Consider a modified, stable, two node Jackson network where server 2 helps server 1 when server 2 is idle. The probability of a large deviation of the number of customers at node one can be calculated using the flat boundary theory of Schwartz and Weiss [Large Deviations Performance Analysis (1994), Chapman and Hall, New York]. Surprisingly, however, these calculations show that the proportion of time spent on the boundary, where server 2 is idle, may be zero. This is in sharp contrast to the unmodified Jackson network which spends a nonzero proportion of time on this boundary.

We extend the Markov additive methodology developed in [Ann. Appl. Probab. 9 (1999) 110–145, Ann. Appl. Probab. 11 (2001) 596–607] to obtain the sharp asymptotics of the steady state probability of a queueing network when one of the nodes gets large. We focus on a new phenomenon we call a bridge. The bridge cases occur when the Markovian part of the twisted Markov additive process is one null recurrent or one transient, while the jitter cases treated in [Ann. Appl. Probab. 9 (1999) 110–145, Ann. Appl. Probab. 11 (2001) 596–607] occur when the Markovian part is (one) positive recurrent. The asymptotics of the steady state is an exponential times a polynomial term in the bridge case, but is purely exponential in the jitter case.

We apply this theory to a modified, stable, two node Jackson network where server two helps server one when server two is idle. We derive the sharp asymptotics of the steady state distribution of the number of customers queued at each node as the number of customers queued at the server one grows large. In so doing we get an intuitive understanding of the companion paper [Ann. Appl. Probab. 15 (2005) 519–541] which gives a large deviation analysis of this problem using the flat boundary theory in the book by Shwartz and Weiss. Unlike the (unscaled) large deviation path of a Jackson network which jitters along the boundary, the unscaled large deviation path of the modified network tries to avoid the boundary where server two helps server one (and forms a bridge). In the fluid limit this bridge does collapse to a straight line, but the proportion of time spent on the flat boundary tends to zero.

This bridge phenomenon is ubiquitous. We also treated the bathroom problem described in the Shwartz and Weiss book and found the bridge case is present. Here we derive the sharp asymptotics of the steady state of the bridge case and we obtain the results consistent with those obtained in [SIAM J. Appl. Math. (1984) 44 1041–1053] using complex variable methods.

Consider the state space model (X_t,Y_t), where (X_t) is a Markov chain, and (Y_t) are the observations. In order to solve the so-called filtering problem, one has to compute ℒ(X_t|Y₁,…,Y_t), the law of X_t given the observations (Y₁,…,Y_t). The particle filtering method gives an approximation of the law ℒ(X_t|Y₁,…,Y_t) by an empirical measure $\frac{1}{n}$∑₁ⁿδ_xi,t. In this paper we establish the moderate deviation principle for the empirical mean $\frac{1}{n}$∑₁ⁿψ(x_i,t) (centered and properly rescaled) when the number of particles grows to infinity, enhancing the central limit theorem. Several extensions and examples are also studied.

We consider the behavior of spatial point processes when subjected to a class of linear transformations indexed by a variable T. It was shown in Ellis [Adv. in Appl. Probab. 18 (1986) 646–659] that, under mild assumptions, the transformed processes behave approximately like Poisson processes for large T. In this article, under very similar assumptions, explicit upper bounds are given for the d₂-distance between the corresponding point process distributions. A number of related results, and applications to kernel density estimation and long range dependence testing are also presented. The main results are proved by applying a generalized Stein–Chen method to discretized versions of the point processes.

Given i.i.d. positive integer valued random variables D₁,…,D_n, one can ask whether there is a simple graph on n vertices so that the degrees of the vertices are D₁,…,D_n. We give sufficient conditions on the distribution of D_i for the probability that this be the case to be asymptotically 0, ½ or strictly between 0 and ½. These conditions roughly correspond to whether the limit of nP(D_i≥n) is infinite, zero or strictly positive and finite. This paper is motivated by the problem of modeling large communications networks by random graphs.

