Hostname: page-component-7b7567c8c-lpg2g Total loading time: 0 Render date: 2026-11-15T13:58:09.801Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of:Stochastic processes

Published online by Cambridge University Press:  30 January 2018

Johannes Kugler and
Vitali Wachtel
Johannes Kugler*
Affiliation:
University of Munich
Vitali Wachtel*
Affiliation:
University of Munich
*
∗ Postal address: Mathematical Institute, University of Munich, Theresienstrasse 39, D-80333, Munich, Germany.
∗ Postal address: Mathematical Institute, University of Munich, Theresienstrasse 39, D-80333, Munich, Germany.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

Consider a random walk S n = ∑i=0 n X i with negative drift. This paper deals with upper bounds for the maximum M = maxn≥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 = maxn≤τ S n . In addition we test our inequalities in the heavy traffic regime in the case of regularly varying tails.

Keywords

Limit theoremrandom walkrenewal theorem

MSC classification

Primary:60G50: Sums of independent random variables; random walks
Secondary:60G52: Stable processes

Information

Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 4 , December 2013, pp. 1131 - 1146
Copyright
© Applied Probability Trust 

References

Asmussen, S. (1998). Subexponential asymptotics for stochastic processes: extremal behaviour, stationary distributions and first passage probabilities. Ann. Appl. Prob. 8, 354–374 Google Scholar
Asmussen, S. (2000). Ruin Probabilities. World Scientific Publishing Co., Inc., River Edge, NJ.Google Scholar
Borovkov, A. A. (1976). Stochastic Processes in Queueing Theory. Springer, New York.Google Scholar
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, 231–253.Google Scholar
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, 165–169.Google Scholar
Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. II, 2nd edn. John Wiley, New York.Google Scholar
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, 288–313.Google Scholar
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, 37–53.Google Scholar
Foss, S., Korshunov, D. and Zachary, S. (2011). An Introduction to Heavy-Tailed and Subexponential Distributions. Springer, New York.Google Scholar
Fuk, D. Kh. and Nagaev, S. V. (1971). Probability inequalities for sums of independent random variables. Theory Prob. Appl. 16, 643–660.Google Scholar
Kalashnikov, V. (1999). Bounds for ruin probabilities in the presence of large claims and their comparison. N. Amer. Actuarial J. 3, 116–129.Google Scholar
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, 1077–1081.Google Scholar
Lorden, G. (1970). On excess over the boundary. Ann. Math. Statist. 41, 520–527.Google Scholar
Mogul'skii, A. A. (1974). Absolute estimates for moments of certain boundary functionals. Theory Prob. Appl. 18, 340–347.Google Scholar
Nagaev, S. V. (1979). Large deviations of sums of independent random variables. Ann. Prob. 7, 745–789.Google Scholar
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, 645–668.Google Scholar
Richards, A. (2009). On upper bounds for the tail distribution of geometric sums of subexponential random variables. Queueing Systems 62, 229–242.Google Scholar
Veraverbeke, N. (1977). Asymptotic behaviour of Wiener-Hopf factors of a random walk. Stoch. Process. Appl. 5, 27–37.Google Scholar