We study Bernoulli bond percolation on Sierpiński carpet lattices, which is a class of graphs corresponding to generalized Sierpiński carpets. In this paper we give a sufficient condition for the existence of a phase transition on the lattices. The proof is suitable for graphs which have self-similarity. We also discuss the relation between the existence of a phase transition and the isoperimetric dimension.

This paper first recalls some stochastic orderings useful for studying the ℒ-class and the Laplace order in general. We use these orders to show that the higher moments of an ℒ-class distribution need not exist. Using simple sufficient conditions for the Laplace ordering, we give examples of distributions in the ℒ- and ℒα-classes. Moreover, we present explicit sharp bounds on the survival function of a distribution belonging to the ℒ-class of life distributions. The results reveal that the ℒ-class should not be seen as a more comprehensive class of ageing distributions but rather as a large class of life distributions on its own.

We consider a finite-capacity queueing system where arriving customers offer rewards which are paid upon acceptance into the system. The gatekeeper, whose objective is to ‘maximize’ rewards, decides if the reward offered is sufficient to accept or reject the arriving customer. Suppose the arrival rates, service rates, and system capacity are changing over time in a known manner. We show that all bias optimal (a refinement of long-run average reward optimal) policies are of threshold form. Furthermore, we give sufficient conditions for the bias optimal policy to be monotonic in time. We show, via a counterexample, that if these conditions are violated, the optimal policy may not be monotonic in time or of threshold form.

We consider a nonneutral population genetics model with parent-independent mutations and two selective classes. We calculate the stationary distribution of the type of the common ancestor of a sample of genes from this model. The expected fitness of any ancestor (including the most recent common ancestor of any sample) is shown to be greater than the expected fitness of a randomly chosen gene from the population. The process of mutations to the common ancestor is also analysed. Our results are related to, but more general than, results obtained from diffusion theory.

Consider an optimization problem for a company with the following parameters: a constant liability payment rate (δ), an average return (μ) and a risk (σ) proportional to the size of the business unit, and an internal competition factor (α) between different units. The goal is to maximize the expected present value of the total dividend distributions, via controls (Ut, Zt), where Ut is the size of the business unit and Zt is the total dividend payoff up to time t. We formulate this as a stochastic control problem for a diffusion process Xt and derive an explicit solution by solving the corresponding Hamilton-Jacobi-Bellman equation. The resulting optimal control policy involves a mixture of a nonlinear control for Ut and a singular control for Zt. The optimal strategies are different for the cases δ < 0 and δ = 0. When δ > 0, it is optimal to play bold: the initial optimal investment size should be proportional to the debt rate δ. Under this optimal rule, however, the probability of bankruptcy in finite time is 1. When δ = 0, i.e. when the company is free of debt, the probability of going broke in finite time reduces to 0. Moreover, when δ = 0, the value function is singular at X0 = 0. Our analytical result shows considerable consistency with daily business practices. For instance, it shows that ‘too many people is counter-productive’. In fact, the maximal optimal size of the business unit should be inversely proportional to α. This eliminates the redundant and simplistic technical assumption of a known uniform upper bound on the size of the firm.

Probability generation functions of waiting time distributions of runs and patterns have been used successfully in various areas of statistics and applied probability. In this paper, we provide a simple way to obtain the probability generating functions for waiting time distributions of compound patterns by using the finite Markov chain imbedding method. We also study the characters of waiting time distributions for compound patterns. A computer algorithm based on Markov chain imbedding technique has been developed for automatically computing the distribution, probability generating function, and mean of waiting time for a compound pattern.

Ease of computation of Fréchet-optimal lower bounds, given numerical values of the binomial moments Sij, i, j = 1, 2, is demonstrated. A sufficient condition is given for an explicit bivariate bound of Dawson-Sankoff structure to be Fréchet optimal. An example demonstrates that in the bivariate case even the multiplicative structure of the Sij does not guarantee a Dawson-Sankoff structure for Fréchet-optimal bounds. A final section is used to illuminate the nature of Fréchet optimality by using generalized explicit bounds. This note is a sequel to both Chen and Seneta (1995) and Chen (1998).

We consider the estimation of Markov transition matrices by Bayes’ methods. We obtain large and moderate deviation principles for the sequence of Bayesian posterior distributions.

