Geodesic
Cylinder Symmetric Measures With the Tail Property
Publications de l'Institut Mathématique, _N_S_80 (2006) no. 94, p. 7

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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].
DOI : 10.2298/PIM0694007B
Classification : 26A12 60B10 26A12
Keywords: exceedance, exponential family, flat, Lie group, Pareto, regular variation, risk scenario 
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Guus Balkema. Cylinder Symmetric Measures With the Tail Property. Publications de l'Institut Mathématique, _N_S_80 (2006) no. 94, p. 7 . doi: 10.2298/PIM0694007B

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