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Small drift limit theorems for random walks

Part of: Limit theorems Stochastic processes

Published online by Cambridge University Press:  04 April 2017

Ernst Schulte-Geers  and
Wolfgang Stadje
Ernst Schulte-Geers*
Affiliation:
Bundesamt für Sicherheit in der Informationstechnik
Wolfgang Stadje*
Affiliation:
University of Osnabrück
*
* Postal address: Bundesamt für Sicherheit in der Informationstechnik (BSI), Godesberger Allee 185–189, 53175 Bonn, Germany. Email address: ernst.schulte-geers@bsi.bund.de
** Postal address: Institute of Mathematics, University of Osnabrück, 49069 Osnabrück, Germany. Email address: wstadje@uos.de
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Abstract

We show analogs of the classical arcsine theorem for the occupation time of a random walk in (−∞,0) in the case of a small positive drift. To study the asymptotic behavior of the total time spent in (−∞,0) we consider parametrized classes of random walks, where the convergence of the parameter to 0 implies the convergence of the drift to 0. We begin with shift families, generated by a centered random walk by adding to each step a shift constant a>0 and then letting a tend to 0. Then we study families of associated distributions. In all cases we arrive at the same limiting distribution, which is the distribution of the time spent below 0 of a standard Brownian motion with drift 1. For shift families this is explained by a functional limit theorem. Using fluctuation-theoretic formulae we derive the generating function of the occupation time in closed form, which provides an alternative approach. We also present a new form of the first arcsine law for the Brownian motion with drift.

Keywords

Random walktransientoccupation timearcsine lawsmall driftlimit distribution

MSC classification

Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60F05: Central limit and other weak theorems
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 1 , March 2017 , pp. 199 - 212
Copyright
Copyright © Applied Probability Trust 2017 

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References

[1] Billingsley, P. (1999).Convergence of Probability Measures, 2nd edn.John Wiley,New York.Google Scholar
[2] Borodin, A. N. and Salminen, P. (2002).Handbook of Brownian Motion - Facts and Formulae, 2nd edn.Birkhäuser,Basel.CrossRefGoogle Scholar
[3] Boxma, O. J. and Cohen, J. W. (1999).Heavy-traffic analysis for the GI/G/1 queue with heavy-tailed distributions. Queues with heavy-tailed distributions.Queueing Syst. Theory Appl. 33,177–204.Google Scholar
[4] Brown, M., Peköz, E. A. and Ross, S. M. (2010).Some results for skipfree random walk.Prob. Eng. Inf. Sci. 24,491–507.Google Scholar
[5] Chung, K. L. and Hunt, G. A. (1949).On the zeros of ∑(±1) n .Ann. Math. 50,385–400.CrossRefGoogle Scholar
[6] Doney, R. A. and Yor, M. (1998).On a formula of Takács for Brownian motion with drift.J. Appl. Prob. 35,272–280.Google Scholar
[7] Feller, W. (1971).An Introduction to Probability Theory and Its Applications, Vol. II, 2nd edn.John Wiley,New York.Google Scholar
[8] Hoffmann-Jørgensen, J. (1999).The arcsine law.J. Theoret. Prob. 12,131–145.Google Scholar
[9] Imhof, J.-P. (1986).On the time spent above a level by Brownian motion with negative drift.Adv. Appl. Prob. 18,1017–1018.CrossRefGoogle Scholar
[10] Imhof, J.-P. (1999).On some equalities of laws for Brownian motion with drift.J. Appl. Prob. 36,682–687.Google Scholar
[11] Karlin, S. and Taylor, H. M. (1975).A First Course in Stochastic Processes, 2nd edn.Academic Press,New York.Google Scholar
[12] Kingman, J. F. C. (1965).The heavy traffic approximation in the theory of queues. In Proceedings of Symposium on Congestion Theory, eds W. L. Smith and W. E. Wilkinson,University of North Carolina Press,Chapel Hill, pp.137–159.Google Scholar
[13] Kosiński, K. M., Boxma, O. J. and Zwart, A. P. (2011).Convergence of the all-time supremum of a Lévy process in the heavy-traffic regime.Queueing Systems 67,295–304.CrossRefGoogle Scholar
[14] Lachal, A. (2012).Sojourn time in ℤ+ for the Bernoulli random walk on ℤ.ESAIM Prob. Statist. 16,324–351.Google Scholar
[15] Marchal, P. (1998).Temps d’occupation de (0,∞) pour les marches aléatoires.Stoch. Stoch. Reports 64,267–282.Google Scholar
[16] Prohorov, J. V. (1963).Transition phenomena in queueing processes. I.Litovsk. Mat. Sb. 3,199–205 (in Russian).Google Scholar
[17] Resnick, S. and Samorodnitsky, G. (2010).A heavy traffic limit theorem for workload processes with heavy tailed service requirements.Management Sci. 46,1236–1248.CrossRefGoogle Scholar
[18] Salminen, P. (1988).On the first hitting time and the last exit time of Brownian motion to/from a moving boundary.Adv. Appl. Prob. 20,411–426.Google Scholar
[19] Shneer, S. and Wachtel, V. (2010).Heavy-traffic analysis of the maximum of an asymptotically stable random walk.Teor. Verojatn i Primenen. 55,335–344. (in Russian). English translation: Theory Prob. Appl. (2011) 55,332–341.Google Scholar
[20] Takács, L. (1996).On a generalization of the arc-sine law.Ann. Appl. Prob. 6,1035–1040.CrossRefGoogle Scholar