The Annals of Applied Probability

Cramér’s estimate for a reflected Lévy process

R. A. Doney and R. A. Maller

Full-text: Open access


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.”

Article information

Ann. Appl. Probab., Volume 15, Number 2 (2005), 1445-1450.

First available in Project Euclid: 3 May 2005

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G51: Processes with independent increments; Lévy processes 60G17: Sample path properties

Maximum of reflected process maximal segmental score Poisson limit theorem high excursions


Doney, R. A.; Maller, R. A. Cramér’s estimate for a reflected Lévy process. Ann. Appl. Probab. 15 (2005), no. 2, 1445--1450. doi:10.1214/105051605000000016.

Export citation


  • 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.
  • Asmussen, S. (1982). Applied Probability and Queueing. Wiley, New York.
  • Bertoin, J. (1996). L$\acute\mathrme$vy Processes. Cambridge Univ. Press.
  • Bertoin, J. and Doney, R. A. (1994). Cramér's estimate for Lévy processes. Statist. Probab. Lett. 21 363--365.
  • Erickson, K. B. (1973). The strong law of large numbers when the mean is undefined. Trans. Amer. Math Soc. 185 371--381.
  • Feller, W. E. (1971). An Introduction to Probability Theory and Its Applications 2, 2nd ed. Wiley, New York.
  • Iglehart, D. L. (1972). Extreme values in the $\mathitGI/G/1$ queue. Ann. Math. Statist. 43 627--635.
  • Karlin, S. and Dembo, A. (1992). Limit distributions of maximal segmental score among Markov-dependent partial sums. Adv. in Appl. Probab. 24 113--140.