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Cramér’s estimate for Lévy processes. (English) Zbl 0809.60085
One of the most useful results in applied probability is Cramér’s estimate, which gives the asymptotic behaviour of the tail of the distribution of the all-time maximum of a random walk with negative drift. This estimate is an immediate consequence of the renewal theorem applied to the renewal process formed by the ascending ladder heights in the associated random walk. The authors show that exactly the same method works for Lévy processes, which, being processes with stationary and independent increments, are the continuous time analogues of random walks. The only complication is that the set of increasing ladder heights is not usually discrete, but coincides with the range of a certain subordinator. However, the authors observe that the potential function of this subordinator coincides with the renewal function of a certain renewal process, so that they can still appeal to the renewal theorem.
Reviewer: Z.Rychlik (Lublin)

60J99 Markov processes
Full Text: DOI
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