Annals of Applied Probability

The maximum on a random time interval of a random walk with long-tailed increments and negative drift

Serguei Foss and Stan Zachary

Full-text: Open access

Abstract

We study the asymptotics for the maximum on a random time interval of a random walk with a long-tailed distribution of its increments and negative drift. We extend to a general stopping time a result by Asmussen, simplify its proof and give some converses.

Article information

Source
Ann. Appl. Probab., Volume 13, Number 1 (2003), 37-53.

Dates
First available in Project Euclid: 16 January 2003

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1042765662

Digital Object Identifier
doi:10.1214/aoap/1042765662

Mathematical Reviews number (MathSciNet)
MR1951993

Zentralblatt MATH identifier
1045.60039

Subjects
Primary: 60G70: Extreme value theory; extremal processes
Secondary: 60K30: Applications (congestion, allocation, storage, traffic, etc.) [See also 90Bxx] 60K25: Queueing theory [See also 68M20, 90B22]

Keywords
Ruin probability long-tailed distribution subexponential distribution

Citation

Foss, Serguei; Zachary, Stan. The maximum on a random time interval of a random walk with long-tailed increments and negative drift. Ann. Appl. Probab. 13 (2003), no. 1, 37--53. doi:10.1214/aoap/1042765662. https://projecteuclid.org/euclid.aoap/1042765662


Export citation

References

  • ASMUSSEN, S. (1998). Subexponential asy mptotics for stochastic processes: Extremal behavior, stationary distributions and first passage probabilities. Ann. Appl. Probab. 8 354-374.
  • ASMUSSEN, S. (1999). Semi-Markov queues with heavy tails. In Semi-Markov Models and Applications (J. Janssen and N. Limnios, eds.). Kluwer, Dordrecht.
  • ASMUSSEN, S., FOSS, S. G. and KORSHUNOV, D. A. (2002). Asy mptotics for sums of random variables with local subexponential behaviour. Unpublished manuscript.
  • ASMUSSEN, S., KALASHNIKOV, V., KONSTANTINIDES, D., KLÜPPELBERG, C. and TSITI
  • ASHVILI, G. (2001). A local limit theorem for random walk maxima with heavy tails. Unpublished manuscript.
  • BOROVKOV, A. A. and BOROVKOV, K. A. (2001). On large deviation probabilities for random walks. I. Regularly varying distribution tails. II. Regularly exponential distribution tails. Theory Probab. Appl. 46 209-232 (in Russian).
  • EMBRECHTS, P., KLÜPPELBERG, C. and MIKOSCH, T. (1997). Modelling Extremal Events. Springer, Berlin.
  • EMBRECHTS, P. and VERAVERBEKE, N. (1982). Estimates for the probability of ruin with special emphasis on the possibility of large claims. Insurance Math. Econom. 1 55-72.
  • FELLER, W. (1971). An Introduction to Probability Theory and Its Applications. Wiley, New York.
  • GREINER, M., JOBMANN, M. and KLÜPPELBERG, C. (1999). Telecommunication traffic, queueing models, and subexponential distributions. Queueing Sy stems Theory Appl. 33 125-152.
  • HEATH, D., RESNICK, S. and SAMORODNITSKY, G. (1997). Patterns of buffer overflow in a class of queues with long memory in the input stream. Ann. Appl. Probab. 7 1021-1057.
  • KLÜPPELBERG, C. (1988). Subexponential distributions and integrated tails. J. Appl. Probab. 35 325-347.
  • KORSHUNOV, D. A. (1997). On distribution tail of the maximum of a random walk. Stochastic Process. Appl. 72 97-103.
  • KORSHUNOV, D. A. (2001). Large-deviation probabilities for maxima of sums of independent random variables with negative mean and subexponential distribution. Theory Probab. Appl. 46 387-397 (in Russian).
  • SIGMAN, K. (1999). A primer on heavy-tailed distributions. Queueing Sy stems Theory Appl. 33 261-275.
  • VERAVERBEKE, N. (1977). Asy mptotic behavior of Wiener-Hopf factors of a random walk. Stochastic Process. Appl. 5 27-37.