  A family of self-similar and translation-invariant random sup-measures with
long-range dependence are investigated. They are shown to arise as the limit of
the empirical random sup-measure of a stationary heavy-tailed process, inspired
by an infinite urn scheme, where same values are repeated at several random
locations. The random sup-measure reflects the long-range dependence nature of
the original process, and in particular characterizes how locations of extremes
appear as long-range clusters represented by random closed sets. A limit
theorem for the corresponding point-process convergence is established.
