Ï㽶ÊÓƵ

Title: Univariate and multivariate extremes of extendible random vectors

Abstract:

In its most common form extreme value theory is concerned with the limiting distribution of location-scale transformed block-maxima Mn=max(X1,…,Xn) of a sequence of identically distributed random variables (Xi), i≥1. In case the members of the sequence (Xi) are independent, the weak limiting behaviour of Mn is adequately described by the classical Fisher-Tippett-Gnedenko theorem. In this presentation we are interested in the case of dependent random variables (Xi) while retaining a common marginal distribution function F for all Xi, i∈ℕ. Complementary to the well established extreme value theory in a time series setting we consider a framework in which the dependence between (extreme) events does not decay over time. This approach is facilitated by highlighting the connection between block-maxima and copula diagonals in an asymptotic context. The main goal of this presentation is to discuss a generalization of the Fisher–Tippett–Gnedenko theorem in this setting, leading to limiting distributions that are not in the class of generalized extreme value distributions. This result is exemplified for popular dependence structures related to extreme value, Archimedean and Archimax copulas. Focusing on the class of hierarchical Archimedean copulas the results can further be extended to the multivariate setting. Finally, we illustrate the resulting limit laws and discuss their properties.

(it is joint work with Marius Hofert and Johanna G. Neslehova).

Klaus Herrmann is a Postdoctoral Fellow in the Department of Mathematics and Statistics at McGill. His supervisor is Professor Johanna Neslehova.