ABSTRACT
Continuing the study initiated in Chapter 9 of Book II, this chapter again has two main themes. The first theme is large deviation theory, which starts off summarizing the main result and open questions of Book II. The section then introduces and exemplifies the Chernoff bound, which requires the existence of the moment generating function. Following an analysis of properties of this bound, and introducing tilted distributions and their relevant properties, the section concludes with the Cramér-Chernoff theorem, which conclusively settles the open questions of Book II. The second major section is on extreme value theory and focuses on two matters. The first is a study of the Hill estimator for the extreme value index γ for γ>0, the index values most commonly encountered in finance applications. This estimator is introduced and exemplified in the context of Pareto distributions, and the Hill result of convergence with probability 1 derived, along with a variety of related results. For this, earlier developments on order statistics will play a prominent role, as does a representation theorem of Karamata. The second major investigation is into the Pickands-Balkema-de Haan theorem, which is a result on the limiting distribution of certain conditional tail distributions. This final result was approximated in the Book II development, but here can be derived in detail with another representation theorem of Karamata. By example, it is then shown that the convergence promised by this result need not be fast.
