Advances in Applied Probability

Series expansions for the all-time maximum of α-stable random walks

Clifford Hurvich and Josh Reed

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We study random walks whose increments are α-stable distributions with shape parameter 1<α<2. Specifically, assuming a mean increment size which is negative, we provide series expansions in terms of the mean increment size for the probability that the all-time maximum of an α-stable random walk is equal to 0 and, in the totally skewed-to-the-left case of skewness parameter β=-1, for the expected value of the all-time maximum of an α-stable random walk. Our series expansions generalize previous results for Gaussian random walks. Key ingredients in our proofs are Spitzer's identity for random walks, the stability property of α-stable random variables, and Zolotarev's integral representation for the cumulative distribution function of an α-stable random variable. We also discuss an application of our results to a problem arising in queueing theory.

Article information

Adv. in Appl. Probab., Volume 48, Number 3 (2016), 744-767.

First available in Project Euclid: 19 September 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60K25: Queueing theory [See also 68M20, 90B22]

Doubly stochastic process multi-state life insurance models credit risk stochastic mortality stochastic interest


Hurvich, Clifford; Reed, Josh. Series expansions for the all-time maximum of α-stable random walks. Adv. in Appl. Probab. 48 (2016), no. 3, 744--767.

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  • Abramowitz, M. and Stegun, I. A. (1964). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. U.S. Government Printing Office, Washington, D.C.
  • Asmussen, S. (2003). Applied Probability and Queues, 2nd edn. Springer, New York.
  • Billingsley, P. (1999). Convergence of Probability Measures, 2nd edn. John Wiley, New York.
  • Billingsley, P. (2012). Probability and Measure. John Wiley, Hoboken, NJ.
  • Blanchet, J. and Glynn, P. (2006). Complete corrected diffusion approximations for the maximum of a random walk. Ann. Appl. Prob. 16, 951–983.
  • Broadie, M., Glasserman, P. and Kou, S. (1997). A continuity correction for discrete barrier options. Math. Finance 7, 325–349.
  • Broadie, M., Glasserman, P. and Kou, S. G. (1999). Connecting discrete and continuous path-dependent options. Finance Stoch. 3, 55–82.
  • Chang, J. T. and Peres, Y. (1997). Ladder heights, Gaussian random walks and the Riemann zeta function. Ann. Prob. 25, 787–802.
  • Chung, K. L. (2001). A Course in Probability Theory, 3rd edn. Academic Press, San Diego, CA.
  • Cohen, J. W. and Boxma, O. J. (1983). Boundary Value Problems in Queueing System Analysis. North-Holland, Amsterdam.
  • Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. II, 2nd edn. John Wiley, New York.
  • Glasserman, P. and Liu, T.-W. (1997). Corrected diffusion approximations for a multistage production-inventory system. Math. Operat. Res. 22, 186–201.
  • Gorenflo, R., Kilbas, A. A., Mainardi, F. and Rogosin, S. V. (2014). Mittag–Leffler Functions, Related Topics and Applications. Springer, Heidelberg.
  • Janssen, A. J. E. M. (2014). Personal communcation.
  • Janssen, A. J. E. M. and van Leeuwaarden, J. S. H. (2007). Cumulants of the maximum of the Gaussian random walk. Stoch. Process. Appl. 117, 1928–1959.
  • Janssen, A. J. E. M. and van Leeuwaarden, J. S. H. (2007). On Lerch's transcendent and the Gaussian random walk. Ann. Appl. Prob. 17, 421–439.
  • Janssen, A. J. E. M., van Leeuwaarden, J. S. H. and Zwart, B. (2008). Corrected asymptotics for a multi-server queue in the Halfin–Whitt regime. Queueing Systems 58, 261–301.
  • Janssen, A. J. E. M., van Leeuwaarden, J. S. H. and Zwart, B. (2011). Refining square-root safety staffing by expanding Erlang C. Operat. Res. 59, 1512–1522.
  • Jelenković, P., Mandelbaum, A. and Momčilović, P. (2004). Heavy traffic limits for queues with many deterministic servers. Queueing Systems 47, 53–69.
  • Kiefer, J. and Wolfowitz, J. (1955). On the theory of queues with many servers. Trans. Amer. Math. Soc. 78, 1–18.
  • Kiefer, J. and Wolfowitz, J. (1956). On the characteristics of the general queueing process, with applications to random walk. Ann. Math. Stat. 27, 147–161.
  • Kingman, J. F. C. (1965). The heavy traffic approximation in the theory of queues. In Proceedings of the Symposium on Congestion Theory, University of North Carolina Press, Chapel Hill, NC, pp. 137–169.
  • Lang, S. (1999). Complex Analysis, 4th edn. Springer, New York.
  • Lieb, E. H. and Loss, M. (2001). Analysis, 2nd edn. American Mathematical Society, Providence, RI.
  • Nolan, J. P. (1997). Numerical calculation of stable densities and distribution functions. Commun. Statist. Stoch. Models 13, 759–774.
  • Owen, W. L. (1973). An estimate for $E(|S_n|)$ for variables in the domain of normal attraction of a stable law of index $\alpha, 1<\alpha< 2$. Ann. Prob. 1, 1071–1073.
  • Riemann, B. (1859). Ueber die anzahl der primzahlen unter einer gegebenen grösse. Ges. Math. Werke Wissenschaftlicher Nachlaß 2, 145–155.
  • Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Chapman & Hall, New York.
  • Siegmund, D. (1979). Corrected diffusion approximations in certain random walk problems. Adv. Appl. Prob. 11, 701–719.
  • Siegmund, D. (1985). Sequential Analysis: Tests and Confidence Intervals. Springer, New York.
  • Spitzer, F. (1956). A combinatorial lemma and its application to probability theory. Trans. Amer. Math. Soc. 82, 323–339.
  • Whitt, W. (2002). Stochastic-Process Limits. Springer, New York.
  • Wolff, R. W. (1989). Stochastic Modeling and the Theory of Queues. Prentice Hall, Englewood Cliffs, NJ.
  • Zhang, B., van Leeuwaarden, J. S. H. and Zwart, B. (2012). Staffing call centers with impatient customers: refinements to many-server asymptotics. Operat. Res. 60, 461–474.
  • Zolotarev, V. M. (1986). One-Dimensional Stable Distributions. American Mathematical Society, Providence, RI.