Thesis (Ph. D.)--University of Chicago, Department of Statistics, August 2003. / Includes bibliographical references. Also available on the Internet.

Large deviation theory has experienced much development and interest in the last two decades. A large deviation principle is the exponential decay of the probability of increasingly rare events and the computation of a rate or entropy function which measures the rate of decay. Within the probability literature there has been much use made of these rates in diverse applications. These large deviation principles have been discovered for independent and identically distributed random variables, as well as random vectors and these have been extended to some cases of weak dependence. In this thesis we prove large deviation principles for finite dimensional distributions of scaling limits of random measures. Functional approaches to large deviation theory using test functions as dual objects to random measures are also developed. These results are applied to some important classes of models, in particular Poisson point processes, Poisson center cluster processes and doubly stochastic point processes. / Graduation date: 1991

The theory of large deviations can help to shed light on systems in non-equilibrium statistical mechanics and, more generically, on non-reversible stochastic processes. For this purpose, we target trajectories in space time rather than static configurations and study time-extensive observables. This suggests that the details of the evolution law such as the presence of time correlations take on a major role. In this thesis, we investigate selected models with stochastic dynamics that incorporate memory by means of different mechanisms, devise a numerical approach for such models, and quantify to what extent the memory affects the large deviation functionals. The results are relevant for real-world situations, where simplified memoryless (Markovian) models may not always be appropriate. After an original introduction to the mathematics of stochastic processes, we explore, analytically and numerically, an open-boundary zero-range process which incorporates memory by means of hidden variables that affect particle congestion. We derive the exact solution for the steady state of the one-site system, as well as a mean-field approximation for larger one-dimensional lattices. Then, we focus on the large deviation properties of the particle current in such a system. This reveals that the time correlations can be apparently absorbed in a memoryless description for the steady state and the small fluctuation regime. However, they can dramatically alter the probability of rare currents. Different regimes are separated by dynamical phase transitions. Subsequently, we address systems in which the memory cannot be encoded in hidden variables or the waiting-time distributions depend on the whole trajectory. Here, the difficulty in obtaining exact analytical results is exacerbated. To tackle these systems, we have proposed a version of the so-called 'cloning' algorithm for the evaluation of large deviations that can be applied consistently for both Markovian and non-Markovian dynamics. The efficacy of this approach is confirmed by numerical results for some of the rare non-Markovian models whose large deviation functions can be obtained exactly. We finally adapt this machinery to a technological problem, specifically the performance evaluation of communication systems, where temporal correlations and large deviations are important.

Thesis (Ph. D.)--Oregon State University, 1991. / Typescript (photocopy). Includes bibliographical references (leaves 71-73). Also available on the World Wide Web.

by Ma Yiguang. / "April 2000." / Thesis (Ph.D.)--Chinese University of Hong Kong, 2000. / Includes bibliographical references (p. 117-[124]). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Mode of access: World Wide Web. / Abstracts in English and Chinese.

Understand the behavior of queueing networks in heavy tra c is very important due to its importance in evaluating the network performance in related applications. However, in many cases, the stationary distributions of such networks are intractable. Based on di usion limits of queueing networks, we can use Re ected Brownian Motion (RBM) processes as reasonable approximations. As such, we are interested in obtaining the stationary distribution of RBM. Unfortunately, these distributions are also in most cases intractable. However, the tail behavior (large deviations) of RBM may give insight into the stationary distribution. Assuming that a large deviations principle holds, we need only solve the corresponding variational problem to obtain the rate function. Our research is mainly focused on how to solve variational problems in the case of rotationally symmetric (RS) data. The contribution of this dissertation primarily consists of three parts. In the rst part we give out the speci c stability condition for the RBM in the octant in the RS vi case. Although the general stability conditions for RBM in the octant has been derived previously, we simplify these conditions for the case we consider. In the second part we prove that there are only two types of possible solutions for the variational problem. In the last part, we provide a simple computational method. Also we give an example under which a spiral path is the optimal solution. / text

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Thesis (Ph. D.)--University of Texas at Austin, 2007. / Vita. Includes bibliographical references.

We explore the effect of multiscale structure on weakly interacting diffusions through two main projects. In the first, we consider a collection of weakly interacting diffusion processes moving in a two-scale locally periodic environment. We study the large deviations principle of the empirical distribution of the particles' positions in the combined limit as the number of particles grow to infinity and the time-scale separation parameter goes to zero simultaneously. We make use of weak convergence methods providing a convenient representation for the large deviations rate function, which allow us to characterize the effective controlled mean field dynamics. In addition, we obtain equivalent representations for the large deviations rate function of the form of Dawson-G\"artner which hold even in the case where the diffusion matrix depends on the empirical measure and when the particles undergo averaging in addition to the propagation of chaos. In the second, we consider a fully-coupled slow-fast system of McKean-Vlasov SDEs with full dependence on the slow and fast component and on the law of the slow component and derive convergence rates to its homogenized limit. We do not make periodicity assumptions, but we impose conditions on the fast motion to guarantee ergodicity. In the course of the proof we obtain related ergodic theorems and we gain results on the regularity of Poisson type of equations and of the associated Cauchy-Problem on the Wasserstein space that are of independent interest.

We study the totally asymmetric exclusion process on the positive integers with a single particle source at the origin. Liggett (1975) has shown that the long term behaviour of this process has a phase transition: If the particle production rate at the source and the initial density are below certain critical values, the stationary measure is a product measure, otherwise the stationary measure is spatially correlated. Following the approach of Derrida et al. (1993) it was shown by Grosskinsky (2004) that these correlations can be described by means of a matrix product representation. In this thesis we derive a large deviation principle with explicit rate function for the particle density in a macroscopic box based on this representation. The novel and rigorous technique we develop for this problem combines spectral theoretical and combinatorial ideas and has the potential to be applicable to other models described by matrix products.