| This dissertation consists of results in two distinct areas of probability theory. One is the extreme value theory, the other is the central limit theorem..;In the extreme value theory, the focus is on max-stable processes. Such processes play an increasingly important role in characterizing and modeling extremal phenomena in finance, environmental sciences and statistical mechanics. In particular, the association of sum- and max-stable processes and the decomposability of sum-and max-stable processes are investigated. Besides, the conditional distributions of max-stable processes are also studied, and a computationally efficient algorithm is developed. This algorithm has many potential applications in prediction of extremal phenomena.;In the central limit theorem, the asymptotic normality for partial slims of stationary random fields is studied, with a focus on the projective conditions on the dependence. Such conditions, easy to check for many stochastic processes and random fields, have recently drawn many attentions for (one-dimensional) time series models in statistics and econometrics. Here, the focus is on (high-dimensional) stationary random fields. In particular, general central limit theorems for stationary random fields and orthomartingales are established. The method is then extended to establish the asymptotic normality for the kernel density estimator of linear random fields. |