This paper extends earlier work by Cox and Durrett, who studied the coalescence times for two lineages in the stepping stone model on the two-dimensional torus. We show that the genealogy of a sample of size n is given by a time change of Kingman’s coalescent. With DNA sequence data in mind, we investigate mutation patterns under the infinite sites model, which assumes that each mutation occurs at a new site. Our results suggest that the spatial structure of the human population contributes to the haplotype structure and a slower than expected decay of genetic correlation with distance revealed by recent studies of the human genome.

We give computable bounds on the rate of convergence of the transition probabilities to the stationary distribution for a certain class of geometrically ergodic Markov chains. Our results are different from earlier estimates of Meyn and Tweedie, and from estimates using coupling, although we start from essentially the same assumptions of a drift condition toward a “small set.” The estimates show a noticeable improvement on existing results if the Markov chain is reversible with respect to its stationary distribution, and especially so if the chain is also positive. The method of proof uses the first-entrance–last-exit decomposition, together with new quantitative versions of a result of Kendall from discrete renewal theory.

We study a large family of competing spatial growth models. In these models the vertices in ℤ^d can take on three possible states {0,1,2}. Vertices in states 1 and 2 remain in their states forever, while vertices in state 0, which are adjacent to a vertex in state 1 (or state 2), can switch to state 1 (or state 2). We think of the vertices in states 1 and 2 as infected with one of two infections, while the vertices in state 0 are considered uninfected. In this way these models are variants of the Richardson model. We start the models with a single vertex in state 1 and a single vertex in state 2. We show that with positive probability state 1 reaches an infinite number of vertices and state 2 also reaches an infinite number of vertices. This extends results and proves a conjecture of Häggström and Pemantle [J. Appl. Probab. 35 (1998) 683–692]. The key tool is applying the ergodic theorem to stationary first passage percolation.

We introduce a linear space of finitely additive measures to treat the problem of optimal expected utility from consumption under a stochastic clock and an unbounded random endowment process. In this way we establish existence and uniqueness for a large class of utility-maximization problems including the classical ones of terminal wealth or consumption, as well as the problems that depend on a random time horizon or multiple consumption instances. As an example we explicitly treat the problem of maximizing the logarithmic utility of a consumption stream, where the local time of an Ornstein–Uhlenbeck process acts as a stochastic clock.

Let {ξ(n)}_n∈ℤ be a two-color random scenery, that is, a random coloring of ℤ in two colors, such that the ξ(i)’s are i.i.d. Bernoulli variables with parameter ½. Let {S(n)}_n∈ℕ be a symmetric random walk starting at 0. Our main result shows that a.s., ξ○S (the composition of ξ and S) determines ξ up to translation and reflection. In other words, by observing the scenery ξ along the random walk path S, we can a.s. reconstruct ξ up to translation and reflection. This result gives a positive answer to the question of H. Kesten of whether one can a.s. detect a single defect in almost every two-color random scenery by observing it only along a random walk path.

This paper studies a diffusion model that arises as the limit of a queueing system scheduling problem in the asymptotic heavy traffic regime of Halfin and Whitt. The queueing system consists of several customer classes and many servers working in parallel, grouped in several stations. Servers in different stations offer service to customers of each class at possibly different rates. The control corresponds to selecting what customer class each server serves at each time. The diffusion control problem does not seem to have explicit solutions and therefore a characterization of optimal solutions via the Hamilton–Jacobi–Bellman equation is addressed. Our main result is the existence and uniqueness of solutions of the equation. Since the model is set on an unbounded domain and the cost per unit time is unbounded, the analysis requires estimates on the state process that are subexponential in the time variable. In establishing these estimates, a key role is played by an integral formula that relates queue length and idle time processes, which may be of independent interest.

We consider random sequential adsorption processes where the initially empty sites of a graph are irreversibly occupied, in random order, either by monomers which block neighboring sites, or by dimers. We also consider a process where initially occupied sites annihilate their neighbors at random times.

We verify that these processes are well defined on infinite graphs, and derive forward equations governing joint vacancy/occupation probabilities. Using these, we derive exact formulae for occupation probabilities and pair correlations in Bethe lattices. For the blocking and annihilation processes we also prove positive correlations between sites an even distance apart, and for blocking we derive rigorous lower bounds for the site occupation probability in lattices, including a lower bound of 1/3 for Z². We also give normal approximation results for the number of occupied sites in a large finite graph.

