Asymptotic behaviour of first passage time distributions for subordinators
Abstract
In this paper we establish local estimates for the first passage time of a subordinator under the assumption that it belongs to the Feller class, either at zero or infinity, having as a particular case the subordinators which are in the domain of attraction of a stable distribution, either at zero or infinity. To derive these results we first obtain uniform local estimates for the one dimensional distribution of such a subordinator, which sharpen those obtained by Jain and Pruitt in 1987. In the particular case of a subordinator in the domain of attraction of a stable distribution the results are the analogue of the results obtained by the authors for non-monotone L\'evy processes. For subordinators an approach different to that used for non-monotone L\'evy processes is necessary because the excursion techniques are not available and also because typically in the non-monotone case the tail distribution of the first passage time has polynomial decrease, while in the subordinator case it is exponential.
Keywords
Cite
@article{arxiv.1306.1503,
title = {Asymptotic behaviour of first passage time distributions for subordinators},
author = {Ronald A. Doney and Victor Rivero},
journal= {arXiv preprint arXiv:1306.1503},
year = {2014}
}
Comments
This version is substantially different from the previous one. A mistake in the main theorem has been fixed, in doing so we improved the method of proof and obtained sharper results