Skip to main content

Small drift limit theorems for random walks

  • Ernst Schulte-Geers (a1) and Wolfgang Stadje (a2)

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.

Corresponding author
* Postal address: Bundesamt für Sicherheit in der Informationstechnik (BSI), Godesberger Allee 185–189, 53175 Bonn, Germany. Email address:
** Postal address: Institute of Mathematics, University of Osnabrück, 49069 Osnabrück, Germany. Email address:
Hide All
[1] Billingsley, P. (1999).Convergence of Probability Measures, 2nd edn.John Wiley,New York.
[2] Borodin, A. N. and Salminen, P. (2002).Handbook of Brownian Motion - Facts and Formulae, 2nd edn.Birkhäuser,Basel.
[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. 33,177204.
[4] Brown, M., Peköz, E. A. and Ross, S. M. (2010).Some results for skipfree random walk.Prob. Eng. Inf. Sci. 24,491507.
[5] Chung, K. L. and Hunt, G. A. (1949).On the zeros of ∑(±1) n .Ann. Math. 50,385400.
[6] Doney, R. A. and Yor, M. (1998).On a formula of Takács for Brownian motion with drift.J. Appl. Prob. 35,272280.
[7] Feller, W. (1971).An Introduction to Probability Theory and Its Applications, Vol. II, 2nd edn.John Wiley,New York.
[8] Hoffmann-Jørgensen, J. (1999).The arcsine law.J. Theoret. Prob. 12,131145.
[9] Imhof, J.-P. (1986).On the time spent above a level by Brownian motion with negative drift.Adv. Appl. Prob. 18,10171018.
[10] Imhof, J.-P. (1999).On some equalities of laws for Brownian motion with drift.J. Appl. Prob. 36,682687.
[11] Karlin, S. and Taylor, H. M. (1975).A First Course in Stochastic Processes, 2nd edn.Academic Press,New York.
[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.137159.
[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 67,295304.
[14] Lachal, A. (2012).Sojourn time in ℤ+ for the Bernoulli random walk on ℤ.ESAIM Prob. Statist. 16,324351.
[15] Marchal, P. (1998).Temps d’occupation de (0,∞) pour les marches aléatoires.Stoch. Stoch. Reports 64,267282.
[16] Prohorov, J. V. (1963).Transition phenomena in queueing processes. I.Litovsk. Mat. Sb. 3,199205 (in Russian).
[17] Resnick, S. and Samorodnitsky, G. (2010).A heavy traffic limit theorem for workload processes with heavy tailed service requirements.Management Sci. 46,12361248.
[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. 20,411426.
[19] Shneer, S. and Wachtel, V. (2010).Heavy-traffic analysis of the maximum of an asymptotically stable random walk.Teor. Verojatn i Primenen. 55,335344. (in Russian). English translation: Theory Prob. Appl. (2011) 55,332341.
[20] Takács, L. (1996).On a generalization of the arc-sine law.Ann. Appl. Prob. 6,10351040.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Applied Probability
  • ISSN: 0021-9002
  • EISSN: 1475-6072
  • URL: /core/journals/journal-of-applied-probability
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed