable assumptions we compute the moment generating function of the time of failure of the system and the distribution of the number of shocks required to cause failure of the system.

Some recent constructions for the generation of dependent sequences of identically distributed negative exponential random variables with specific correlation structures are generalized. This is achieved by attributing a correlation structure to the binary sequence which controls the generation of the exponentials. The procedure causes the autocorrelation function of the exponential sequence to copy that of the binary sequence and thus be extended to include negative values and other values beyond the usual range.

There is a well-known connection between the asymptotic behaviour of the tail of the distribution of the increasing ladder height and the integrated tail of the step distribution of a random walk which either drifts to –∞, or oscillates and whose decreasing ladder height has finite mean. We establish a similar connection in a local sense; this means that in the lattice case we link the asymptotic behaviours of the mass function of the ladder height distribution and of the tail of the step distribution. We deduce the asymptotic behaviour of the mass function of the maximum of the walk, when this is finite, and also treat the non-lattice case.

Necessary and sufficient conditions for stability, imposed firstly on the initial structure and the sequence of recruitment, and secondly on the initial structure and the sequence of expansion are provided in forms of two theorems. Also the limiting behaviour of the expected relative grade sizes is studied if we drop the conditions for stability imposed on the initial structure and keep the same sequence of expansion. Finally we examine the limiting behaviour of the expected grade sizes if we drop the assumption of a continuously expanding system.

A class of limit theorems involving asymptotic normality is derived for stationary processes whose spectral density has a singular behavior near frequency zero. Generally these processes have ‘long-range dependence’ but are generated from strongly mixing processes by a fractional integral or derivative transformation. Some related remarks are made about random solutions of the Burgers equation.

The stationary version of the infinitely many neutral alleles diffusion model, as studied by Ethier and Kurtz, is constructed and analyzed using the theory of Dirichlet forms. We prove that the ‘boundary' of the state space is hit by the process iff θ < 1, where θ is the mutation parameter.

We obtain bounds for the distribution of the number of comparisons needed by Hoare's randomized selection algorithm FIND and give a new proof for Grübel and Rösler's (1996) result on the convergence of this distribution. Our approach is based on the construction and analysis of a suitable associated Markov chain. Some numerical results for the quantiles of the limit distributions are included, leading for example to the statement that, for a set S with n elements and n large, FIND will need with probability 0.9 about 4.72 x n comparisons to find the median of S.

We consider the optimal stopping problem of maximizing the probability of stopping on any of the last m successes of a sequence of independent Bernoulli trials of length n, where m and n are predetermined integers such that 1 ≤ m < n. The optimal stopping rule of this problem has a nice interpretation, that is, it stops on the first success for which the sum of the m-fold multiplicative odds of success for the future trials is less than or equal to 1. This result can be viewed as a generalization of Bruss' (2000) odds theorem. Application will be made to the secretary problem. For more generality, we extend the problem in several directions in the same manner that Ferguson (2008) used to extend the odds theorem. We apply this extended result to the full-information analogue of the secretary problem, and derive the optimal stopping rule and the probability of win explicitly. The asymptotic results, as n tends to ∞, are also obtained via the planar Poisson process approach.

We consider tandem queues which have a general arrival process. The queueing system consists of s (s ≧ 2) single-server service stations and the servers have exponential service-time distributions. Firstly we give a new proof for the fact that the departure process does not depend on the particular allocation of the servers to the stations. Secondly, considering the service rates, we prove that the departure process becomes stochastically faster as the homogeneity of the servers increases in the sense of a given condition. It turns out that, given the sum of the service rates, the departure process is stochastically fastest in the case where the servers are homogeneous.

We construct a process with gamma increments, which has a given convex autocorrelation function and asymptotically a self-similar limit. This construction validates the use of long-range dependent t and variance-gamma subordinator models for actual financial data as advocated in Heyde and Leonenko (2005) and Finlay and Seneta (2006), in that it allows for noninteger-valued model parameters to occur as found empirically by data fitting.

We consider a Wiener process between a reflecting and an absorbing barrier and derive a series solution for the transition density of the process and for the density of the time to absorption. If the drift is towards the reflecting barrier, the variance is not too large, and the distance of the barriers is not too small, the leading term of the series derives from imaginary solutions of the basic eigenvalue equation of this problem. It is shown that these leading terms often make the dominant contribution to the complete series. Finally, we consider previous attempts by Fürth [3], Cox and Miller [1], and by Goel and Richter-Dyn [5] to solve the stated problem and point out, in some detail, why their solutions are wrong.

For independent random lifelengths of the units in use stochastic comparisons of the number of failures and removal in [0,s] under age and block replacement policies are performed. A new concept of NBU (NWU) in sequence is introduced.

We consider a model of N interacting two-colour Friedman urns. The interaction model considered is such that the reinforcement of each urn depends on the fraction of balls of a particular colour in that urn as well as the overall fraction of balls of that colour in all the urns combined together. We show that the urns synchronize almost surely and that the fraction of balls of each colour converges to the deterministic limit of one-half, which matches with the limit known for a single Friedman urn. Furthermore, we use the notion of stable convergence to obtain limit theorems for fluctuations around the synchronization limit.

The changes in diversity of competitive strategies in a Maynard Smith population model with mixed strategies are related to the changes in population mean strategy. The effects of slight mutations in strategy frequencies, and of slight perturbations of the contest payoff rules are then investigated, and found to increase and decrease diversity respectively (to a third-order approximation). A relation among mutational effects, payoff perturbation effects and stable population diversity is suggested.

A quantity of particular interest in the study of (road) traffic jams is the total waiting time X of all vehicles involved in a given hold-up (Gaver (1969): see note following (2.3) below and the first paragraph of Section 5). With certain assumptions on the process this random variable X is the same as the sum of waiting times of customers in a busy period of a GI/G/1 queueing system, and it is the object of this paper and its sequel to study the random variable in the queueing theory context.

Although his work on the application of invariance concepts to the sequential testing of composite hypotheses is better known, Cox (1963) has also outlined a large sample approach to the same problem. His method is based on Bartlett's (1946) recognition that the sequence of maximum likelihood estimates (MLE) of the parameter of interest, calculated from an increasing number of observations, resembles asymptotically a random walk of normally distributed variables. However, the large sample theory needed to justify this approach rigorously is left largely implicit. At the end of his paper, Cox suggests that these methods may be extended to yield a sequential comparison of survival curves (Armitage (1959)), a suggestion which has been reiterated as a research problem in the monograph of Wetherill (1966).

Let {X k , k ≥ 1} be a sequence of independently and identically distributed random variables with common subexponential distribution function concentrated on (−∞, ∞), and let τ be a nonnegative and integer-valued random variable with a finite mean and which is independent of the sequence {X k , k ≥ 1}. This paper investigates asymptotic behavior of the tail probabilities P(· > x) and the local probabilities P(x < · ≤ x + h) of the quantities and for n ≥ 1, and their randomized versions X (τ), S τ and S (τ), where X 0 = 0 by convention and h > 0 is arbitrarily fixed.

The ruin probability in the classical Brownian risk model can be explicitly calculated for both finite and infinite time horizon. This is not the case for the simultaneous ruin probability in the two-dimensional Brownian risk model. Relying on asymptotic theory, we derive in this contribution approximations for both simultaneous ruin probability and simultaneous ruin time for the two-dimensional Brownian risk model when the initial capital increases to infinity.