Title: Asymptotic Theory and Applications of Random Functions
Description: Random functions is the central component in many statistical and probabilistic problems. This dissertation presents theoretical analysis and computation for random functions and its applications in statistics. This dissertation consists of two parts. The first part is on the topic of classic continuous random fields. We present asymptotic analysis and computation for three non-linear functionals of random fields. In Chapter 1, we propose an efficient Monte Carlo algorithm for computing P{sup_T f(t)>b} when b is large, and f is a Gaussian random field living on a compact subset T. For each pre-specified relative error ɛ, the proposed algorithm runs in a constant time for an arbitrarily large $b$ and computes the probability with the relative error ɛ. In Chapter 2, we present the asymptotic analysis for the tail probability of ∫_T e^{σf(t)+μ(t)}dt under the asymptotic regime that σ tends to zero. In Chapter 3, we consider partial differential equations (PDE) with random coefficients, and we develop an unbiased Monte Carlo estimator with finite variance for computing expectations of the solution to random PDEs.

Moreover, the expected computational cost of generating one such estimator is finite. In this analysis, we employ a quadratic approximation to solve random PDEs and perform precise error analysis of this numerical solver. The second part of this dissertation focuses on topics in statistics. The random functions of interest are likelihood functions, whose maximum plays a key role in statistical inference. We present asymptotic analysis for likelihood based hypothesis tests and sequential analysis. In Chapter 4, we derive an analytical form for the exponential decay rate of error probabilities of the generalized likelihood ratio test for testing two general families of hypotheses. In Chapter 5, we study asymptotic properties of the generalized sequential probability ratio test, the stopping rule of which is the first boundary crossing time of the generalized likelihood ratio statistic. We show that this sequential test is asymptotically optimal in the sense that it achieves asymptotically the shortest expected sample size as the maximal type I and type II error probabilities tend to zero.

These results have important theoretical implications in hypothesis testing, model selection, and other areas where maximum likelihood is employed.