The Annals of Applied Probability

Cramér’s estimate for the reflected process revisited

R. A. Doney and Philip S. Griffin

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Abstract

The reflected process of a random walk or Lévy process arises in many areas of applied probability, and a question of particular interest is how the tail of the distribution of the heights of the excursions away from zero behaves asymptotically. The Lévy analogue of this is the tail behaviour of the characteristic measure of the height of an excursion. Apparently, the only case where this is known is when Cramér’s condition hold. Here, we establish the asymptotic behaviour for a large class of Lévy processes, which have exponential moments but do not satisfy Cramér’s condition. Our proof also applies in the Cramér case, and corrects a proof of this given in Doney and Maller [Ann. Appl. Probab. 15 (2005) 1445–1450].

Article information

Source
Ann. Appl. Probab., Volume 28, Number 6 (2018), 3629-3651.

Dates
Received: August 2017
Revised: April 2018
First available in Project Euclid: 8 October 2018

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1538985631

Digital Object Identifier
doi:10.1214/18-AAP1399

Mathematical Reviews number (MathSciNet)
MR3861822

Zentralblatt MATH identifier
06994402

Subjects
Primary: 60G51: Processes with independent increments; Lévy processes 60F10: Large deviations

Keywords
Reflected Lévy process Cramér’s estimate excursion height excursion measure close to exponential convolution equivalence

Citation

Doney, R. A.; Griffin, Philip S. Cramér’s estimate for the reflected process revisited. Ann. Appl. Probab. 28 (2018), no. 6, 3629--3651. doi:10.1214/18-AAP1399. https://projecteuclid.org/euclid.aoap/1538985631


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References

  • [1] Bertoin, J. (1996). Lévy Processes. Cambridge Tracts in Mathematics 121. Cambridge Univ. Press, Cambridge.
  • [2] Bertoin, J. and Doney, R. A. (1994). Cramér’s estimate for Lévy processes. Statist. Probab. Lett. 21 363–365.
  • [3] Bertoin, J. and Doney, R. A. (1996). Some asymptotic results for transient random walks. Adv. in Appl. Probab. 28 207–226.
  • [4] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1989). Regular Variation. Encyclopedia of Mathematics and Its Applications 27. Cambridge Univ. Press, Cambridge.
  • [5] Braverman, M. (2005). On a class of Lévy processes. Statist. Probab. Lett. 75 179–189.
  • [6] Doney, R. A. (2007). Fluctuation Theory for Lévy Processes. Lecture Notes in Math. 1897. Springer, Berlin.
  • [7] Doney, R. A. and Maller, R. A. (2005). Cramér’s estimate for a reflected Lévy process. Ann. Appl. Probab. 15 1445–1450.
  • [8] Embrechts, P. and Goldie, C. M. (1982). On convolution tails. Stochastic Process. Appl. 13 263–278.
  • [9] Griffin, P. S. (2016). Sample path behavior of a Lévy insurance risk process approaching ruin, under the Cramér–Lundberg and convolution equivalent conditions. Ann. Appl. Probab. 26 360–401.
  • [10] Iglehart, D. L. (1972). Extreme values in the $GI/G/1$ queue. Ann. Math. Stat. 43 627–635.
  • [11] Klüppelberg, C., Kyprianou, A. E. and Maller, R. A. (2004). Ruin probabilities and overshoots for general Lévy insurance risk processes. Ann. Appl. Probab. 14 1766–1801.
  • [12] Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Universitext. Springer, Berlin.
  • [13] Sato, K. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics 68. Cambridge Univ. Press, Cambridge.
  • [14] Watanabe, T. (2008). Convolution equivalence and distributions of random sums. Probab. Theory Related Fields 142 367–397.