In this paper we study the convergence of stochastic processes related to a random partial differential equation (PDE with random coefficients) of heat equation propagation type in a Kolmogorov's random velocity field. Then we are able to improve the results of Avellanda and Majda in the case of "shear-flow" advection-diffusion because we prove a convergence in law of the solution of the RPDE instead of just convergence of the moments.

Extreme value theory for a class of stochastic volatility models, in which the logarithm of the conditional variance follows a Gaussian linear process, is developed. A result for the asymptotic tail behavior of the transformed stochastic volatility process is established and used to prove that the suitably normalized extremes converge in distribution to the double exponential (Gumbel) distribution. Explicit normalizing constants are obtained, and point process convergence is discussed.

Transaction costs preclude the construction of hedging strategies for general contingent claims. Leland introduced the concept of diffusion limits of hedging strategies in a small transaction cost limit. This paper establishes such diffusion limits for very general hedging strategies and also establishes leading order asymptotic expressions for the replication error. In addition to subsuming previously considered temporal strategies, the results in this paper yield new results, namely expressions for replication errors of stock price strategies and a variety of "renewal" strategies. Most importantly, this paper provides a unified methodology for calculating hedging strategies and replication errors in the small transaction cost limit. This is an essential component of optimization methods, when, for example one is trying to minimize replication error for a given initial portfolio value.

Let D be the Wiener sausage of width $\varepsilon$ around two-sided Brownian motion. The components of two-dimensional reflected Brownian motion in D converge to one-dimensional Brownian motion and iterated Brownian motion, respectively, as $\varepsilon$ goes to 0.

The probability that a stochastic process with negative drift exceed a value a often has a renewal-theoretic approximation as $a \to \infty$. Except for a process of iid random variables, this approximation involves a constant which is not amenable to analytic calculation. Naive simulation of this constant has the drawback of necessitating a choice of finite a, thereby hurting assessment of the precision of a Monte Carlo simulation estimate, as the effect of the discrepancy between a and $\infty$ is usually difficult to evaluate.

Here we suggest a new way of representing the constant. Our approach enables simulation of the constant with prescribed accuracy. We exemplify our approach by working out the details of a sequential power one hypothesis testing problem of whether a sequence of observations is iid standard normal against the alternative that the sequence is AR(1). Monte Carlo results are reported.

It is widely accepted that the Gaussian assumption is too restrictive to model either financial or some important macroeconomic variables, because their distributions exhibit asymmetry and heavy tails. In this paper we develop the asymptotic theory for econometric cointegration processes under the assumption of infinite variance innovations with different distributional tail behavior. We extend some of the results of Park and Phillips which were derived under the assumption of finite variance errors.

This study introduces a Markov network process called a string-net. Its state is the vector of quantities of customers or units that move among the nodes, and a transition of the network consists of a string of instantaneous vector increments in the state. The rate of such a string transition is a product of a transition-initiation rate and a string-generation rate. The main result characterizes the stationary distribution of a string-net. Key parameters in this distribution satisfy certain "polynomial traffic equations" involving the string-generation rates. We identify sufficient conditions for the existence of a solution of the polynomial equations, and we relate these equations to a partial balance property and throughputs of the network. Other results describe the stationary behavior of a large class of string-nets in which the vectors in the strings are unit vectors and a string-generation rate is a product of Markov routing probabilities. This class includes recently studied open networks with Jackson-type transitions augmented by transitions in which a signal (or negative customer) deletes units at nodes in one or two stages. The family of string-nets contains essentially all Markov queueing network processes, aside from reversible networks, that have known formulas for their stationary distributions. We discuss old and new variations of Jackson networks with batch services, concurrent or multiple-unit movements of units, state-dependent routings and multiple types of units and routes.

This paper is concerned with dynamic scheduling in a queueing system that has two independent Poisson input streams, two servers, deterministic service times and linear holding costs. One server can process both classes of incoming jobs, but the other can process only one class, and the service time for the shared job class is different depending on which server is involved. A bound on system performance is developed in terms of a single pooled resource, or super-server, whose capabilities combine those of the original two servers. Thereafter, attention is focused on the heavy traffic regime, where the combined capacity of the two servers is approximately equal to the total input rate. We construct a discrete-review control policy and show that if its parameters are chosen correctly as one approaches the heavy traffic limit, then its cost performance approaches the bound associated with a single pooled resource. Thus the discrete-review policy is proved to be asymptotically optimal in the heavy traffic limit. Although resource pooling in heavy traffic has been observed to occur in other network scheduling problems, there have been very few studies that rigorously proved the pooling phenomenon, or that proved the asymptotic optimality of a specific policy. Our discrete-review policy is obtained by applying a general method, called the BIGSTEP method in an earlier paper, to the parallel-server model.

