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

Part of: Stochastic processes Special processes

Published online by Cambridge University Press:  19 September 2016

Clifford Hurvich  and
Josh Reed
Clifford Hurvich*
Affiliation:
Leonard N. Stern School of Business New York University
Josh Reed*
Affiliation:
Leonard N. Stern School of Business New York University
*
* Postal address: Leonard N. Stern School of Business, New York University, 44 West 4th St., New York, NY 10012, USA.
* Postal address: Leonard N. Stern School of Business, New York University, 44 West 4th St., New York, NY 10012, USA.
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Abstract

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.

Keywords

Random walkqueueingheavy tail

MSC classification

Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60K25: Queueing theory
Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue 3 , September 2016 , pp. 744 - 767
Copyright
Copyright © Applied Probability Trust 2016 

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References

[1] 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.Google Scholar
[2] Asmussen, S. (2003).Applied Probability and Queues,2nd edn.Springer,New York.Google Scholar
[3] Billingsley, P. (1999).Convergence of Probability Measures,2nd edn.John Wiley,New York.CrossRefGoogle Scholar
[4] Billingsley, P. (2012).Probability and Measure.John Wiley,Hoboken, NJ.Google Scholar
[5] Blanchet, J. and Glynn, P. (2006).Complete corrected diffusion approximations for the maximum of a random walk.Ann. Appl. Prob. 16,951–983.CrossRefGoogle Scholar
[6] Broadie, M.,Glasserman, P. and Kou, S. (1997).A continuity correction for discrete barrier options.Math. Finance 7,325–349.CrossRefGoogle Scholar
[7] Broadie, M.,Glasserman, P. and Kou, S. G. (1999).Connecting discrete and continuous path-dependent options.Finance Stoch. 3,55–82.CrossRefGoogle Scholar
[8] Chang, J. T. and Peres, Y. (1997).Ladder heights, Gaussian random walks and the Riemann zeta function.Ann. Prob. 25,787–802.CrossRefGoogle Scholar
[9] Chung, K. L. (2001).A Course in Probability Theory,3rd edn.Academic Press,San Diego, CA.Google Scholar
[10] Cohen, J. W. and Boxma, O. J. (1983).Boundary Value Problems in Queueing System Analysis.North-Holland,Amsterdam.Google Scholar
[11] Feller, W. (1971).An Introduction to Probability Theory and Its Applications, Vol. II,2nd edn.John Wiley,New York.Google Scholar
[12] Glasserman, P. and Liu, T.-W. (1997).Corrected diffusion approximations for a multistage production-inventory system.Math. Operat. Res. 22,186–201.CrossRefGoogle Scholar
[13] Gorenflo, R.,Kilbas, A. A.,Mainardi, F. and Rogosin, S. V. (2014).Mittag‒Leffler Functions, Related Topics and Applications.Springer,Heidelberg.CrossRefGoogle Scholar
[14] Janssen, A. J. E. M. (2014). Personal communcation.Google Scholar
[15] 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.CrossRefGoogle Scholar
[16] 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.CrossRefGoogle Scholar
[17] 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.CrossRefGoogle Scholar
[18] 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.CrossRefGoogle Scholar
[19] Jelenković, P.,Mandelbaum, A. and Momčilović, P. (2004).Heavy traffic limits for queues with many deterministic servers.Queueing Systems 47,53–69.CrossRefGoogle Scholar
[20] Kiefer, J. and Wolfowitz, J. (1955).On the theory of queues with many servers.Trans. Amer. Math. Soc. 78,1–18.CrossRefGoogle Scholar
[21] 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.CrossRefGoogle Scholar
[22] 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.Google Scholar
[23] Lang, S. (1999).Complex Analysis,4th edn.Springer,New York.CrossRefGoogle Scholar
[24] Lieb, E. H. and Loss, M. (2001).Analysis,2nd edn.American Mathematical Society,Providence, RI.Google Scholar
[25] Nolan, J. P. (1997).Numerical calculation of stable densities and distribution functions.Commun. Statist. Stoch. Models 13,759–774.CrossRefGoogle Scholar
[26] Owen, W. L. (1973).An estimate for E(|S n |) for variables in the domain of normal attraction of a stable law of index α, 1<α<2.Ann. Prob. 1,1071–1073.CrossRefGoogle Scholar
[27] Riemann, B. (1859).Ueber die anzahl der primzahlen unter einer gegebenen grösse.Ges. Math. Werke Wissenschaftlicher Nachlaß 2,145–155.Google Scholar
[28] Samorodnitsky, G. and Taqqu, M. S. (1994).Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance.Chapman & Hall,New York.Google Scholar
[29] Siegmund, D. (1979).Corrected diffusion approximations in certain random walk problems.Adv. Appl. Prob. 11,701–719.CrossRefGoogle Scholar
[30] Siegmund, D. (1985).Sequential Analysis: Tests and Confidence Intervals.Springer,New York.CrossRefGoogle Scholar
[31] Spitzer, F. (1956).A combinatorial lemma and its application to probability theory.Trans. Amer. Math. Soc. 82,323–339.CrossRefGoogle Scholar
[32] Whitt, W. (2002).Stochastic-Process Limits.Springer,New York.CrossRefGoogle Scholar
[33] Wolff, R. W. (1989).Stochastic Modeling and the Theory of Queues.Prentice Hall,Englewood Cliffs, NJ.Google Scholar
[34] 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.CrossRefGoogle Scholar
[35] Zolotarev, V. M. (1986).One-Dimensional Stable Distributions.American Mathematical Society,Providence, RI.CrossRefGoogle Scholar