On a basis for 'Peaks over Threshold' modeling

'Peaks over Thresholds' ('POT') models commonly used e.g. in hydrology, assume that peak values of an i.i.d. or stationary sequence Xi above a high value u, occur at Poisson points, and the excess values of the peak above u are independent with an arbitrary common d.f. G. Motivation for these models has been provided by Smith (1985, 1987), by using Pareto-type approximations of Pickands (1975) for distributions of such excess values. These works strongly suggest that the Pareto family provides the appropriate class of distributions G for the POT model. In the present paper we consider the point process of excess values of peaks above a high level u and demonstrate that this converges in distribution to a Compound Poisson Process as u-->[infinity] under appropriate assumptions. It is shown that the multiplicity distribution of this limit (i.e. the limiting distribution of excess values of peaks) must belong to the Pareto family and detailed forms are given for the normalizing constants involved. This exhibits the POT model specifically as a limit for the point process of excesses of peaks and delineates the distributions involved.