Some notes on multivariate generalized Pareto distributions

The investigation of multivariate generalized Pareto distributions (GPDs) has begun only recently and there are slightly varying definitions of GPDs available. In this article we investigate the one from Section 5.1 of Falk et al. [Laws of Small Numbers: Extremes and Rare Events, second ed., Birkhäuser, Basel, 2004], which does not differ in the area of interest from those of other authors. We first give an interpretation of the case of independence in terms of the peaks-over-threshold approach. This case is also used in dimension d = 3 by Falk et al. [Laws of Small Numbers: Extremes and Rare Events, second ed., Birkhäuser, Basel, 2004] as a counterexample to show that GP functions are not necessarily distribution functions on their entire support. We generalize this counterexample to an arbitrary dimension d ≥ 3 and demonstrate also that other GP functions show this behavior. Finally we show that different GPDs can lead to the same conditional probability measure in the area of interest.