SMALL DRIFT LIMIT THEOREMS FOR RANDOM WALKS

Autor(en): Schulte-Geers, Ernst
Stadje, Wolfgang 
Stichwörter: arcsine law; BROWNIAN-MOTION; limit distribution; Mathematics; occupation time; Random walk; small drift; Statistics & Probability; TIME; transient
Erscheinungsdatum: 2017
Herausgeber: CAMBRIDGE UNIV PRESS
Journal: JOURNAL OF APPLIED PROBABILITY
Volumen: 54
Ausgabe: 1
Startseite: 199
Seitenende: 212
Zusammenfassung: 
We show analogs of the classical arcsine theorem for the occupation time of a random walk in (-infinity, 0) in the case of a small positive drift. To study the asymptotic behavior of the total time spent in (-infinity, 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.
ISSN: 00219002
DOI: 10.1017/jpr.2016.95

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