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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: [email protected]
** Postal address: Institute of Mathematics, University of Osnabrück, 49069 Osnabrück, Germany. Email address: [email protected]
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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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