This paper develops bounds on the distribution function of the empirical mean for irreducible finite-state Markov chains. One approach, explored by Gillman, reduces this problem to bounding the largest eigenvalue of a perturbation of the transition matrix for the Markov chain. By using estimates on eigenvalues given in Kato's book Perturbation Theory for Linear Operators, we simplify the proof of Gillman and extend it to nonreversible finite-state Markov chains and continuous time. We also set out another method, directly applicable to some general ergodic Markov kernels having a spectral gap.

This paper concerns the asymptotic distributions of "tail array" sums of the form $\Sigma \Psi_n (X_i - u_n)$ where ${X_i}$ is a strongly mixing stationary sequence, $\Psi_n$ are real functions which are constant for negative arguments, $\Psi_n (x) = \Psi_n (X_+)$ and ${u_n}$ are levels with $u_n \to \infty$. Compound Poisson limits for rapid convergence of $u_n \to \infty (nP{X_1 > u_n} \to \tau < \infty)$ are considered briefly and a more detailed account given for normal limits applicable to slower rates $(nP(X_1 > u_n) \to \infty)$. The work is motivated by (1) the modeling of "damage" due to very high and moderately high extremes and (2) the provision of probabilistic theory for application to problems of "tail inference" for stationary sequences.

We derive a general asymptotic formula for the variance of the number of maxima in a set of independent and identically distributed random vectors in $\mathbb{R}^d$, where the components of each vector are independently and continuously distributed. Applications of the results to algorithmic analysis are also indicated.

For a simulated annealing process $X_t$ on S with transition rates $q_{ij}(t) = p_{ij} \exp (-(U(i, j))/T(t))$ where $i, j \epsilon S$ and $T(t) \downarrow 0$ in a suitable way, we study the exit distribution $P_{t,i}(X_{\tau} = a)$ and mean exit time $E_{t,i}(\tau)$ of $X_t$ from a cycle c as $t \to \infty$. A cycle is a particular subset of S whose precise definition will be given in Section 1. Here $\tau$ is the exit time of the process from c containing i and a is an arbitrary state not in c. We consider the differential (backward) equation of $P_{t,i}(X_{\tau} = a)$ and $E_{t,i}(\tau)$ and show that $\lim_{t\to\infty}P_{t,i}(X_{\tau} = a)/\exp (-U(c, a) - T(t))$ and $\lim_{t\to\infty E_{t,i}(\tau)/\exp(V(c)/T(t))$ exist and are independent of $i \epsilon c$. The constant $(U(c, a))$ is usually referred to as the cost from c to a and $V(c), (\leq U(c, a))$ is the minimal cost coming out of c. We also obtain estimates of $|P_{t,i}(X_{\tau} = a) - P_{t,j}(X_{\tau} = a)|$ and $|E_{t,i}(\tau)|$ for $i \not= j$ as $t \to \infty$. As an application, we shall show that similar results hold for the family of Markov processes with transition rates $q_{ij}^{\varepsilon} = p_{ij} \exp(-U(i, j)/\varepsilon)$ where $\varepsilon > 0$ is small.

We derive upper and lower bounds for the spectral gap of the random energy model under Metropolis dynamics which are sharp in exponential order. They are based on the variational characterization of the gap. For the lower bound, a Poincaré inequality derived by Diaconis and Stroock is used. The scaled asymptotic expression is a linear function of the temperature. The corresponding function for a global version of the dynamics exhibits phase transition instead.

We also study the dependence of lower order terms on the volume. In the global dynamics, we observe a phase transition. For the local dynamics, the expressions we have, which are possibly not sharp, do not change their order of dependence on the volume as the temperature changes.

We consider a percolation configuration on a general lattice in which edges are included independently with probability p. We study the rigidity properties of the resulting configuration, in the sense of generic rigidity in d dimensions. We give a mathematically rigorous treatment of the problem, starting with a definition of an infinite rigid component. We prove that, for a broad class of lattices, there exists an infinite rigid component for some p strictly below unity. For the particular case of two-dimensional rigidity on the two-dimensional triangular lattice, we prove first that the critical probability for rigidity percolation lies strictly above that for connectivity percolation and second that the infinite rigid component (when it exists) is unique for all but countably many values of p. We conjecture that this uniqueness in fact holds for all p. Some of our arguments could be applied to two-dimensional lattices in more generality.