Renewal processes (nondecreasing partial-sum processes) generated by infinitely divisible life times are used as stepping stones between general nondecreasing partial-sum processes and nondecreasing Lévy processes (subordinators). In this way, it is easy to conjecture the limit distributions of the ‘undershoot’ and ‘overshoot’ at the passage of a high level by subordinators. These conjectures are then proved by Lévy-process methods.
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