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Small drift limit theorems for random walks

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

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Corresponding author
* 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
References
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Journal of Applied Probability
  • ISSN: 0021-9002
  • EISSN: 1475-6072
  • URL: /core/journals/journal-of-applied-probability
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