  The convergence of a sequence of point processes with dependent points,
defined by a symmetric function of iid high-dimensional random vectors, to a
Poisson random measure is proved. This also implies the convergence of the
joint distribution of a fixed number of upper order statistics. As applications
of the result a generalization of maximum convergence to point process
convergence is given for simple linear rank statistics, rank-type U-statistics
and the entries of sample covariance matrices.
