  Recent studies have demonstrated an interesting connection between the
asymptotic behavior at ruin of a L\'evy insurance risk process under the
Cram\'er-Lundberg and convolution equivalent conditions. For example, the
limiting distributions of the overshoot and the undershoot are strikingly
similar in these two settings. This is somewhat surprising since the global
sample path behavior of the process under these two conditions is quite
different. Using tools from excursion theory and fluctuation theory, we provide
a means of transferring results from one setting to the other which, among
other things, explains this connection and leads to new asymptotic results.
This is done by describing the evolution of the sample paths from the time of
the last maximum prior to ruin until ruin occurs.

Probability