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

Sample path behavior of a L\'evy insurance risk process approaching ruin, under the Cram\'er-Lundberg and convolution equivalent conditions