Publications de l'Institut Mathématique, Nouvelle Série Vol. 80(94), pp. 7–27 (2006) 

CYLINDER SYMMETRIC MEASURES WITH THE TAIL PROPERTYGuus BalkemaDepartment of Mathematics, University of Amsterdam, Amsterdam, The NetherlandsAbstract: A Pareto distribution has the property that any tail of the distribution has the same shape as the original distribution. The exponential distribution and the uniform distribution have the tail property too. The tail property characterizes the univariate generalized Pareto distributions. There are three classes of univariate GPDs: Pareto distributions, power laws, and the exponential distribution. All these distributions extend to infinite measures. The tail property translates into a group of symmetries for these infinite measures: translations for the exponential law; multiplications for the Pareto and power laws. In the multivariate case, for cylinder symmetric measures in dimension $d\ge3$, there are seven classes of measures with the tail property, corresponding to five symmetry groups. The second part of this paper establishes this classification. The first part introduces the probabilistic setting, and discusses the associated geometric theory of multiparameter regular variation. We prove a remarkable result about a class of multiparameter slowly varying functions introduced in Ostrogorski [1995]. Keywords: exceedance; exponential family; flat; Lie group; Pareto; regular variation; risk scenario Classification (MSC2000): 26A12; 60B10; 26A12 Full text of the article: (for faster download, first choose a mirror)
Electronic fulltext finalized on: 10 Oct 2006. This page was last modified: 4 Dec 2006.
© 2006 Mathematical Institute of the Serbian Academy of Science and Arts
