A new approach to multivariate extreme value theory : f-implicit max-infinitely divisible distributions and f-implicit max-stable processes

摘要

Let X1,...,Xn be independent and identically distributed random vectors in R^d and f:R^d -> [0,∞) a suitable function being referred to as the loss function. Further, let k(n) = argmax(f(X1),…, f(Xn)). Referring to [SchSt14, Definition 4.1], recall that a random vector X in R^d is f-implicit max-stable if for all n >= 1 there exist an>0 such that an^{-1}X{k(n)} and X are equal in distribution, with X1,...,Xn being independent copies of X. Now, the aim is to expand on this notion and to advance it. To this end, a new mathematical framework called f-implicit extreme value theory, which is closely related to multivariate extreme value theory but yet different as to the study of extremes, is developed. More precisely, adopting the approach suggested in [SchSt14], the idea of focusing on extreme loss events rather than extreme values is pursued. The motivation behind this stems from some kind of inverse problem where one wants to determine the extremal behavior of an R^d-valued random vector X when only explicitly observing the extremal loss f(X). In the first part of the thesis, some basics constituting the fundament of all further deliberations are introduced. In particular, this includes a specific (inner) binary operation on R^d called f-implicit max-operation, an astute convolution concept being referred to as f-implicit max-convolution and a distinctive partial order named f-implicit max-order. Finally, various possibilities to estimate the distribution of the f-implicit maximum X{k(n)} of X1,...,Xn are provided.Equipped with these aspects, the notion of f-implicit max-infinite divisibility is developed, thus extending the class of f-implicit max-stable distributions. Here, it is proved that all random vectors coming under one of two specific classes of random vectors are f-implicit max-infinitely divisible. To this end, the notion of f-implicit max-convolution semigroups is applied.The third part of the thesis deals with the class of f-implicit max-stable processes being the analogue of max-stable processes. In order to provide non-trivial examples of such processes, the ingenious concepts of f-implicit sup-measures and f-implicit extremal integrals are established.The thesis concludes with several suggestions for additional research possibilities which might refine the novel field of f-implicit extreme value theory.