  We prove limit theorems of an entirely new type for certain long memory
regularly varying stationary infinitely divisible random processes. These
theorems involve multiple phase transitions governed by how long the memory is.
Apart from one regime, our results exhibit limits that are not among the
classical extreme value distributions. Restricted to the one-dimensional case,
the distributions we obtain interpolate, in the appropriate parameter range,
the $\alpha$-Fr\'echet distribution and the skewed $\alpha$-stable
distribution. In general, the limit is a new family of stationary and
self-similar random sup-measures with parameters $\alpha\in(0,\infty)$ and
$\beta\in(0,1)$, with representations based on intersections of independent
$\beta$-stable regenerative sets. The tail of the limit random sup-measure on
each interval with finite positive length is regularly varying with index
$-\alpha$. The intriguing structure of these random sup-measures is due to
intersections of independent $\beta$-stable regenerative sets and the fact that
the number of such sets intersecting simultaneously increases to infinity as
$\beta$ increases to one. The results in this paper extend substantially
previous investigations where only $\alpha\in(0,2)$ and $\beta\in(0,1/2)$ have
been considered.
