  We consider regular variation for marked point processes with independent
heavy-tailed marks and prove a single large point heuristic: the limit measure
is concentrated on the cone of point measures with one single point. We then
investigate successive hidden regular variation removing the cone of point
measures with at most $k$ points, $k\geq 1$, and prove a multiple large point
phenomenon: the limit measure is concentrated on the cone of point measures
with $k+1$ points. We show how these results imply hidden regular variation in
Skorokhod space of the associated risk process. Finally, we provide an
application to risk theory in a reinsurance model where the $k$ largest claims
are covered and we study the asymptotic behavior of the residual risk.
