In first-passage percolation (FPP) models, the passage time Tℓ from the origin to the point ℓeℓ satisfies f(ℓ) := ETℓ = μℓ + o(ℓ½+ε), where μ ∊ (0,∞) denotes the time constant. Yet, for lattice FPP, it is not known rigorously that f(ℓ) is eventually monotonically increasing. Here, for the Poisson-based Euclidean FPP of Howard and Newman (Prob. Theory Relat. Fields108 (1997), 153–170), we prove an explicit formula for f′(ℓ). In all dimensions, for certain values of the model's only parameter we have f′(ℓ) ≥ C > 0 for large ℓ.

Scherk's graph is a subgraph of the three-dimensional lattice. It was shown by Markvorsen, McGuinness and Thomassen (1992) that Scherk's graph is transient. Consider the Bernoulli bond percolation in Scherk's graph. We prove that the infinite cluster is transient for p > ½ and is recurrent for p < ½. This implies the well-known result of Grimmett, Kesten and Zhang (1993) on the transience of the infinite cluster of the Bernoulli bond percolation in the three-dimensional lattice for p > ½. On the other hand, Scherk's graph exhibits a new dichotomy in the supercritical region.

A new procedure that generates the transient solution of the first moment of the state of a Markovian queueing network with state-dependent arrivals, services, and routeing is developed. The procedure involves defining a partial differential equation that relates an approximate multivariate cumulant generating function to the intensity functions of the network. The partial differential equation then yields a set of ordinary differential equations which are numerically solved to obtain the first moment.

In this paper, the instantaneous availability of a system maintained under periodic inspection is investigated using random walk models. Two cases are considered. In the first model, the system is repaired or modified and it is assumed to be as good as new upon periodic inspection and maintenance. In the second model, the system is not modified after the inspection if the system is still working, and the condition of the system is assumed to be the same as that before the inspection. For both models the failures only can be found through the inspection. Perfect repair or replacement of a failed system is assumed to be carried out, but the time it takes can be constant or of a random length. The relationship between this problem and the random walk model in a two-dimensional plane is described. Several new results are also shown.

We present a new inspection policy useful when testing is needed to detect failures of a single-unit system. It is supposed that tests may fail and give an erroneous result. The inspection policy minimizing cost per unit of time for an infinite time span is also discussed. In addition, we study the behaviour of the optimum policy for some time to failure distributions often assumed in reliability: exponential and Pareto.

For most repairable systems, the number N(t) of failed components at time t appears to be a good quality parameter, so it is critical to study this random function. Here the components are assumed to be independent and both their lifetime and their repair time are exponentially distributed. Moreover, the system is considered new at time 0. Our aim is to compare the random variable N(t) with N(∞), especially in terms of total variation distance. This analysis is used to prove a cut-off phenomenon in the same way as Ycart (1999) but without the assumption of identical components.

The stochastic logistic model has been studied in various contexts, including epidemiology, population biology, chemistry and sociology. Among the model predictions, the quasistationary distribution and the mean time to extinction are of major interest for most applications, and a number of approximation formulae for these quantities have been derived. In this paper, previous approximation formulae are improved for two mathematically tractable cases: at the limit of the number of individuals N → ∞ (with relative error of the approximations of the order 𝒪(1/N)), and at the limit of the basic reproduction ratio R0 → ∞ (with relative error of the approximations of the order 𝒪(1/R0)). The mathematically rigorous formulae are then extended heuristically for other values of N and R0 > 1.

Let be the scan statistic of window size r for a sequence of n bistate trials . The scan statistic Sn(r) has been successfully used in various fields of applied probability and statistics, and its distribution has been studied extensively in the literature. Currently, all existing formulae for the distribution of Sn(r) are rather complex, and they can only be numerically implemented when is a sequence of Bernoulli trials, the window size r is less than 20 and the length of the sequence n is not too large. Hence, these formulae have been limiting the practical applications of the scan statistic. In this article, we derive a simple and effective formula for the distribution of Sn(r) via the finite Markov chain embedding technique to overcome some of the limitations of the existing complex formulae. This new formula can be applied when is either a sequence of Bernoulli trials or a sequence of Markov dependent bistate trials. Selected numerical examples are given to illustrate our results.

We consider a continuous-time Markov additive process (Jt,St) with (Jt) an irreducible Markov chain on E = {1,…,s}; it is known that (St/t) satisfies the large deviations principle as t → ∞. In this paper we present a variational formula H for the rate function κ∗ and, in some sense, we have a composition of two large deviations principles. Moreover, under suitable hypotheses, we can consider two other continuous-time Markov additive processes derived from (Jt,St): the averaged parameters model (Jt,St(A)) and the fluid model (Jt,St(F)). Then some results of convergence are presented and the variational formula H can be employed to show that, in some sense, the convergences for (Jt,St(A)) and (Jt,St(F)) are faster than the corresponding convergences for (Jt,St).

