## 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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