We consider a single server system with N input flows. We assume that each flow has stationary increments and satisfies a sample path large deviation principle, and that the system is stable. We introduce the largest weighted delay first (LWDF) queueing discipline associated with any given weight vector α=(α₁,...,α_N). We show that under the LWDF discipline the sequence of scaled stationary distributions of the delay \(\hat{w}_{i}\) of each flow satisfies a large deviation principle with the rate function given by a finite- dimensional optimization problem. We also prove that the LWDF discipline is optimal in the sense that it maximizes the quantity $$\min_{i= 1,\space ...,\space N}\left[α_i \lim_{n\to\infty}\frac{−1}{n}\log P(\hat{w}_i>n)\right],$$ within a large class of work conserving disciplines.

Using a slight modification of the framework of Bramson [7] and Williams [54], we prove heavy traffic limit theorems for six families of multiclass queueing networks. The first three families are single-station systems operating under first-in-first-out (FIFO), generalized-head-of-the-line proportional processor sharing (GHLPPS) and static buffer priority (SBP) service disciplines. The next two families are reentrant lines that operate under first-buffer-first-serve (FBFS) and last-buffer-first-serve (LBFS) service disciplines; the last family consists of certain two-station, five-class networks operating under an SBP service discipline. Some of these heavy traffic limits have appeared earlier in the literature; our new proofs demonstrate the significant simplifications that can be achieved in the present setting.

We study a Hopfield model whose number of patterns M grows to infinity with the system size N,in such a way that M(N)² log M(N)/N tends to zero. In this model the unbiased Gibbs state in volume N can essentially be decomposed into M(N) pairs of disjoint measures. We investigate the distributions of the corresponding weights,and show,in particular, that these weights concentrate for any given N very closely to one of the pairs, with probability tending to 1. Our analysis is based upon a new result on the asymptotic distribution of order statistics of certain correlated exchangeable random variables.

Consider the system of particles on Z^d where particles are of two types, A and B, and execute simple random walks in continuous time. Particles do not interact with their own type, but when a type A particle meets a type B particle, both disappear. Initially, particles are assumed to be distributed according to homogeneous Poisson random fields, with equal intensities for the two types. This system serves as a model for the chemical reaction A ∑ B → inert. In Bramson and Lebowitz [7], the densities of the two types of particles were shown to decay asymptotically like 1/t^d/4 for d<4 and 1/t for d > 4, as t → ∞. This change in behavior from low to high dimensions corresponds to a change in spatial structure. In d<4, particle types segregate, with only one type present locally. After suitable rescaling, the process converges to a limit, with density given by a Gaussian process. In d>4, both particle types are, at large times, present locally in concentrations not depending on the type, location or realization. In d=4, both particle types are present locally, but with varying concentrations. Here, we analyze this behavior in d<4; the behavior for d=4 will be handled in a future work by the authors.

We study a nonlinear filtering problem in which the signal to be estimated is conditioned by the observations. The main results establish pathwise uniqueness for the unnormalized filter equation and uniqueness in law for the normalized and unnormalized filter equations.

We address the maximization problem of expected utility from terminal wealth. The special feature of this paper is that we consider a financial market where price process of risky assets follows a stochastic volatility model and we require that investors observe just the vector of stock prices. Using stochastic filtering techniques and adapting martingale duality methods in this partially observed incomplete model, we characterize the value function and the optimal portfolio policies. We study in detail the Bayesian case, when risk premia of the stochastic volatility model are unobservable random variables with known prior distribution. We also consider the case of unobservable risk premia modelled by linear Gaussian processes.

We address the problem of determining in an optimal way the sequence of times at which a firm can enter or exit an economic activity. In particular, we consider an investment model which involves production scheduling as well as a sequence of entry and exit decisions. The pricing of an investment conforming with this model gives rise to a stochastic impulse control problem that we explicitly solve. Our solution takes qualitatively different forms, depending on the problem's data.

We find a Lyapunov-type sufficient condition for discrete-time Markov chains on a countable state space including an absorbing set to almost surely reach this absorbing set and to asymptotically stabilize conditional on nonabsorption. This result is applied to Bienaymè-Galton-Watson-like branching processes in which the offspring distribution depends on the current population size. This yields a generalization of the Yaglom limit. The techniques used mainly rely on the spectral theory of linear operators on Banach spaces and especially on the notion of quasi-compact linear operator.

We consider a process in which information is transmitted from a given root node on a noisy -dary tree network T. We start with a uniform symbol taken from an alphabet \(\mathcal{A}\). Each edge of the tree is an independent copy of some channel (Markov chain) M, where M is irreducible and aperiodic on \(\mathcal{A}\). The goal is to reconstruct the symbol at the root from the symbols at the nth level of the tree. This model has been studied in information theory, genetics and statistical physics. The basic question is: is it possible to reconstruct (some information on)the root? In other words, does the probability of correct reconstruction tend to \(1 /{\mathcal{A}}\) as n →∞?

It is known that reconstruction is possible if dλ²₂(M) > 1, where λ₂(M) is the second eigenvalue of M. Moreover,in this case it is possible to reconstruct using a majority algorithm which ignores the location of the data at the boundary of the tree. When M is a symmetric binary channel, this threshold is sharp. In this paper we show that, both for the binary asymmetric channel and for the symmetric channel on many symbols, it is sometimes possible to reconstruct even when dλ²₂(M) < 1. This result indicates that, for many (maybe most) tree-indexed Markov chains, the location of the data on the boundary plays a crucial role in reconstruction problems.