  This paper is concerned with the optimal approximation of a given
multivariate Dirac mixture, i.e., a density comprising weighted Dirac
distributions on a continuous domain, by an equally weighted Dirac mixture with
a reduced number of components. The parameters of the approximating density are
calculated by minimizing a smooth global distance measure, a generalization of
the well-known Cram\'{e}r-von Mises Distance to the multivariate case. This
generalization is achieved by defining an alternative to the classical
cumulative distribution, the Localized Cumulative Distribution (LCD), as a
characterization of discrete random quantities (on continuous domains), which
is unique and symmetric also in the multivariate case. The resulting
approximation method provides the basis for various efficient nonlinear state
and parameter estimation methods.
