Extremal Properties Of Markov Chains And The Conditional Extreme Value Model

Multivariate extreme value theory has proven useful for modeling multivariate data in fields such as finance and environmental science, where one is interested in accounting for the tendency of observations to exceed an extremely high (or low) threshold. Recent work has developed extremal models by studying the conditional distribution of a random vector, conditional on one of the components becoming extreme. This provides a way to handle situations such as asymptotic dependence, where traditional techniques may be uninformative. In this thesis, we explore the implications of the assumption that such a conditional distribution is well approximated by a limiting probability distribution when the conditioning component is extreme. We consider a version of the conditional distribution specified by a transition function. If the transition kernel of a Markov chain satisfies our assumption, then a process known as the tail chain approximates the Markov chain over extreme states. We characterize the class of chains which admit such an approximation, and investigate the properties of the tail chain in relation to the distinction between extreme and non-extreme states. We find that, in general, the tail chain approximates a portion of the original process we term the "extremal component". We further derive the limit in distribution of a point process consisting of normalized Markov chain observations, expressing the limit in terms of the tail chain. We also consider the case where a transition function satisfying our assumption describes the dependence structure of a random vector. We establish conditions under which a conditional extreme value model is appropriate, and derive the form of the limiting measure