The Annals of Applied Probability

Tail asymptotics for the maximum of perturbed random walk

Victor F. Araman and Peter W. Glynn

Full-text: Open access

Abstract

Consider a random walk S=(Sn:n≥0) that is “perturbed” by a stationary sequence (ξn:n≥0) to produce the process (Sn+ξn:n≥0). This paper is concerned with computing the distribution of the all-time maximum M=max {Sk+ξk:k≥0} of perturbed random walk with a negative drift. Such a maximum arises in several different applications settings, including production systems, communications networks and insurance risk. Our main results describe asymptotics for ℙ(M>x) as x→∞. The tail asymptotics depend greatly on whether the ξn’s are light-tailed or heavy-tailed. In the light-tailed setting, the tail asymptotic is closely related to the Cramér–Lundberg asymptotic for standard random walk.

Article information

Source
Ann. Appl. Probab., Volume 16, Number 3 (2006), 1411-1431.

Dates
First available in Project Euclid: 2 October 2006

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1159804986

Digital Object Identifier
doi:10.1214/105051606000000268

Mathematical Reviews number (MathSciNet)
MR2260068

Zentralblatt MATH identifier
1118.60073

Subjects
Primary: 60K25: Queueing theory [See also 68M20, 90B22] 60F17: Functional limit theorems; invariance principles 68M20: Performance evaluation; queueing; scheduling [See also 60K25, 90Bxx] 90F35

Keywords
Perturbed random walk Cramér–Lundberg approximation tail asymptotics coupling heavy tails

Citation

Araman, Victor F.; Glynn, Peter W. Tail asymptotics for the maximum of perturbed random walk. Ann. Appl. Probab. 16 (2006), no. 3, 1411--1431. doi:10.1214/105051606000000268. https://projecteuclid.org/euclid.aoap/1159804986


Export citation

References

  • Araman, V. F. (2002). The maximum of perturbed random walk: Limit theorems and applications. Ph.D. dissertation, Management Science and Engineering, Stanford Univ.
  • Araman, V. F. and Glynn, P. W. (2005). Heavy-traffic diffusion approximations for the maximum of a perturbed random walk. Adv. in Appl. Probab. 37 663–680.
  • Asmussen, S. (2003). Applied Probability and Queues, 2nd ed. Springer, New York.
  • Bucklew, J. A. (1990). Large Deviation Techniques in Decision, Simulation, and Estimation. Wiley, New York.
  • Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes Characterization and Convergence. Wiley, New York.
  • Feller, W. (1971). An Introduction to Probability Theory and Its Applications 2, 2nd ed. Wiley, New York.
  • Gerber, H. U. (1970). An extension of the renewal theorem and Its application in the collective theory of risk. Skand. Aktuarietidskr. 205–210.
  • Glasserman, P. (1997). Bounds and asymptotics for planning critical safety stocks. Oper. Res. 45 244–257.
  • Glasserman, P. and Liu, T. (1997). Corrected diffusion approximations for a multistage production-inventory system. Math. Oper. Res. 22 186–201.
  • Gut, A. (1992). First-passage times for perturbed random walks. Sequential Anal. 11 149–179.
  • Kaplan, R. (1970). A dynamic inventory model with stochastic lead times. Management Sci. 16 491–507.
  • Lai, T. L. and Siegmund, D. (1977). A nonlinear renewal theory with applications to sequential analysis. I. Ann. Statist. 5 946–954.
  • Lai, T. L. and Siegmund, D. (1979). A nonlinear renewal theory with applications to sequential analysis. II. Ann. Statist. 7 60–76.
  • Ney, P. (1981). A refinement of the coupling method in renewal theory. Stochastic Process. Appl. 11 11–26.
  • Sahin, I. (1983). On the continuous-review $(s,S)$ inventory model under compound renewal demand and random lead times. J. Appl. Probab. 20 213–219.
  • Schlegel, S. (1998). Ruin probabilities in perturbed risk models. Insurance Math. Econom. 22 93–104.
  • Schmidli, H. (1995). Cramér–Lundberg approximations for ruin probabilities of risk processes perturbed by diffusion. Insurance Math. Econom. 16 135–149.
  • Thorisson, H. (2000). Coupling, Stationarity and Regeneration. Springer, New York.
  • Woodroofe, M. (1982). Nonlinear Renewal Theory. SIAM, Philadelphia.
  • Zipkin, P. (1986). Stochastic leadtimes in continuous time inventory models. Naval Res. Logist. Quart. 33 763–774.