Motivated by regularities observed in time series of returns on speculative assets, we develop an asymptotic theory of GARCH(1,1) processes {y_k} defined by the equations y_k=σ_kɛ_k, σ_k²=ω+αy_k−1²+βσ_k−1² for which the sum α+β approaches unity as the number of available observations tends to infinity. We call such sequences near-integrated. We show that the asymptotic behavior of near-integrated GARCH(1,1) processes critically depends on the sign of γ:=α+β−1. We find assumptions under which the solutions exhibit increasing oscillations and show that these oscillations grow approximately like a power function if γ≤0 and exponentially if γ>0. We establish an additive representation for the near-integrated GARCH(1,1) processes which is more convenient to use than the traditional multiplicative Volterra series expansion.

This paper studies a very general urn model stimulated by designs in clinical trials, where the number of balls of different types added to the urn at trial n depends on a random outcome directed by the composition at trials 1,2,…,n−1. Patient treatments are allocated according to types of balls. We establish the strong consistency and asymptotic normality for both the urn composition and the patient allocation under general assumptions on random generating matrices which determine how balls are added to the urn. Also we obtain explicit forms of the asymptotic variance–covariance matrices of both the urn composition and the patient allocation. The conditions on the nonhomogeneity of generating matrices are mild and widely satisfied in applications. Several applications are also discussed.

In this paper we investigate the speed of convergence of the fluctuations of a general class of Feynman–Kac particle approximation models. We design an original approach based on new Berry–Esseen type estimates for abstract martingale sequences combined with original exponential concentration estimates of interacting processes. These results extend the corresponding statements in the classical theory and apply to a class of branching and genealogical path-particle models arising in nonlinear filtering literature as well as in statistical physics and biology.

We analyze a (1+1)-dimension directed random walk model of a polymer dipped in a medium constituted by two immiscible solvents separated by a flat interface. The polymer chain is heterogeneous in the sense that a single monomer may energetically favor one or the other solvent. We focus on the case in which the polymer types are periodically distributed along the chain or, in other words, the polymer is constituted of identical stretches of fixed length. The phenomenon that one wants to analyze is the localization at the interface: energetically favored configurations place most of the monomers in the preferred solvent and this can be done only if the polymer sticks close to the interface.

We investigate, by means of large deviations, the energy–entropy competition that may lead, according to the value of the parameters (the strength of the coupling between monomers and solvents and an asymmetry parameter), to localization. We express the free energy of the system in terms of a variational formula that we can solve. We then use the result to analyze the phase diagram.

We introduce a spatial stochastic process on the lattice Z^d to model mass extinctions. Each site of the lattice may host a flock of up to N individuals. Each individual may give birth to a new individual at the same site at rate ϕ until the maximum of N individuals has been reached at the site. Once the flock reaches N individuals, then, and only then, it starts giving birth on each of the 2d neighboring sites at rate λ(N). Finally, disaster strikes at rate 1, that is, the whole flock disappears. Our model shows that, at least in theory, there is a critical maximum flock size above which a species is certain to disappear and below which it may survive.

Let Y be an Ornstein–Uhlenbeck diffusion governed by a stationary and ergodic Markov jump process X: dY_t=a(X_t)Y_t dt+σ(X_t) dW_t, Y₀=y₀. Ergodicity conditions for Y have been obtained. Here we investigate the tail propriety of the stationary distribution of this model. A characterization of either heavy or light tail case is established. The method is based on a renewal theorem for systems of equations with distributions on ℝ.

Fudenberg and Harris’ stochastic version of the classical replicator dynamics is considered. The behavior of this diffusion process in the presence of an evolutionarily stable strategy is investigated. Moreover, extinction of dominated strategies and stochastic stability of strict Nash equilibria are studied. The general results are illustrated in connection with a discrete war of attrition. A persistence result for the maximum effort strategy is obtained and an explicit expression for the evolutionarily stable strategy is derived.