We extend certain exponential decay results of subcritical percolation to a class of locally dependent random graphs, introduced by Kuulasmaa as models for spatial epidemics on $\mathbb{Z}^d$. In these models, infected individuals eventually die (are removed) and are not replaced. We combine these results with certain continuity and rescaling arguments in order to improve our knowledge of the phase diagram of a modified epidemic model in which new susceptibles are born at some positive rate. In particular, we show that, throughout an intermediate phase where the infection rate lies between two certain critical values, no coexistence is possible for sufficiently small positive values of the recovery rate. This result provides a converse to results of Durrett and Neuhauser and Andjel and Schinazi. We show also that such an intermediate phase indeed exists for every $d \geq 1$ (i.e., that the two critical values mentioned above are distinct). An important technique is the general version of the BK inequality for disjoint occurrence, proved in 1994 by Reimer.

An interacting particle model for load transferring in parallel architectures is defined. In the case of an infinite lattice the model is proved to be ergodic and to converge exponentially fast to its equilibrium. When the architecture is that of a complete graph, the total number of loads behaves as a birth and death process, and explicit upper bounds on the benefits that can be expected from a transferring policy are derived. Experimental results for different types of architectures are presented and compared to the solution of the mean field equations. There is fairly good agreement between the two for quantities of practical interest.

Consider a reflected random walk $W_{n+1} = (W_n + X_n)^+$, where $X_0, X_1,\dots$ are i.i.d. with negative mean and subexponential with common distribution F. It is shown that the probability that the maximum within a regenerative cycle with mean $\mu$ exceeds x is approximately $\mu\bar{F}(x)$ as $x \to \infty$, and thereby that $\max(W_0, \dots, W_n)$ has the same asymptotics as $\max(X_0, \dots, X_n)$ as $n \to \infty$. In particular, the extremal index is shown to be $\theta = 0$, and the point process of exceedances of a large level is studied. The analysis extends to reflected Lévy processes in continuous time, say, stable processes. Similar results are obtained for a storage process with release rate $r(x)$ at level x and subexponential jumps (here the extremal index may be any value in $[0, \infty]$; also the tail of the stationary distribution is found. For a risk process with premium rate $r(x)$ at level x and subexponential claims, the asymptotic form of the infinite-horizon ruin probability is determined. It is also shown by example $[r(x) = a + bx$ and claims with a tail which is either regularly varying, Weibull- or lognormal-like] that this leads to approximations for finite-horizon ruin probabilities. Typically, the conditional distribution of the ruin time given eventual ruin is asymptotically exponential when properly normalized.

We investigate how to tune a generalized simulated annealing algorithm with piecewise constant cooling schedule to get an optical convergence exponent. The optimal convergence exponent of generalized simulated annealing algorithms has been computed by Catoni and Trouvé. It is reached only with triangular sequences of temperatures, meaning that different finite sequences are used, depending on the time resource available for computations (expressed by an overall number of iterations). We show first that it is possible to get close to the optimal convergence exponent uniformly over suitably bounded families of energy landscapes using a fixed number of temperature steps. Then we show that, letting the number of steps increase with the time resource, we can build a cooling schedule which is universally robust with respect to the convergence exponent: a fixed triangular sequence of temperatures gives an optimal convergence exponent for any energy landscape. Piecewise constant temperature sequences are often used in practice: in favourable cases, the use of the same temperature during a large number of iterations allows tabulating the exponential penalties appearing in the transition matrix, thus sparing a significant amount of computer time. The proofs we give rely on Freidlin and Wentzell's closed formulas for the exit time and point from subdomains of time homogeneous Markov chains.

Theoretical work on Markov chain Monte Carlo (MCMC) algorithms has so far mainly concentrated on the properties of simple algorithms, such as the Gibbs sampler, or the full-dimensional Hastings-Metropolis algorithm. In practice, these simple algorithms are used as building blocks for more sophisticated methods, which we shall refer to as hybrid samplers. It is often hoped that good convergence properties (e.g., geometric ergodicity, etc.) of the building blocks will imply similar properties of the hybrid chains. However, little is rigorously known.

In this paper, we concentrate on two special cases of hybrid samplers. In the first case, we provide a quantitative result for the rate of convergence of the resulting hybrid chain. In the second case, concerning the combination of various Metropolis algorithms, we establish geometric ergodicity.

