Skip to main content Accessibility help
×
Home

Upper Bounds for the Maximum of a Random Walk with Negative Drift

  • Johannes Kugler (a1) and Vitali Wachtel (a1)

Abstract

Consider a random walk S n = ∑ i=0 n X i with negative drift. This paper deals with upper bounds for the maximum M = max n≥1 S n of this random walk in different settings of power moment existences. As is usual for deriving upper bounds, we truncate summands. Therefore, we use an approach of splitting the time axis by stopping times into intervals of random but finite length and then choose a level of truncation on each interval. Hereby, we can reduce the problem of finding upper bounds for M to the problem of finding upper bounds for M τ = max n≤τ S n . In addition we test our inequalities in the heavy traffic regime in the case of regularly varying tails.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Upper Bounds for the Maximum of a Random Walk with Negative Drift
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Upper Bounds for the Maximum of a Random Walk with Negative Drift
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Upper Bounds for the Maximum of a Random Walk with Negative Drift
      Available formats
      ×

Copyright

Corresponding author

Postal address: Mathematical Institute, University of Munich, Theresienstrasse 39, D-80333, Munich, Germany.

References

Hide All
[1] Asmussen, S. (1998). Subexponential asymptotics for stochastic processes: extremal behaviour, stationary distributions and first passage probabilities. Ann. Appl. Prob. 8, 354374
[2] Asmussen, S. (2000). Ruin Probabilities. World Scientific Publishing Co., Inc., River Edge, NJ.
[3] Borovkov, A. A. (1976). Stochastic Processes in Queueing Theory. Springer, New York.
[4] Borovkov, A. A. and Foss, S. G. (2000). Estimates for overshooting an arbitrary boundary by a random walk and their applications. Theory Prob. Appl. 44, 231253.
[5] Denisov, D. (2005). A note on the asymptotics for the maximum on a random time interval of a random walk. Markov Process. Relat. Fields 11, 165169.
[6] Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. II, 2nd edn. John Wiley, New York.
[7] Foss, S. and Puhalskii, A. A. (2011). On the limit law of a random walk conditioned to reach a high level. Stoch. Process. Appl. 121, 288313.
[8] Foss, S. and Zachary, S. (2003). The maximum on a random time interval of a random walk with long-tailed increments and negative drift. Ann. Appl. Prob. 13, 3753.
[9] Foss, S., Korshunov, D. and Zachary, S. (2011). An Introduction to Heavy-Tailed and Subexponential Distributions. Springer, New York.
[10] Fuk, D. Kh. and Nagaev, S. V. (1971). Probability inequalities for sums of independent random variables. Theory Prob. Appl. 16, 643660.
[11] Kalashnikov, V. (1999). Bounds for ruin probabilities in the presence of large claims and their comparison. N. Amer. Actuarial J. 3, 116129.
[12] Korshunov, D. (2005). The critical case of the Cramér-Lundberg theorem on the asymptotics of the distribution of the maximum of a random walk with negative drift. Siberian Math. J. 46, 10771081.
[13] Lorden, G. (1970). On excess over the boundary. Ann. Math. Statist. 41, 520527.
[14] Mogul'skii, A. A. (1974). Absolute estimates for moments of certain boundary functionals. Theory Prob. Appl. 18, 340347.
[15] Nagaev, S. V. (1979). Large deviations of sums of independent random variables. Ann. Prob. 7, 745789.
[16] Olvera-Cravioto, M., Blanchet, J. and Glynn, P. (2011). On the transition from heavy traffic to heavy tails for the M/G/1 queue: the regularly varying case. Ann. Appl. Prob. 21, 645668.
[17] Richards, A. (2009). On upper bounds for the tail distribution of geometric sums of subexponential random variables. Queueing Systems 62, 229242.
[18] Veraverbeke, N. (1977). Asymptotic behaviour of Wiener-Hopf factors of a random walk. Stoch. Process. Appl. 5, 2737.

Keywords

MSC classification

Upper Bounds for the Maximum of a Random Walk with Negative Drift

  • Johannes Kugler (a1) and Vitali Wachtel (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed