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Renewal theory and level passage by subordinators. (English) Zbl 0965.60053

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 the limit distributions are conjectured of the ‘undershoot’ and ‘overshoot’ at the passage of a high level by subordinators. These conjectures are then proved by Lévy-process methods.

MSC:

60G51 Processes with independent increments; Lévy processes
60K05 Renewal theory
60E07 Infinitely divisible distributions; stable distributions

References:

[1] Bertoin, J., 1996. Lévy Processes. Cambridge University Press, Cambridge.; Bertoin, J., 1996. Lévy Processes. Cambridge University Press, Cambridge. · Zbl 0861.60003
[2] Feller, W., 1971. An Introduction to Probability Theory and its Applications, vol. 2, second ed. Wiley, New York.; Feller, W., 1971. An Introduction to Probability Theory and its Applications, vol. 2, second ed. Wiley, New York. · Zbl 0219.60003
[3] van Harn, K.; Steutel, F. W., Infinite divisibility and the waiting-time paradox, Commun. Statist. Stochastic Models, 11, 3, 527-540 (1995) · Zbl 0843.60020
[4] Winter, B. B., Joint simulation of backward and forward recurrence times in a renewal process, J. Appl. Prob., 26, 404-407 (1989) · Zbl 0676.60081
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