  Multivariate peaks over thresholds modeling based on generalized Pareto
distributions has up to now only been used in few and mostly 2-dimensional
situations. This paper contributes theoretical understanding, physically based
models, inference tools, and simulation methods to support routine use, with an
aim at higher dimensions. We derive a general point process model for extreme
episodes in data, and show how conditioning the distribution of extreme
episodes on threshold exceedance gives four basic representations of the family
of generalized Pareto distributions. The first representation is constructed on
the real scale of the observations. The second one starts with a model on a
standard exponential scale which then is transformed to the real scale. The
third and fourth are reformulations of a spectral representation proposed in A.
Ferreira and L. de Haan [Bernoulli 20 (2014) 1717--1737]. Numerically tractable
forms of densities and censored densities are found and give tools for flexible
parametric likelihood inference. New simulation algorithms, explicit formulas
for probabilities and conditional probabilities, and conditions which make the
conditional distribution of weighted component sums generalized Pareto are
derived.
