Doney, R. A.; Maller, R. A. Cramér’s estimate for a reflected Lévy process. (English) Zbl 1069.60045 Ann. Appl. Probab. 15, No. 2, 1445-1450 (2005). Summary: The natural analogue for a Lévy process of Cramér’s estimate for a reflected random walk is a statement about the exponential rate of decay of the tail of the characteristic measure of the height of an excursion above the minimum. We establish this estimate for any Lévy process with finite negative mean which satisfies Cramér’s condition, and give an explicit formula for the limiting constant. Just as in the random walk case, this leads to a Poisson limit theorem for the number of “high excursions”. Cited in 2 ReviewsCited in 10 Documents MSC: 60G51 Processes with independent increments; Lévy processes 60G17 Sample path properties Keywords:Maximum of reflected process; maximal segmental score; Poisson limit theorem; high excursions × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] Asmussen, S. (1982). Conditioned limit theorems relating a random walk to its associate, with applications to risk reserve processes and the \(\mathitGI/\mathitGI/1\) queue. Adv. in Appl. Probab. 14 143–170. · Zbl 0501.60076 · doi:10.2307/1426737 [2] Asmussen, S. (1982). Applied Probability and Queueing . Wiley, New York. · Zbl 0501.60076 [3] Bertoin, J. (1996). L \(\acute\mathrme\) vy Processes . Cambridge Univ. Press. [4] Bertoin, J. and Doney, R. A. (1994). Cramér’s estimate for Lévy processes. Statist. Probab. Lett. 21 363–365. · Zbl 0809.60085 · doi:10.1016/0167-7152(94)00032-8 [5] Erickson, K. B. (1973). The strong law of large numbers when the mean is undefined. Trans. Amer. Math Soc. 185 371–381. · Zbl 0304.60016 · doi:10.2307/1996445 [6] Feller, W. E. (1971). An Introduction to Probability Theory and Its Applications 2 , 2nd ed. Wiley, New York. · Zbl 0219.60003 [7] Iglehart, D. L. (1972). Extreme values in the \(\mathitGI/G/1\) queue. Ann. Math. Statist. 43 627–635. · Zbl 0238.60072 · doi:10.1214/aoms/1177692642 [8] Karlin, S. and Dembo, A. (1992). Limit distributions of maximal segmental score among Markov-dependent partial sums. Adv. in Appl. Probab. 24 113–140. · Zbl 0767.60017 · doi:10.2307/1427732 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.