The paper presents a method of computing the extremal index of a real-valued, higher-order (kth-order, $k \geq 1$) stationary Markov chain ${X_n}$. The method is based on the assumption that the joint distribution of $k +1$ consecutive variables is in the domain of attraction of some multivariate extreme value distribution. We introduce limiting distributions of some rescaled stationary transition kernels, which are used to define a new $k -1$th-order Markov chain ${Y_n}$, say. Then, the kth-order Markov chain ${Z_n}$ defined by $Z_n = Y_1 + \dots + Y_n$ is used to derive a representation for the extremal index of ${X_n}$. We further establish convergence in distribution of multilevel exceedance point processes for ${X_n}$ in terms of ${Z_n}$. The representations for the extremal index and for quantities characterizing the distributional limits are well suited for Monte Carlo simulation.

In the paper we study interacting particle approximations of discrete time and measure-valued dynamical systems. These systems have arisen in such diverse scientific disciplines as physics and signal processing. We give conditions for the so-called particle density profiles to converge to the desired distribution when the number of particles is growing. The strength of our approach is that is applicable to a large class of measure-valued dynamical systems arising in engineering and particularly in nonlinear filtering problems. Our second objective is to use these results to solve numerically the nonlinear filtering equation. Examples arising in fluid mechanics are also given.

Using the Malliavin calculus, we give a unified treatment of the so-called perturbation analysis of dynamic systems. Several applications are also given.

An on-off process is a 0-1 process $\xi_t$ in which consecutive 0-periods ${T_{0, n}}$ alternate with 1-periods ${T_{1, n}}(n = 1, 2,\dots)$. The on and off time sequences are independent, each consisting of i.i.d. r.v.s. By the superposed flow, we mean the process $Z_t = \Sigma_{\ell=1}^N r^{\ell}\xi_t^{\ell}$, where $r^{\ell} > 0$ and ${\xi_t^1}, \dots,{\xi_t^N}$ are independent on-off flows. The process $\xi_t^{\ell}$ is not Markovian; however, with the age component $\eta_t^{\ell}$, the process $w_t^{\ell} = (\xi_t^{\ell}, \eta_t^{\ell})$ is a piecewise deterministic Markov process. In this paper we study the buffer content process for which the tail of its steady-state distribution $\Psi (b)$ fulfills inequality $C_- e^{\gammab} \leq \Psi (b) \leq C_+ e^{-\gammab}, where $\gamma > 0$ is the solution of some basic nonlinear system of equations.

This study establishes limiting distributions for customer waiting times and queue lengths in treelike networks with single-server nodes. The main result characterizes the limiting distributions when the network data (interarrival times, service times and routes) is "asymptotically stationary." This is a weak condition covering a variety of networks including standard ones where the network data is stationary, regenerative, Markovian, satisfies coupling, and so on. The dependencies in the network data may be customer centered or node centered. The proof is based on two preliminary results that are of interest by themselves. The first one justifies the existence of the waiting time and queue length processes on the entire time axis for any network whose service capacity has been adequate to handle all the customers as one looks back to the "beginning of time." This is a sample-path generalization of a result of Loynes for a queueing system with stationary data. The second preliminary result is a characterization of functionals of sequences that preserve the asymptotic stationarity property. This is somewhat analogous to continuous-mapping principles for weak convergence. We also present functional central limit theorems for the waiting time processes in a network when the partial sums of the network data obey a heavy-traffic functional limit property. The limiting waiting time sequence is a functional of a process that is typically a multivariate Brownian motion, or a process with stationary increments and long range dependence such as a fractional Brownian motion.

In a state-dependent queueing network, arrival and service rates, as well as routing probabilities, depend on the vector of queue lengths. For properly normalized such networks, we derive functional laws of large numbers (FLLNs) and functional central limit theorems (FCLTs). The former support fluid approximations and the latter support diffusion refinements.

The fluid limit in FLLN is the unique solution to a multidimensional autonomous ordinary differential equation with state-dependent reflection. The diffusion limit in FCLT is the unique strong solution to a stochastic differential equation with time-dependent reflection.

Examples are provided that demonstrate how such approximations facilitate the design, analysis and optimization of various manufacturing, service, communication and other systems.