If Z is the branching mechanism for a supercritical Galton-Watson tree with a single progenitor and E[ZlogZ] < ∞, then there is a branching measure μ defined on ∂Γ, the set of all paths ξ which have a unique node ξ|n at each generation n. We use the natural metric ρ(ξ,η) = e−n, where n = max{k : ξ|k = η|k}, and observe that the local dimension index is d(μ,ξ) = limn→∞ log(μB(ξ|n))/(-n) = α = logm, for μ-almost every ξ. Our objective is to consider the exceptional points where the above display may fail. There is a nontrivial ‘thin’ spectrum for ̄d(μ,ξ) when p1 = P{Z = 1} > 0 and Z has finite moments of all positive orders. Because ̱d(μ,ξ) = a for all ξ, we obtain a ‘thick’ spectrum by introducing the ‘right’ power of a logarithm. In both cases, we find the Hausdorff dimension of the exceptional sets.

The paper is concerned with filtering the cardinality of a branching process observing the cardinality of a subpopulation. In this model, both the processes, state and observation are pure jump processes and may have common jump times. Preliminary properties are studied in the tree framework. A recursive structure for the filtering equation is proved in the supercritical case.

In this article we obtain rates of convergence to equilibrium of marked Hawkes processes in two situations. Firstly, the stationary process is the empty process, in which case we speak of the rate of extinction. Secondly, the stationary process is the unique stationary and nontrivial marked Hawkes process, in which case we speak of the rate of installation. The first situation models small epidemics, whereas the results in the second case are useful in deriving stopping rules for simulation algorithms of Hawkes processes with random marks.

In this paper we obtain the rate of convergence to equilibrium of a class of interacting marked point processes, introduced by Kerstan, in two different situations. Indeed, we prove the exponential and subexponential ergodicity of such a class of stochastic processes. Our results are an extension of the corresponding results of Brémaud, Nappo and Torrisi. The generality of the dynamics which we take into account allows the application to the so-called loss networks, and multivariate birth and death processes.

The dynamics of opinion transformation is modeled by a neural network with a nonnegative matrix of connections. Noise is introduced at each site, and the limit of the stationary distributions of the resulting Markov chains as the noise goes to zero is taken as an indication of what configurations will be seen. An algorithm for computing this limit is given, and a number of examples are worked out. Some of the mathematical ideas developed, such as visible states, time scales, and a calculus of indexed probabilities, are of independent interest.

Patch clamp recordings from ion channels often show periods of repetitive activity, known as bursts, which are noticeably separated from each other by periods of inactivity. Depending on the type of channel, such recordings may exhibit (conductance) levels between the closed (zero) level and the fully open level. Properties of bursts are less subject to problems that arise from recording than are properties for individual sojourns at different levels, and study of bursting behaviour provides important information about the finer structure of the underlying channel gating process. For a general finite state space continuous-time Markov chain model allowing one or more nonzero conductance levels, the present paper establishes results about the semi-Markov structure of a single burst and of a sequence of bursts, and uses this in a unified approach to properties of both theoretical and empirical bursts. The distribution and moments of particular burst properties, including the total charge transfer, the total sojourn time and the total number of visits to specified conductance levels during a burst, are derived. Various extensions are also described.

Let (Xt) and (Yt) be continuous-time Markov chains with countable state spaces E and F and let K be an arbitrary subset of E x F. We give necessary and sufficient conditions on the transition rates of (Xt) and (Yt) for the existence of a coupling which stays in K. We also show that when such a coupling exists, it can be chosen to be Markovian and give a way to construct it. In the case E=F and K ⊆ E x E, we see how the problem of construction of the coupling can be simplified. We give some examples of use and application of our results, including a new concept of lumpability in Markov chains.

This paper presents an algorithmic procedure to calculate the delay distribution of a type k customer in a first-come-first-served (FCFS) discrete-time queueing system with multiple types of customers, where each type has different service requirements (the MMAP[K]/PH[K]/1 queue). First, we develop a procedure, using matrix analytical methods, to handle arrival processes that do not allow batch arrivals to occur. Next, we show that this technique can be generalized to arrival processes that do allow batch arrivals to occur. We end the paper by presenting some numerical examples.

In this note, we give new proofs of the closure property of ageing classes NBUC and NBU(2) under convolution to make up the gaps in the proofs of Cao and Wang (1991) and of Li and Kochar (2001).

The Laplace transform is obtained for the joint length and height of an M/M/∞ queue excursion above occupancy level C.