In this paper, a single channel FIFO fluid queue with an infinite buffer space and a long-range dependent input is studied. The input traffic is modeled by an average input rate plus a standard fractional Brownian motion as the fluctuation. Lower and upper bounds are derived for the tail distribution of the transient queue length at time T, which result in a logarithmic characterization of the asymptotic behavior of the tail distribution. Furthermore, the exact asymptotic is also obtained. It is observed that the transient queue length under fractional Brownian input does not suffer from the heavy-tail property as does the steady-state queue length. The results are used to compute the equivalent bandwidth requirement for ATM broadband connections with fractional Brownian traffic feed and finite connection holding time.

An urn contains m minus balls and p plus balls, and we draw balls from this urn one at a time randomly without replacement until we wish to stop. Let Pn and Mn denote the respective numbers of plus balls and minus balls drawn by time n and define Z0 = 0, Zn = Pn - Mn, 1 ≤ n ≤ m + p. The main problem of this paper is to stop with maximum probability on the maximum of the trajectory formed by . This problem is closely related to the celebrated ballot problem, so that we obtain some identities concerning the ballot problem and then derive the optimal stopping rule explicitly. Some related modifications are also studied.

In this paper we consider the stochastic wave equation in one spatial dimension driven by a two-parameter Gaussian noise which is white in time and has general spatial covariance. We give conditions on the spatial covariance of the driving noise sufficient for the string to have finite expected energy and calculate this energy as a function of time. We show that these same conditions on the spatial covariance of the driving noise are also sufficient to guarantee that the energy of the string has a version which is continuous almost surely.

Simulated annealing is a popular and much studied method for maximizing functions on finite or compact spaces. For noncompact state spaces, the method is still sound, but convergence results are scarce. We show here how to prove convergence in such cases, for Markov chains satisfying suitable drift and minorization conditions.

General and unifying approaches are discussed for computing fundamental characteristics of both continuous-time and discrete-time birth-death processes. In particular, an exponential family framework is used to derive explicit expressions, in terms of continued fractions, for joint generating functions of first-passage times and a whole collection of associated random quantities, and a random sum representation is used to obtain formulae for means, variances and covariances of stopped reward functions defined on a birth-death process.

We give a general construction of sequential games among multiple players, as well as a construction of the composition of sequential games. We obtain new properties of the optimal class of win-by-k games, including closure under composition and independence between the winner of the game and the number of points played. We obtain new results on the asymptotic efficiency of the n-point, win-by-k games.

We prove the law of large numbers for random walks in random environments on the d-dimensional integer lattice Zd. The environment is described in terms of a stationary random field of transition probabilities on the lattice, possessing a certain drift property, modeled on the Kalikov condition. In contrast to the previously considered models, we admit possible correlation of transition probabilities at different sites, assuming however that they become independent at finite distances. The possible dependence of sites makes impossible a direct application of the renewal times technique of Sznitman and Zerner.

The paper studies the impact of a broadly understood trend, which includes a change point in mean and monotonic trends studied by Bhattacharya et al. (1983), on the asymptotic behaviour of a class of tests designed to detect long memory in a stationary sequence. Our results pertain to a family of tests which are similar to Lo's (1991) modified R/S test. We show that both long memory and nonstationarity (presence of trend or change points) can lead to rejection of the null hypothesis of short memory, so that further testing is needed to discriminate between long memory and some forms of nonstationarity. We provide quantitative description of trends which do or do not fool the R/S-type long memory tests. We show, in particular, that a shift in mean of a magnitude larger than N-½, where N is the sample size, affects the asymptotic size of the tests, whereas smaller shifts do not do so.

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

Consider a Navier-Stokes incompressible turbulent fluid in R2. Let x(t) denote the position coordinate of a moving vortex with initial circulation Γ0 > 0 in the fluid, subject to a force F. Define x(t) as a stochastic process with continuous sample paths described by a stochastic differential equation. Assuming a suitable notion of weak rotationality, it is shown that the stochastic equation is equivalent to a linear partial differential equation for the complex function ψ, i∂ψ/∂t = [-Γ0Δ + F] ψ, where |ψ|2 = ρ(x,t), ρ being the probability density function of finding the vortex centre in position x at time t.

Let Xt be an n-dimensional diffusion process and S(t) be a set-valued function. Suppose Xt is invisible when it is hidden by S(t), but we can see the process exactly otherwise. In this paper, we derive the optimal estimator E[f(X1) | Xs1Xs∉S(s), 0 ≤ s ≤ 1] for a bounded Borel function f. We illustrate some computations for Gauss-Markov processes.