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First-passage time asymptotics over moving boundaries for random walk bridges

Part of: Stochastic processes Limit theorems

Published online by Cambridge University Press:  26 July 2018

Fiona Sloothaak ,
Vitali Wachtel  and
Bert Zwart
Fiona Sloothaak*
Affiliation:
Eindhoven University of Technology
Vitali Wachtel*
Affiliation:
University of Augsburg
Bert Zwart*
Affiliation:
CWI and Eindhoven University of Technology
*
* Postal address: Eindhoven University of Technology, Postbus 513, 5600 MB Eindhoven, The Netherlands. Email address: f.sloothaak@tue.nl
** Postal address: Institut für Mathematik, University of Augsburg, D-86135 Augsburg, Germany. Email address: vitali.wachtel@math.uni-augsburg.de
*** Postal address: CWI, Science Park 123, 1098 XG Amsterdam, The Netherlands. Email address: bert.zwart@cwi.nl
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Abstract

We study the asymptotic tail behavior of the first-passage time over a moving boundary for a random walk conditioned to return to zero, where the increments of the random walk have finite variance. Typically, the asymptotic tail behavior may be described through a regularly varying function with exponent -½, where the impact of the boundary is captured by the slowly varying function. Yet, the moving boundary may have a stronger effect when the tail is considered at a time close to the return point of the random walk bridge, leading to a possible phase transition depending on the order of the distance between zero and the moving boundary.

Keywords

Random walkfirst-passage timemoving boundarybridge

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 55 , Issue 2 , June 2018 , pp. 627 - 651
Copyright
Copyright © Applied Probability Trust 2018 

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