## Electronic Journal of Probability

### Functional Limit Theorems for Lévy Processes Satisfying Cramér's Condition

#### Abstract

We consider a Lévy process that starts from $x&lt;0$ and conditioned on having a positive maximum. When Cramér's condition holds, we provide two weak limit theorems as $x$ goes to $-\infty$ for the law of the (two-sided) path shifted at the first instant when it enters $(0,\infty)$, respectively shifted at the instant when its overall maximum is reached. The comparison of these two asymptotic results yields some interesting identities related to time-reversal, insurance risk, and self-similar Markov processes.

#### Article information

Source
Electron. J. Probab., Volume 16 (2011), paper no. 73, 2020-2038.

Dates
Accepted: 31 October 2011
First available in Project Euclid: 1 June 2016

https://projecteuclid.org/euclid.ejp/1464820242

Digital Object Identifier
doi:10.1214/EJP.v16-930

Mathematical Reviews number (MathSciNet)
MR2851054

Zentralblatt MATH identifier
1244.60049

Rights

#### Citation

Barczy, Matyas; Bertoin, Jean. Functional Limit Theorems for Lévy Processes Satisfying Cramér's Condition. Electron. J. Probab. 16 (2011), paper no. 73, 2020--2038. doi:10.1214/EJP.v16-930. https://projecteuclid.org/euclid.ejp/1464820242

#### References

• Asmussen, SÃ¸ren. Conditioned limit theorems relating a random walk to its associate, with applications to risk reserve processes and the $GI/G/1$ queue. Adv. in Appl. Probab. 14 (1982), no. 1, 143–170.
• Asmussen, SÃ¸ren. Ruin probabilities. Advanced Series on Statistical Science & Applied Probability, 2. World Scientific Publishing Co., Inc., River Edge, NJ, 2000. xii+385 pp. ISBN: 981-02-2293-9
• Bertoin, Jean. Splitting at the infimum and excursions in half-lines for random walks and Lévy processes. Stochastic Process. Appl. 47 (1993), no. 1, 17–35.
• Bertoin, Jean. Lévy processes. Cambridge Tracts in Mathematics, 121. Cambridge University Press, Cambridge, 1996. x+265 pp. ISBN: 0-521-56243-0
• Bertoin, J.; Doney, R. A. Cramér's estimate for Lévy processes. Statist. Probab. Lett. 21 (1994), no. 5, 363–365.
• Bertoin, Jean; Savov, Mladen. Some applications of duality for Lévy processes in a half-line. Bull. Lond. Math. Soc. 43 (2011), no. 1, 97–110.
• Bertoin, Jean; Yor, Marc. Exponential functionals of Lévy processes. Probab. Surv. 2 (2005), 191–212.
• Chaumont, L. Conditionings and path decompositions for Lévy processes. Stochastic Process. Appl. 64 (1996), no. 1, 39–54.
• Chaumont, L.; Doney, R. A. On Lévy processes conditioned to stay positive. Electron. J. Probab. 10 (2005), no. 28, 948–961 (electronic).
• Chaumont, L., Kyprianou, A., Pardo, J.C. and Rivero, V. Fluctuation theory and exit systems for positive self-similar Markov processes. To appear in Ann. Probab.
• Fitzsimmons, P. J. On the existence of recurrent extensions of self-similar Markov processes. Electron. Comm. Probab. 11 (2006), 230–241.
• Griffin, P.S. Convolution equivalent Lévy processes and first passage times. Preprint (2011).
• Griffin, P.S., and Maller, R.A. Path decomposition of ruinous behaviour for a general Lévy insurance risk process. To apper in Ann. Appl. Probab.
• Lamperti, John. Semi-stable Markov processes. I. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 22 (1972), 205–225.
• Klüppelberg, Claudia; Kyprianou, Andreas E.; Maller, Ross A. Ruin probabilities and overshoots for general Lévy insurance risk processes. Ann. Appl. Probab. 14 (2004), no. 4, 1766–1801.
• Kyprianou, Andreas E. Introductory lectures on fluctuations of Lévy processes with applications. Universitext. Springer-Verlag, Berlin, 2006. xiv+373 pp. ISBN: 978-3-540-31342-7; 3-540-31342-7
• Rivero, Víctor. Recurrent extensions of self-similar Markov processes and Cramér's condition. Bernoulli 11 (2005), no. 3, 471–509.
• Rivero, Víctor. Recurrent extensions of self-similar Markov processes and Cramér's condition. II. Bernoulli 13 (2007), no. 4, 1053–1070.
• Vuolle-Apiala, J. Itô excursion theory for self-similar Markov processes. Ann. Probab. 22 (1994), no. 2, 546–565.
• Whitt, Ward. Some useful functions for functional limit theorems. Math. Oper. Res. 5 (1980), no. 1, 67–85.
• Williams, David. Path decomposition and continuity of local time for one-dimensional diffusions. I. Proc. London Math. Soc. (3) 28 (1974), 738–768.